Algebraic Topology on Polyhedra - TU Berlin · Algebraic Topology on Polyhedra ... functions...
Transcript of Algebraic Topology on Polyhedra - TU Berlin · Algebraic Topology on Polyhedra ... functions...
Algebraic Topology on Polyhedrafrom Linear Finite Elements
Max Wardetzky
(with K Hildebrandt and K Polthier)
Free University Berlin amp
DFG Research Center ldquoMatheonrdquo
Oberwolfach March 6 2006
Letrsquos start with a concrete problem
Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0
bull no vorticesbull no sinkssources
No maybeNo
Letrsquos start with a concrete problem
Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0
Letrsquos start with a concrete problem
Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0
locally integrable byharmonic fcts
Applications Mesh parameterization (cf GuYau) texture mapping shape matching shape morphing hellip
Harmonic vector fields
Strategybull discretize deRham complexbull get model for cohomology Hsup1bull discretize Hodge decompositionbull get harmonic vector fieldsbull compute basis of harmonic vector fields
Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Its deRham complex is
d is Cartan outer differential (no metric)
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Its deRham complex is
This is a chain complex
d is Cartan outer differential (no metric)
Review smooth deRham complex
functions 1-forms 2-forms
vector fields
(Mg) smooth surface Make use of metric g
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
vector fields
sharp Hodge star
Review smooth deRham complex
vector fields
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Review deRham cohomology
encodes global topology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Local integrability iff locally
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
encodes global topology
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Strategy for discretization
Local integrability iff locally
encodes global topology
1st step2nd step
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Letrsquos start with a concrete problem
Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0
bull no vorticesbull no sinkssources
No maybeNo
Letrsquos start with a concrete problem
Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0
Letrsquos start with a concrete problem
Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0
locally integrable byharmonic fcts
Applications Mesh parameterization (cf GuYau) texture mapping shape matching shape morphing hellip
Harmonic vector fields
Strategybull discretize deRham complexbull get model for cohomology Hsup1bull discretize Hodge decompositionbull get harmonic vector fieldsbull compute basis of harmonic vector fields
Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Its deRham complex is
d is Cartan outer differential (no metric)
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Its deRham complex is
This is a chain complex
d is Cartan outer differential (no metric)
Review smooth deRham complex
functions 1-forms 2-forms
vector fields
(Mg) smooth surface Make use of metric g
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
vector fields
sharp Hodge star
Review smooth deRham complex
vector fields
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Review deRham cohomology
encodes global topology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Local integrability iff locally
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
encodes global topology
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Strategy for discretization
Local integrability iff locally
encodes global topology
1st step2nd step
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Letrsquos start with a concrete problem
Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0
Letrsquos start with a concrete problem
Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0
locally integrable byharmonic fcts
Applications Mesh parameterization (cf GuYau) texture mapping shape matching shape morphing hellip
Harmonic vector fields
Strategybull discretize deRham complexbull get model for cohomology Hsup1bull discretize Hodge decompositionbull get harmonic vector fieldsbull compute basis of harmonic vector fields
Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Its deRham complex is
d is Cartan outer differential (no metric)
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Its deRham complex is
This is a chain complex
d is Cartan outer differential (no metric)
Review smooth deRham complex
functions 1-forms 2-forms
vector fields
(Mg) smooth surface Make use of metric g
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
vector fields
sharp Hodge star
Review smooth deRham complex
vector fields
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Review deRham cohomology
encodes global topology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Local integrability iff locally
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
encodes global topology
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Strategy for discretization
Local integrability iff locally
encodes global topology
1st step2nd step
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Letrsquos start with a concrete problem
Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0
locally integrable byharmonic fcts
Applications Mesh parameterization (cf GuYau) texture mapping shape matching shape morphing hellip
Harmonic vector fields
Strategybull discretize deRham complexbull get model for cohomology Hsup1bull discretize Hodge decompositionbull get harmonic vector fieldsbull compute basis of harmonic vector fields
Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Its deRham complex is
d is Cartan outer differential (no metric)
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Its deRham complex is
This is a chain complex
d is Cartan outer differential (no metric)
Review smooth deRham complex
functions 1-forms 2-forms
vector fields
(Mg) smooth surface Make use of metric g
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
vector fields
sharp Hodge star
Review smooth deRham complex
vector fields
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Review deRham cohomology
encodes global topology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Local integrability iff locally
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
encodes global topology
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Strategy for discretization
Local integrability iff locally
encodes global topology
1st step2nd step
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Harmonic vector fields
Strategybull discretize deRham complexbull get model for cohomology Hsup1bull discretize Hodge decompositionbull get harmonic vector fieldsbull compute basis of harmonic vector fields
Problem Compute space of harmonic vector fieldsfor meshes of genus gt 0
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Its deRham complex is
d is Cartan outer differential (no metric)
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Its deRham complex is
This is a chain complex
d is Cartan outer differential (no metric)
Review smooth deRham complex
functions 1-forms 2-forms
vector fields
(Mg) smooth surface Make use of metric g
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
vector fields
sharp Hodge star
Review smooth deRham complex
vector fields
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Review deRham cohomology
encodes global topology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Local integrability iff locally
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
encodes global topology
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Strategy for discretization
Local integrability iff locally
encodes global topology
1st step2nd step
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Its deRham complex is
d is Cartan outer differential (no metric)
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Its deRham complex is
This is a chain complex
d is Cartan outer differential (no metric)
Review smooth deRham complex
functions 1-forms 2-forms
vector fields
(Mg) smooth surface Make use of metric g
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
vector fields
sharp Hodge star
Review smooth deRham complex
vector fields
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Review deRham cohomology
encodes global topology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Local integrability iff locally
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
encodes global topology
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Strategy for discretization
Local integrability iff locally
encodes global topology
1st step2nd step
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Its deRham complex is
d is Cartan outer differential (no metric)
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Its deRham complex is
This is a chain complex
d is Cartan outer differential (no metric)
Review smooth deRham complex
functions 1-forms 2-forms
vector fields
(Mg) smooth surface Make use of metric g
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
vector fields
sharp Hodge star
Review smooth deRham complex
vector fields
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Review deRham cohomology
encodes global topology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Local integrability iff locally
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
encodes global topology
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Strategy for discretization
Local integrability iff locally
encodes global topology
1st step2nd step
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Its deRham complex is
This is a chain complex
d is Cartan outer differential (no metric)
Review smooth deRham complex
functions 1-forms 2-forms
vector fields
(Mg) smooth surface Make use of metric g
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
vector fields
sharp Hodge star
Review smooth deRham complex
vector fields
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Review deRham cohomology
encodes global topology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Local integrability iff locally
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
encodes global topology
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Strategy for discretization
Local integrability iff locally
encodes global topology
1st step2nd step
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Review smooth deRham complex
functions 1-forms 2-forms
vector fields
(Mg) smooth surface Make use of metric g
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
vector fields
sharp Hodge star
Review smooth deRham complex
vector fields
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Review deRham cohomology
encodes global topology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Local integrability iff locally
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
encodes global topology
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Strategy for discretization
Local integrability iff locally
encodes global topology
1st step2nd step
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Review smooth deRham complex
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
vector fields
sharp Hodge star
Review smooth deRham complex
vector fields
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Review deRham cohomology
encodes global topology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Local integrability iff locally
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
encodes global topology
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Strategy for discretization
Local integrability iff locally
encodes global topology
1st step2nd step
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Review smooth deRham complex
vector fields
functions 1-forms 2-forms
(Mg) smooth surface Make use of metric g
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Review deRham cohomology
encodes global topology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Local integrability iff locally
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
encodes global topology
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Strategy for discretization
Local integrability iff locally
encodes global topology
1st step2nd step
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Review deRham cohomology
encodes global topology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Local integrability iff locally
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
encodes global topology
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Strategy for discretization
Local integrability iff locally
encodes global topology
1st step2nd step
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Review deRham cohomology
encodes global topology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Local integrability iff locally
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
encodes global topology
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Strategy for discretization
Local integrability iff locally
encodes global topology
1st step2nd step
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Local integrability iff locally
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
encodes global topology
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Strategy for discretization
Local integrability iff locally
encodes global topology
1st step2nd step
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Review deRham cohomology
First deRham comology of smooth surface (Mg)from grad-curl chain complex
Strategy for discretization
Local integrability iff locally
encodes global topology
1st step2nd step
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constant
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete deRham
space of continuous PL functions on polyhedron
Discrete gradient
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete deRham
space of continuous PL functions on polyhedron
bull Gradients are piecewise constantbull tangential components are equal
along any edge
Local integrability iff locally
Discrete gradient
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete deRham
Discrete curl
(tangential jump)
ifftangential components are equalalong any edge
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete deRham
(tangential jump)
PL functionson vertices
PL functionson edge midpoints
PC vectorfields
Discrete deRham complex
ifftangential components are equalalong any edge
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete deRham
Lagrangeelement
Crouzeix-Raviartrsquo73
If then complex is exact in middle comp
bull ArnoldFalkrsquo89 ldquoA uniformly accurate FEM for Reissner-Mindlin Platerdquobull PolthierPreussrsquo03 ldquoIdentifying Vector Field Singularitieshelliprdquo
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete deRham
In fact complex yields correct cohomology
(dim = 2genus)
(not surprising - think simplicial = closed simplicial 1-forms)
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete Hodge for closed meshes
(adjoint complex with respect to Lsup2 inner products)
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete Hodge for closed meshes
Hodge decomposition is simple linear algebra (since dim is finite)(B Eckmannrsquo44 ldquoHarmonische Funktionenhelliprdquo)
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete Hodge for closed meshes
harmonic vector fieldsDiscrete Hodge decomposition
Lemma Decomposition only depends on choice of spaces and inner product on It does neither
depend on inner products on and nor on choice of operators
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete Harmonic vector fields
dim = 2genus
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete Harmonic vector fields
Solve sparse systemfor u and v
dim = 2genus
Get method for computing
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Computing harmonic fields
dim = 2genus
1 Init H = 0 2 Pick random PC vector field X3 Compute potential and co-potential of X4 Compute harmonic part h of X5 h in H IF TRUE goto 2 ELSE H = span(Hh)6 dim(H) = 2genus IF TRUE exit ELSE goto 2
Complexity (4genus) sparse systems (because odds that step 5 returns TRUE are zero)
Las Vegas algorithm for computing harmonic fields
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Computing harmonic fields
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Harmonic fields cont
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Harmonic fields cont
no tangential jump
X is closed
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Harmonic fields cont
no tangential jump
X is co-closed
X is closed
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Harmonic fields amp harmonic1-forms
(no tangential jump)
X is closed ie X corresponds tosimplicial 1-form
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is co-closed(condition at vertices)X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
this 1-form is closed(condition at faces)X is pcw cnst
this 1-form is co-closed(condition at vertices)X is co-closed
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
this 1-form is co-closed(condition at vertices)
this 1-form is closed(condition at faces)X is pcw cnst harmonic 1-froms on graphs
cf GortlerGotsmanThurston
X is co-closed
Harmonic fields amp harmonic1-forms
X is closed ie X corresponds tosimplicial 1-form
(watch out cotan edge weightscould be negative cf BobenkoSpringborn forgetting non-negative weights by intrinsic edge flips)
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete Hodge Star
Complex multiplication (rotation by )
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete Hodge Star
X has no normal jump across edges
Complex multiplication (rotation by )
hellipdoes not take to itself
Because
X has no tangential jump across edges
X would have to be globally continuous
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Discrete Hodge Star
Complex multiplication (rotation by )
Hodge star
hellipin other words Hodge star of X is harmonic part of J(X)
hellipdoes not take to itself
Cfbull Mercatrsquo01 ldquoDiscrete Riemann Surfaces and the Ising Modelrdquobull Polthierrsquo05 ldquoComputational Aspects of Discrete Minimal Surfacesrdquobull Wilsonrsquo05 ldquoGeometric structures on the cochains of a manifoldrdquo
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
(image are piecewise linear forms)
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57)
Inner product on simplicial co-chains
Adjoint operator
yields Whitney-Hodge decomposition
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
compare Whitney Forms
Simplicial co-chains
Whitney Forms (rsquo57) by construction
deRham map DodziukPatodirsquo76
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Contents
Contentsbull Review of smooth deRham complexbull Discrete deRham complexbull Discrete Hodge splittingbull Computing harmonic vector fieldsbull Glimpse at convergence
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of area
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Convergence What can go wrong
Lantern of Schwarz
hellippointwise convergence wo convergence of areabull pointwise + normal convergence implies convergence of area(cf MorvanThibertrsquo04 for a quantitative result)
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
hellip for a sequence of compact polyhedral surfaces converging to a smooth surface embedded into Euclidean 3-space The smooth surface inherits a sequence of cone metrics via ldquodistance mapsrdquo
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Convergence of Hodge decomposition
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Convergence of Hodge decomposition
Sketch of proofbull Convergence based on Whitney forms (DodziukPatodirsquo76 Wilsonrsquo05)
bull This remains valid for variable (and converging) metrics
bull FEM-Whitney relations
-projection of smooth to piecewise constant vector fields
discrete Hodge
smooth Hodge
Theorem Convergence of positions + normals of surfaces in -conv of discrete Hodge decomposition and Hodge star
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-
Thank you
- Algebraic Topology on Polyhedrafrom Linear Finite Elements
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Letrsquos start with a concrete problem
- Harmonic vector fields
- Contents
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review smooth deRham complex
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Review deRham cohomology
- Contents
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Discrete deRham
- Contents
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Discrete Hodge for closed meshes
- Contents
- Discrete Harmonic vector fields
- Discrete Harmonic vector fields
- Computing harmonic fields
- Computing harmonic fields
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields cont
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Harmonic fields amp harmonic1-forms
- Discrete Hodge Star
- Discrete Hodge Star
- Discrete Hodge Star
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- compare Whitney Forms
- Contents
- Convergence What can go wrong
- Convergence What can go wrong
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
- Convergence of Hodge decomposition
-