A discontinuous Galerkin approximation for a wall–bounded ...
Model-Data Weak Formulation (MDWF): Galerkin Approximation ...
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Model-Data Weak Formulation (MDWF):Galerkin Approximation
RecapitulationUnlimited-Observations StatementLimited-Observations StatementExperimentally Observable SpacesInterpretation: Variational Data Assimilation
57
Model-Data Weak Formulation (MDWF):Galerkin Approximation
RecapitulationUnlimited-Observations StatementLimited-Observations StatementExperimentally Observable SpacesInterpretation: Variational Data Assimilation
58
Abstraction
We consider aphysical system
and associated true state field
utrue ∈ X ,
which we assume is deterministic and stationary.
We wish to predict the field utrue based on
a best-knowledge mathematical model A, f, and
M experimental observations,within an integrated (variational) weak formulation.
59
Model Bias . . .
Given the true state of the physical system
(field) utrue ∈ X ,
define the model bias as residual
g ≡ Autrue − f ∈ Y ,
such that
Autrue = f + g .
Recall Aubk = f : ubk = utrue if and only if g = 0.
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. . .Model Bias
Given that
Autrue = f + g
we must simultaneously estimate
model bias: g ∈ Y , and
state: utrue ∈ X ;
unique decomposition of f = Autrue − g in terms of
g and utrue
requires experimental observations.61
Model-Data Weak Formulation (MDWF):Galerkin Approximation
RecapitulationUnlimited-Observations Statement
PreliminariesSaddle Formulation
Limited-Observations StatementExperimentally Observable SpacesInterpretation: Variational Data Assimilation
62
Model-Data Weak Formulation (MDWF):Galerkin Approximation
RecapitulationUnlimited-Observations Statement
PreliminariesSaddle Formulation
Limited-Observations StatementExperimentally Observable SpacesInterpretation: Variational Data Assimilation
63
Operators and Sesquilinear Forms
Recall
inner products (·, ·)X , (·, ·)Y ,
duality pairings ·, ·X ×X , ·, ·Y ×Y ,
representation operators X : X → X , Y : X → Y ,
associated with our spaces X (Ω),Y(Ω), and
operator A : X → Y , or equivalently
sesquilinear form a : X × Y → C , and
source f ∈ Y ,
associated with our best-knowledge model.64
Model Bias
Recall the definition of model bias
g ≡ Autrue − f ∈ Y
in terms of best-knowledge model and true state.
Now introduce the Y-representation of g
qmod ≡ Y
−1g ∈ Y ;
we shall refer to both g and qmod as model bias.
Recall that model bias can also represent error inthe best-knowledge model A, f, δA, δf.
65
Observable Field (Assumption: Strong)
Recall the “true state” of the physical system,
utrue ∈ X ,
which we assume is deterministic and stationary.
Now express the “observable” state as
uobs ≡ u
true − qobs,
for qobs ∈ X deterministic and stationary:qobs = 0 — perfect observations;qobs = 0 — imperfect observations;
qobs → ∞ — uninformative observations;
qobs represents a measurement-induced perturbation.
66
Weighting Parameter
Introduce non-negative real parameter,ν ∈ R+,0 ,
which can be interpreted (conceptually) as†
trust in best-knowledge modeltrust in experimental observations
.
A priori theory shall suggest
νopt ≡
AqobsYqmodY
as a guideline for choice of ν.Note as approach perfect observations, ν → 0;
as approach uninformative observations, ν → ∞.†We can view ν
−1 as the gain on the data “innovation”.67
Model-Data Weak Formulation (MDWF):Galerkin Approximation
RecapitulationUnlimited-Observations Statement
PreliminariesSaddle Formulation
Limited-Observations StatementExperimentally Observable SpacesInterpretation: Variational Data Assimilation
68
Statement
Given ν ∈ R+,0, find (u,ψ) ∈ X × Y such that
a(u, v)− (ψ, v)Y = f, vY ×Y , ∀v ∈ Y ,
−a(u,φ)− ν(ψ,φ)Y =− a(utrue,φ)
− ν(qmod,φ)Y , ∀φ ∈ Y ,
or
a(u, v)− (ψ, v)Y = f, vY ×Y , ∀v ∈ Y ,
−a(u,φ)− ν(ψ,φ)Y =− a(uobs,φ)− a(qobs,φ)
− ν(qmod,φ)Y , ∀φ ∈ Y ,
since utrue = u
obs + qobs.
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Solution (by Construction)
Proposition 1. The unique solution to our saddle problemis (u,ψ) = (utrue, qmod) ∈ X × Y .
Sketch of proof:
Satisfaction of second equation: trivial.
Satisfaction of first equation (also trivial):
a(u, v)− (ψ, v)Y
= f, vY ×Y ,
from definitions of qmod and g.
Uniqueness follows from our assumptions on A. 70
Solution (by Construction)
Proposition 1. The unique solution to our saddle problemis (u,ψ) = (utrue, qmod) ∈ X × Y .
Sketch of proof:
Satisfaction of second equation: trivial.
Satisfaction of first equation (also trivial):
a(utrue, v)− (qmod, v)Y
= f, vY ×Y ,
from definitions of qmod and g.
Uniqueness follows from our assumptions on A. 71
Solution (by Construction)
Proposition 1. The unique solution to our saddle problemis (u,ψ) = (utrue, qmod) ∈ X × Y .
Sketch of proof:
Satisfaction of second equation: trivial.
Satisfaction of first equation (also trivial):
a(utrue, v)− (qmod, v)Y = Au
true− Y q
mod, vY ×Y
= f, vY ×Y ,
from definitions of qmod and g.
Uniqueness follows from our assumptions on A. 72
Solution (by Construction)
Proposition 1. The unique solution to our saddle problemis (u,ψ) = (utrue, qmod) ∈ X × Y .
Sketch of proof:
Satisfaction of second equation: trivial.
Satisfaction of first equation (also trivial):
a(utrue, v)− (qmod, v)Y = Au
true− Y q
mod, vY ×Y
= Autrue
− g, vY ×Y
= f, vY ×Y ,
from definitions of qmod and g.
Uniqueness follows from our assumptions on A. 73
Solution (by Construction)
Proposition 1. The unique solution to our saddle problemis (u,ψ) = (utrue, qmod) ∈ X × Y .
Sketch of proof:
Satisfaction of second equation: trivial.
Satisfaction of first equation (also trivial):
a(utrue, v)− (qmod, v)Y = Au
true− Y q
mod, vY ×Y
= Autrue
− g, vY ×Y
= f, vY ×Y ,
from definitions of qmod and g.
Uniqueness follows from our assumptions on A. 74
Interpretation
We interpret the first equation
a(u, v)− (ψ, v)Y = f, vY ×Y , ∀v ∈ Y ,
−a(u,φ)− ν(ψ,φ)Y =− a(uobs,φ)− a(qobs,φ)
− ν(qmod,φ)Y , ∀φ ∈ Y ,
as the “model” equation: best-knowledge plus model bias.We interpret the second equation
a(u, v)− (ψ, v)Y = f, vY ×Y , ∀v ∈ Y ,
−a(u,φ)− ν(ψ,φ)Y =−a(uobs,φ)− a(qobs,φ)
−ν(qmod,φ)Y , ∀φ ∈ Y ,
as the “data” equation: model-observation connection. 75
Model-Data Weak Formulation (MDWF):Galerkin Approximation
RecapitulationUnlimited-Observations StatementLimited-Observations StatementExperimentally Observable SpacesInterpretation: Variational Data Assimilation
76
Unlimited-Observations: Impractical
Our infinite-observations saddle
a(u, v)− (ψ, v)Y = f, vY ×Y , ∀v ∈ Y ,
−a(u,φ)− ν(ψ,φ)Y =−
(Auobs,φ)Y
a(uobs,φ) −
(Aqobs,φ)Y a(qobs,φ)
− ν(qmod,φ)Y , ∀φ ∈ Y ,
is not actionable since we do not know1. the observable field, uobs, everywhere;2. (precisely) the observational imperfection, qobs;3. the model bias, qmod.
We can not evaluate the “knowns” for the second equation.77
Limited-Observations: Discretization Procedure
To construct our limited-observations saddle we
1a. project Auobs over appropriate approximationspace YM ⊂ Y of finite dimension M ≥ 0;
1b. search for ψM,ν ≈ ψ in YM to ensureuniqueness of discrete solution (uM,ν,ψM,ν);
2. assume that AqobsY is negligibly small;
3. assume that νqmodY is negligibly small.
The latter two assumptions reflectthe achievable accuracy (for ν ≈ ν
opt).
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Limited-Observations Saddle
Given ν ∈ R+,0, find (uM,ν,ψM,ν) ∈ X × YM such that
a(uM,ν, v)− (ψM,ν, v)Y = f, vY ×Y , ∀v ∈ Y ,
−a(uM,ν,φ)− ν(ψM,ν,φ)Y = −a(uobs,φ), ∀φ ∈ YM ;
recall that uobs is the experimentally observable field.
Note for M = 0 (YM ≡ 0): ψM,ν = 0 and hence
uM,ν = ubk
since
a(ubk, v) = f, vY ×Y , ∀v ∈ Y ;
in absence of data, recover best-knowledge model.79
Model-Data Weak Formulation (MDWF):Galerkin Approximation
RecapitulationUnlimited-Observations StatementLimited-Observations StatementExperimentally Observable SpacesInterpretation: Variational Data Assimilation
80
Observation Functionals [F]
We introduce Mmax observation functionals
om∈ X , m = 1, . . . ,Mmax,
which reflecttransducer position or focus, and
transducer filter characteristics,
associated with the data acquisition procedure.
We define a single observation as
m ∈ 1, . . . ,Mmax → om(uobs) ∈ C ,
where recall uobs = utrue − q
obs is the observable field.81
Example: Gaussian Observation Functionals
Consider Ω ⊂ Rd, H10(Ω) ⊂ X ⊂ H
1(Ω), and define
om(v) = Gauss (v; xc
m, σ) ≡
Ω
1
(2π)d/2σde−
|x−xcm|2
2σ2 v(x)dx
forobservation centers x
cm∈ Ω, m = 1, . . . ,Mmax,
observation filter width σ ∈ R+;
we apply corrections near the domain boundary ∂Ω.
In theory, we may consider σ → 0 but only for d = 1;for real transducers, σ will be finite for any d.
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Hierarchical Spaces Eo
M
Given choices of
observation functionals om
, m = 1, . . . ,Mmax ,
define, for 1 ≤ M ≤ Mmax, Eo
0 ≡ 0
Eo
M≡ spanA−∗
om, 1 ≤ m ≤ M;
note spaces are hierarchical, Eo
M⊂ E
o
M+1.
We say that Eo
Mis experimentally observable
with respect to observation functionals om1≤m≤M .
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Projection of Observable Field
Recall that (uM,ν,ψM,ν) ∈ X × YM satisfies
a(uM,ν, v)− (ψM,ν, v)Y = f, vY ×Y , ∀v ∈ Y ,
−a(uM,ν,φ)− ν(ψM,ν,φ)Y = −
projection of observable field: (Auobs,φ)Y
a(uobs,φ) , ∀φ ∈ YM .
We now require YM = Eo
M: then, for φ ∈ YM ,
Auobs,
M
m=1
αmA−∗om
φ∈YM
Y
=A
∗M
m=1αmA
−∗om, u
obs
X ×X
=M
m=1αm
om, u
obsX ×X
corresponds to M (realizable) single observations.84
Model-Data Weak Formulation (MDWF):Galerkin Approximation
RecapitulationUnlimited-Observations StatementLimited-Observations StatementExperimentally Observable SpacesInterpretation: Variational Data Assimilation
85
Regularization-Misfit Minimization [ZN](restricted to Galerkin approximation)
The state uM,ν ∈ X satisfies
uM,ν = argminw∈X
f − Aw
2Y
regularization
+1νΠMA(uobs − w)2Y
model-observation misfit
,
where the projector ΠM : Y → YM is given by
(ΠMw, v)Y = (w, v)Y , ∀ v ∈ YM , for any w ∈ Y .
For perfect observations and ν → 0 we recovera penalty formulation for constrained estimation.
Saddle: primal-dual Euler-Lagrange equations.††The primal-only Euler-Lagrange equation constitutes a single-field
formulation for uM,ν ∈ X (not well-suited to computation).86
MDWF Galerkin: Analysis
A Priori EstimatesStability Constant: Monotonic ImprovementApproximation Theory: Simple ExampleLimit of Small Model Bias
87
MDWF Galerkin: Analysis
A Priori EstimatesPerfect Observations (P-O) [QV]Imperfect Observations (I-O)
Stability Constant: Monotonic ImprovementApproximation Theory: Simple ExampleLimit of Small Model Bias
88
MDWF Galerkin: Analysis
A Priori EstimatesPerfect Observations (P-O) [QV]Imperfect Observations (I-O)
Stability Constant: Monotonic ImprovementApproximation Theory: Simple ExampleLimit of Small Model Bias
89
Formulation Rappel (I-O) . . .
Recall (u,ψ) ∈ X × Y satisfies
a(u, v)− (ψ, v)Y = f, vY ×Y , ∀v ∈ Y ,
−a(u,φ)− ν(ψ,φ)Y = −a(uobs,φ)− a(qobs,φ)
− ν(qmod,φ)Y , ∀φ ∈ Y ,
and (uM,ν,ψM,ν) ∈ X × YM satisfies
a(uM,ν, v)− (ψM,ν, v)Y = f, vY ×Y , ∀v ∈ Y ,
−a(uM,ν,φ)− ν(ψM,ν,φ)Y = −a(uobs,φ), ∀φ ∈ YM .
Now recall (P-O) ≡ qobs = 0 ⇒ choose ν = 0.
90
. . . Formulation Rappel (P-O) . . .
Then (u,ψ) ∈ X × Y satisfies†
a(u, v)− (ψ, v)Y = f, vY ×Y , ∀v ∈ Y ,
−a(u,φ) = −a(uobs,φ)− a(qobs,φ), ∀φ ∈ Y ,
and (uM,0,ψM,0) ∈ X × YM satisfies
a(uM,0, v)− (ψM,0, v)Y = f, vY ×Y , ∀v ∈ Y ,
−a(uM,0,φ) = −a(uobs,φ), ∀φ ∈ YM .
We henceforth suppress the subscript ν(= 0).
†This nonsymmetric saddle may be converted to a symmetric saddle.91
. . . Formulation Rappel (P-O)
Equivalently, (u,ψ) ∈ X × Y satisfies
(Au, v)Y − (ψ, v)Y = f, vY ×Y , ∀v ∈ Y ,
−(Au,φ)Y = −(Auobs,φ)Y , ∀φ ∈ Y ,
and (uM ,ψM) ∈ X × YM satisfies
(AuM , v)Y − (ψM , v)Y = f, vY ×Y , ∀v ∈ Y ,
−(AuM ,φ)Y = −(Auobs,φ)Y , ∀φ ∈ YM .
Recall A = Y−1A such that
a(w, v) = Aw, vY ×Y = (Aw, v)Y ,
for any w ∈ X , v ∈ Y . 92
A Priori Estimates (P-O): Definitions
Define observation-constrained spacesY⊥
M≡ w ∈ Y : (w, v)Y = 0, ∀v ∈ YM
X⊥
M≡ w ∈ X : (Aw, v)Y
a(w,v)
= 0, ∀v ∈ YM
and an associated inf-sup constant† ν = 0
βM,ν=0 ≡ infw∈X⊥
M
AwYwX
;
note
βM=0,ν=0 = β0 ≡ infw∈X
supv∈Y
Aw, vY ×Y
wXvY> 0.
†The norm · X can be replaced with a semi-norm | · |X .93
A Priori Estimates (P-O)
Proposition 2: The error e ≡ utrue − uM satisfies
AeY = infφ∈YM
qmod
model bias
−φY
or in X -norm
eX ≤1
βM,ν=0inf
φ∈YM
qmod
− φY ;
furthermore
qmod
− ψMY = infφ∈YM
qmod
− φY .
Recall ν = νopt = 0 (for qobs = 0).
94
Proof of Proposition 2 (P-O): Preliminaries
Error Equation: The error e ≡ utrue − uM satisfies
(Ae, v)Y − (qmod− ψM , v)Y = 0, ∀v ∈ Y , EQN1
−(Ae,φ)Y = 0, ∀φ ∈ YM , EQN2
since ψ = qmod.
Projection Operator: Recall ΠM : Y → YM (≡ Eo
M)
(ΠMw, v)Y = (w, v)Y , ∀v ∈ YM , for any w ∈ Y .
Constrained Spaces: Recall
Y⊥
M= w ∈ Y : (w, v)Y = 0, ∀v ∈ YM,
X⊥
M= w ∈ X : Aw ∈ Y
⊥
M.
95
Proof of Proposition 2 (P-O) . . .
To recover the first result
(a) from EQN2, Ae ∈ Y⊥
M, e ∈ X⊥
M;
(b) from EQN1 tested on v ∈ Y⊥
M⊂ Y ,
(Ae, v)Y = (qmod, v)Y , ∀v ∈ Y⊥
M;
(c) from (a) and (b)
Ae = qmod − ΠMq
mod;
(d) from definition of projection
AeY = infφ∈YM
qmod − φY .
Error in bias induces error in state. 96
. . . Proof of Proposition 2 (P-O) . . .
or, in a picture
97
. . . Proof of Proposition 2 (P-O) . . .
or, in a picture
98
. . . Proof of Proposition 2 (P-O) . . .
or, in a picture
99
. . . Proof of Proposition 2 (P-O) . . .
or, in a picture
100
. . . Proof of Proposition 2 (P-O)To recover the second result
(e) from (a), (d), and definition of stability constant†
βM,ν=0 ≡ infw∈X⊥
M
AwYwX
,
we obtainβM,ν=0eX ≤ inf
φ∈YM
qmod
− φY .
To recover the third result
(f) from EQN1 tested on v ∈ Y
qmod − ψM = Ae;
(g) from (c), ψM = ΠMψ.
†We may replace wX with any desired semi-norm |w|X .101
A Priori Estimates (P-O): Contributions
(Recall) Proposition 2:
utrue
− uMX ≤1
βM,ν=0inf
φ∈YM
qmod
− φY .
The state error depends on YM ⊂ YM+1
1. the stability constant:βM,ν=0 (↑) as M (↑);
2. the model-bias best-fit error:inf
φ∈YM
qmod
− φY (↓) as M (↑).
Note implicit dependence on regularity of qmod.102
Output Error Estimation (P-O)
Proposition 3: for any out ∈ X ,
|out(e)| ≤ inf
ζ∈YM
A−∗out
− ζY infφ∈YM
qmod
− φY .
The output error depends on
1. the model-bias best-fit error:
infφ∈YM
qmod
− φY (↓) as M (↑);
2. the adjoint best-fit error:
infζ∈YM
A−∗out
− ζY (↓) as M (↑).
Note for out = o· , |out(e)| = 0.
103
MDWF Galerkin: Analysis
A Priori EstimatesPerfect Observations (P-O) [QV]Imperfect Observations (I-O)
Stability Constant: Monotonic ImprovementApproximation Theory: Simple ExampleLimit of Small Model Bias
104
Energy Norm (I-O)
Define an energy norm ΠM : Y → YM
|||w|||M,ν ≡ (Aw2Y + ν−1ΠMAw2Y)1/2
and a stability constant or semi-norm |w|X
βM,ν ≡ infw∈X
|||w|||M,ν
wX;
note consistency in the ν → 0 limit,
βM,ν→0 = infw∈X
|||w|||M,ν→0
wX→ inf
w∈X⊥M
AwYwX
≡ βM,ν=0
since ΠMAwY is penalized by ν−1 → ∞.
105
A Priori Estimate (I-O): Contributions
Proposition 4: The state estimate satisfies ν = νopt
utrue
− uM,νX ≤
1
βM,νopt
inf
φ∈YM
qmod
− φ2Y + 4qmod
YAqobsY dictates error as M→∞
12.
The state error depends on1. the stability constant: βM,νopt (↑) as M (↑);2. the model-bias best-fit error:
infφ∈YM qmod − φY (↓) as M (↑);3. the observation imperfection: AqobsY .
Note AqmodY does not depend on M .106
A Priori Estimate (I-O): Observations → Perfect
As AqobsY/qmodY → 0 (and νopt → 0),
utrue
− uM,ν=0X ≤1
βM,ν=0inf
φ∈YM
qmod
− φY ;
if infφ∈YM
qmod − φY → 0 as M → ∞ then
utrue − uM,ν=0X → 0 as M → ∞.
Note A, fbk will still play role in convergence:inf-sup constant, βM,ν=0;magnitude and regularity of qmod;experimentally observable spaces YM .
107
MDWF Galerkin: Analysis
A Priori EstimatesStability Constant: Monotonic Improvement
Perfect Observations (P-O)Imperfect Observations (I-O)
Approximation Theory: Simple ExampleLimit of Small Model Bias
108
MDWF Galerkin: Analysis
A Priori EstimatesStability Constant: Monotonic Improvement
Perfect Observations (P-O)Imperfect Observations (I-O)
Approximation Theory: Simple ExampleLimit of Small Model Bias
109
Stabilization (P-O)
Proposition 5. The inf-sup constant
βM,0 ≡ infw∈X⊥
M
AwYwX
is a non-decreasing function of M .
Proof. We note thatYM+1 ⊃ YM ⇒ X
⊥
M+1 ⊂ X⊥
M;
more data implies greater stability.
110
Improvement in Stability: Ideal Case . . .
Consider a generalized SVD of A ∈ L(X ,Y ),
(Aξj,Av)Y = σ2j(ξj, v)X , ∀v ∈ X ,
ηj =1
σjAξj,
forsingular values in R+: σ1 ≤ σ2 ≤ · · · ;trial singular functions in X : (ξm, ξn)X = δmn;test singular functions in Y : (ηm, ηn)Y = δmn.
Note σ1 = β0 > 0 (by assumption).
111
. . . Improvement in Stability: Ideal Case
Proposition 6: For P-O ⇒ ν = 0,
the choice† om= Xξm, 1 ≤ m ≤ M ,
yields
βM,ν=0 = σM+1 :
the first M singular functions are “deflated.”
Proof. Express inf-sup as Rayleigh quotient.
†Note this choice yields non-experimentally observable spaces YM , howeveran SVD Anti-Node computational heuristic will be extracted.
112
MDWF Galerkin: Analysis
A Priori EstimatesStability Constant: Monotonic Improvement
Perfect Observations (P-O)Imperfect Observations (I-O)
Approximation Theory: Simple ExampleLimit of Small Model Bias
113
Monotonicity: Statement (P-O,I-O)
Proposition 7: For any ν ∈ R+0 , hierarchical YM
βM ,ν ≥ βM −1,ν, M = 1, . . . ,M ;
and in particular
βM ,ν ≥ βM=0,ν = β0, M = 1, . . . ,M,
where (recall)
β0 ≡ infw∈X
supv∈Y
Aw, vY ×Y
wXvY
is the “no-data” inf-sup of the best knowledge model.
114
Proposition 5: Proof Sketch
Define the minimizer fixed ν
χM ≡ argminw∈X
|||w|||2M
w2X
;
for any w ∈ X , ΠM Aw2Y≥ ΠM −1Aw2Y , and hence
β2M ≡
|||χM |||2M
χM 2X
=AχM 2
Y+ ν
−1ΠM AχM 2Y
χM 2X
≥AχM 2
Y+ ν
−1ΠM −1AχM 2Y
χM 2X
=|||χM |||2
M −1
χM 2X
≥|||χM −1|||
2M −1
χM −12X
≡ β2M −1.
115
MDWF Galerkin: Analysis
A Priori EstimatesStability Constant: Monotonic ImprovementApproximation Theory: Simple ExampleLimit of Small Model Bias
116
One Space Dimension: Definitions
Consider
Ω =]0, 1[ ;
X = Y = H10(Ω) ;
· X = · Y = | · |H1(Ω) ;
A = Y ;
point-wise observation functionals ;
om≡ δ(·; xc
m), m = 1, . . . ,M ;
uniformly spaced observation centers xcmMm=1 .
Note om
bounded for Ω ⊂ R1.
117
One Space Dimension:Experimentally Observable Spaces YM
Native basis functions for YM : φm = A−∗om
, 1 ≤ m ≤ M .118
One Space Dimension: Model Bias Approximation
Proposition 8: For
qmod ∈ H
2(Ω),
we obtain
infφ∈YM
qmod
− φY ≤ CM−1q
modH2(Ω)
for C independent of M and qmod.
Proof. Note YM ≡ spanA−∗omMm=1 is equivalent to
piecewise linear polynomials over nodes xcmMm=1;
now apply standard approximation theory.†
†Note that qmod = Y−1g vanishes at end points of Ω.
119
MDWF Galerkin: Analysis
A Priori EstimatesStability Constant: Monotonic ImprovementApproximation Theory: Simple ExampleLimit of Small Model Bias
120
Strategy: Stabilization
Recall a priori estimate:
utrue
− uM,νX ≤
1
βM,νopt
inf
φ∈YM
qmod
− φ2Y + 4qmod
YAqobsY dictates error as M→∞
12;
if qmodY utrueX , AqobsY utrueX ,
we can control the state error
(solely) by improvements in βM,νopt.
Best conditions: few small singular values.
121
“Optimal” Spaces for Stabilization (P-O)
Goal: deflate dangerous modes (E-optimality [FM]),but remain experimentally observable.
Algorithm SVD Anti-Node: compute
wmin ≡ arg infw∈X⊥
M
AwYwX
;
locate the most sensitive observation point
xcM+1 = arg sup
x∈Ω|wmin(x)| ;
and set
YM+1 = YM ⊕ (A−∗Gauss( · ; xcM+1, σ)) ;
observe least stable mode.122
“Optimal” Spaces for Stabilization (I-O)
Goal: deflate dangerous modes (E-optimality [FM]),but remain experimentally observable.
Algorithm SVD Anti-Node: compute
wmin ≡ arg infw∈X
|||w|||M,ν
wX;
locate the most sensitive observation point
xcM+1 = arg sup
x∈Ω|wmin(x)| ;
and set
YM+1 = YM ⊕ (A−∗Gauss( · ; xcM+1, σ)) ;
observe least stable mode.123
MDWF: Petrov-Galerkin Approach (in brief)
MotivationFormulationAnalysis
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MDWF: Petrov-Galerkin Approach (in brief)
MotivationFormulationAnalysis
125
ConundrumFor MDWF formulation, in data equation
−a(uM,ν,φ)− ν(ψM,ν,φ)Y = −
projection of observable field: (Auobs,φ)Y
a(uobs,φ) , ∀φ ∈ YM ,
we need YM ≡ Eo
M— experimentally observable.
For Galerkin, in model equation
a(uM,ν, v)− (ψM,ν, v)Y = f, vY ×Y , ∀v ∈ Y ,
we need ψM,ν ∈ Y trial=testM
= YM = Eo
M.
Unfortunately, YM = Eo
Mprovides
good stability: small model bias ,but poor approximation: large model bias .
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MDWF: Petrov-Galerkin Approach (in brief)
MotivationFormulationAnalysis
127
Unlimited Observations
Given ν ∈ R+,0, find (u,ψ) ∈ X × Y such that
a(u, v)− (ψ, v)Y = f, vY ×Y , ∀v ∈ Y ,
−a(u,φ)− ν(ψ,φ)Y =− a(uobs,φ)− a(qobs,φ)
− ν(qmod,φ)Y , ∀φ ∈ Y .
The unlimited-observations formulation is unchangedfrom the case of Galerkin approximation.
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Limited Observations
Given ν ∈ R+,0, find (uM,ν,ψM,ν) ∈ X × Y trialM
a(uM,ν, v)− (ψM,ν, v)Y = f, vY ×Y , ∀v ∈ Y ,
−a(uM,ν,φ)− ν(ψM,ν,φ)Y = −a(uobs,φ), ∀φ ∈ YtestM
;
for
Y testM
≡ Eo
Mexperimentally observable
but
Y trialM
informed only by approximation considerations.
Strategy [DG,DPW]: choose o tomaximize stability from SVD anti-node considerations.
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