A Short Course on Duality, Adjoint Operators, Green’s Functions,...

A Short Course on Duality, Adjoint Operators, Green’sFunctions, andA PosterioriError Analysis

Donald J. Estep

[email protected]

Department of Mathematics

Colorado State University

Fort Collins, CO 80523

Duality, Adjoints, Green’s Functions – p. 1/304

Outline of this course

This course has four parts:

• Basic background and theory of the concepts of duality,adjoint operators, and Green’s functions

• Development of an approach to a posteriori error analysisand adaptive error control based on a generalization of theGreen’s function

• An application of these ideas to using the effective domainof influence in elliptic problems to form a decomposition of asolution

• Extension of the ideas to nonlinear problems.


Duality, Adjoint Operators, and Green’s Functions


The Green’s Function

In the classic problem, u solves

−∆u = f, x ∈ Ω,

u = 0, x ∈ ∂Ω,

where Ω is a domain in Rd with boundary ∂Ω.

The Green’s function φ satisfies

−∆φ(y;x) = δy(x), x ∈ Ω,

φ(y;x) = 0, x ∈ ∂Ω,

δy is the delta function at a point y ∈ Ω.


The Green’s Function

Function representation formula:

u(y) =

∫

Ωδy(x)u(x) dx =

∫

Ω−∆φ(y;x)u(x) dx

=

∫

Ωφ(y;x) · −∆u(x) dx =

∫

Ωφ(y;x)f(x) dx.


Background in linear algebra

X is a vector space with norm ‖ ‖ over the real numbers

An important property of the spaces depends on the notion of aCauchy sequence:

DefinitionA sequence xn in X is a Cauchy sequence if we can makethe distance between elements in the sequence arbitrarily smallby restricting the indices to be large.

More precisely, for every ε > 0 there is an N such that‖xn − xm‖ < ε for all n,m > N .

This is a computable criterion for checking convergence.



ExampleConsider the sequence 1/n∞n=1 in [0, 1]. It is a Cauchysequence since

∣

∣

∣

∣

1

n−

1

m

∣

∣

∣

∣

=

∣

∣

∣

∣

m− n

mn

∣

∣

∣

∣

≤ 2maxm,n

mn=

2

minm,n

can be made arbitrarily small by taking m and n large.

It converges to 0 which is in [0, 1].

Sequences that converge are Cauchy sequences. Cauchysequences do not necessarily converge.

ExampleThe sequence 1/n∞n=1 is a Cauchy sequence in (0, 1) but doesnot converge in (0, 1).



Spaces in which Cauchy sequences converge are greatlypreferred.

DefinitionA Banach space is a vector space with a norm such that everyCauchy sequence converges to a limit in the space.

We also say the space is complete .



ExampleRn with the norms defined for x = (x1, · · · , xn)

>,

‖x‖1 = |x1| + · · · + |xn|

‖x‖2 =(

|x1|2 + · · · + |xn|

2)1/2

‖x‖∞ = max |xi|.

are all Banach spaces.

We use ‖ ‖ = ‖ ‖2 unless noted otherwise.



Spaces of functions:

DefinitionFor an interval [a, b], the space of continuous functions isdenoted C([a, b]), where we take the maximum norm

‖f‖ = maxa≤x≤b

|f(x)|.

C1([a, b]) denotes the space of functions that have continuousfirst derivatives on [a, b], where we use the norm

‖f‖ = maxa≤x≤b

|f(x)| + maxa≤x≤b

|f ′(x)|.



Spaces of functions:

DefinitionFor 1 ≤ p ≤ ∞,

Lp(Ω) = f : f is measurable on Ω and ‖f‖p <∞

where for 1 ≤ p <∞,

‖f‖p =

(∫

Ω‖f‖p dx

)1/p

, 1 ≤ p <∞, and ‖f‖∞ = ess supΩ‖f‖.

L2 is particularly important.



TheoremThe Lp spaces and C([a, b]) are Banach spaces.



Consider two vector spaces X and Y with norms ‖ ‖X and ‖ ‖Y

DefinitionA map or operator L from X to Y is a rule or association thatassigns to each x in X a unique element y in Y .

DefinitionA map L : X → Y is linear if L(αx1 + βx2) = αL(x1) + βL(x2)for all numbers α, β and x1, x2 in X.



ExampleEvery linear map from R

m to Rn is obtained by multiplying

vectors in Rm by a n×m matrix, i.e., they have the form Ax,

where A is a n×m matrix.

ExampleDifferentiation is a linear map from C1([a, b]) to C([a, b]).



The maps we consider also have to behave continuously.

DefinitionA map L : X → Y is continuous if for every sequence xn inX that converges to a limit x in X, i.e., xn → x, we haveL(xn) → L(x).

ExampleLinear maps from R

m to Rn are continuous.



There is an equivalent property for linear maps.

DefinitionA linear map L : X → Y is bounded if there is a constant C > 0such that ‖Lx‖Y ≤ C‖x‖X for all x in X.

TheoremA linear map between normed vector spaces is continuous ifand only if it is bounded.



The set of all linear transformations between two vector spacesX and Y is a vector space.

DefinitionIf X and Y are normed vector spaces, we use L(X,Y ) to denotethe vector space of all bounded linear maps from X to Y .L(X,Y ) is a normed vector space under the operator norm

‖L‖ = sup‖x‖X=1

‖Lx‖Y = supx 6=0

‖Lx‖Y‖x‖X

. (1)



ExampleIf the linear transformation L is given by the n× n matrix A, then

‖L‖1 = ‖A‖1 = max1≤j≤n

n∑

i=1

|aij |

‖L‖2 = ‖A‖2 =√

σ(ATA)

‖L‖∞ = ‖A‖∞ = max1≤i≤n

n∑

j=1

|aij |

where σ(A>A) is the spectral radius of A>A.



TheoremIf X and Y are normed vector spaces and Y is complete, thenL(X,Y ) is complete.


Linear functionals and dual spaces

The concept of duality starts with linear functionals.

DefinitionA linear functional on a vector space X is a linear map from Xto R.

ExampleLet v in R

n be fixed. The map F (x) = v · x = (x, v) is a linearfunctional on R

n.

ExampleConsider C([a, b]). Both I(f) =

∫ ba f(x) dx and F (f) = f(y) for

a ≤ y ≤ b are linear functionals.



A linear functional provides a “low dimensional snapshot” of avector.

ExampleIn the example above with F (x) = (x, v), consider v = ei, the ith

standard basis function. Then F (x) = xi where x = (x1, · · · , xn).

We can take v = (1, 1, · · · , 1)/n and compute the average of thecomponents of a given input vector.



Exampleδy gives a linear functional on sufficiently smooth, real valuedfunctions via

F (u) = u(y) =

∫

Ωδy(x)u(x) dx.

Another linear functional is the average value of an integrablefunction,

F (u) =1

vol. of Ω

∫

Ωu(x) dx.

We can view the formulas defining the Fourier coefficients of afunction as a set of linear functionals.



We are interested in the continuous linear functionals.

DefinitionIf X is a normed vector space, the space L(X,R) of boundedlinear functionals on X is called the dual space of or on or toX, and is denoted by X∗.

The dual space is a normed vector space under the dual normdefined for y ∈ X∗ as

‖y‖X∗ = supx∈X

‖x‖X=1

|y(x)| = supx∈Xx 6=0

|y(x)|

‖x‖.



ExampleConsider X = R

n. Every vector v in Rn is associated with a

linear functional Fv(·) = (·, v). This functional is clear boundedsince |(x, v)| ≤ ‖v‖ ‖x‖ (The “C” in the definition is ‖v‖).

A classic result in linear algebra is that all linear functionals onRn have this form, i.e., we can make the identification

(Rn)∗ ' Rn.



ExampleFor C([a, b]), consider I(f) =

∫ ba f(x) dx. It is easy to compute

‖I‖C([a,b])∗ = supf∈C([a,b])max |f |=1

∣

∣

∣

∣

∫ b

af(x) dx

∣

∣

∣

∣

by looking at a picture.



1

-1

ab

possible functions

Computing the dual norm of the integration functional.

The maximum value for I(f) is clearly given by f = 1 or f = −1,and we get ‖I‖C([a,b])∗ = b− a.



ExampleRecall Hölder’s inequality for f ∈ Lp(Ω) and g ∈ Lq(Ω) with1

p+

1

q= 1 for 1 ≤ p, q ≤ ∞ is

‖fg‖L1(Ω) ≤ ‖f‖Lp(Ω)‖g‖Lq(Ω).



Each g in Lq(Ω) is associated with a bounded linear functional

on Lp(Ω) when1

p+

1

q= 1 and 1 ≤ p, q ≤ ∞ by

F (f) =

∫

Ωg(x)f(x) dx.

We can “identify” (Lp)∗ with Lq when 1 < p, q <∞, The casesp = 1, q = ∞ and p = ∞, q = 1 are trickier.

The case L2 is special in that we can identify (L2)∗ with L2.



The dual space is the collection of “reasonable” possiblesamples.

An important characteristic of a dual space is how much we canreveal about a vector by considering samples in the dual space.

ExampleBy considering the set of n functionals corresponding to takingthe inner product with e1, · · · , en, we can “reconstruct” anygiven vector in R

n by looking at the functional values.

The question of whether or not we can “recover” a vector ucompletely by computing sufficiently many linear functionalsdepends heavily on properties of the underlying space.In practice, we will often be content with one or just a few“snapshots”.



We might wonder about the number of bounded linearfunctionals that exist on an arbitrary normed vector space.

The celebrated Hahn-Banach theorem says there is a greatabundance of them.

Let x0 6= 0 be a fixed element of Banach space X.

The set X0 = αx0 : α ∈ R forms a vector subspace of X.

The linear functional F (αx0) = α is defined on X0 and isbounded since |F (αx0)| = |α| = ‖αx0‖/‖x‖.

So, we have found a bounded linear functional on a subspace ofX.

Can we extend this to be defined on all of X?



Issues:• What does it mean to extend?• Does the norm increase upon extension?• What if there is an infinite number of extensions involved?

TheoremHahn-Banach Let X be a Banach space and X0 a subspace ofX. Suppose that F0(x) is a bounded linear functional defined onX0. There is a linear functional F defined on X such thatF (x) = F0(x) for x in X0 and ‖F‖ = ‖F0‖.



Dual spaces are connected to distribution theory and weakconvergence.

The dual space can be better behaved than the original normedvector space.

TheoremIf X is a normed vector space over R, then X∗ is a Banachspace (whether or not X is a Banach space).



Useful notation:

DefinitionIf x is in X and y is in X∗, we denote the value

y(x) =< x, y > .

This is called the bracket notation .



The norms on X and its dual X∗ are closely related. Recall thatif y ∈ X∗, then

‖y‖X∗ = supx∈X

‖x‖X=1

|y(x)| = supx∈Xx 6=0

|y(x)|

‖x‖.



DefinitionThe generalized Cauchy inequality is

|< x, y >| ≤ ‖x‖X ‖y‖X∗ , x ∈ X, y ∈ X∗.

TheoremIf X is a Banach space, then

‖x‖X = supy∈X∗

y 6=0

|y(x)|

‖y‖X∗

= supy∈X∗

‖y‖X∗=1

|y(x)|

for all x in X.


Hilbert spaces and duality

ExampleRn with the standard Euclidean norm ‖ ‖ = ‖ ‖2 can be identified

with its dual space.

L2 can be identified with its dual space.

Both of these spaces are Hilbert spaces.



One way to get a normed vector space is to place an innerproduct (i.e., dot product) on the space.

TheoremIf X has an inner product (x, y), then it is a normed vector spacewith norm ‖x‖ = (x, x)1/2 for x in X.

DefinitionA vector space with an inner product that is a Banach spacewith respect to the associated norm is called a Hilbert space.

ExampleRn with ‖ ‖ = ‖ ‖2 and L2 are both Hilbert spaces.



If X is a Hilbert space with inner product (x, y), then each y ∈ Xdetermines a linear functional Fy(x) =< x, y >= (x, y) for x in X.

This functional is bounded by Cauchy’s inequality,|(x, y)| ≤ ‖x‖ ‖y‖.

The Riesz Representation theorem says this is the only kind oflinear functional on a Hilbert space.

TheoremRiesz Representation For every bounded linear functional Fon a Hilbert space X, there is a unique element y in X such that

F (x) = (x, y), all x ∈ X and ‖y‖X∗ = supx∈Xx 6=0

|F (x)|

‖x‖.



A Hilbert space is isometric to its dual space.

DefinitionTwo normed vector spaces X and Y are isometric if there is alinear 1-1 and onto map L : X → Y such that ‖L(x)‖Y = ‖x‖Xfor all x in X.

We often replace the bracket notation and the generalizedCauchy inequality by the inner product and the “real” Cauchyinequality without comment.



The Sobolev spaces are Hilbert spaces based on L2.

ExampleFor k = 1, 2, 3, · · · , we define Hk(Ω) to be the distributionfunctions in L2(Ω) whose partial derivatives of order k and lessare also distributions in L2(Ω).



Definitionindex notation: For α = (α1, · · · , αn) with integer coefficients,we define

D =

(

∂

∂x1, · · · ,

∂

∂xn

)

, Dα =∂|α|

∂xα1

1 · · · ∂xαnn

with |α| = α1 + · · · + αn.

Hk(Ω) = u,Dαu ∈ L2(Ω), |α| ≤ k. The inner products andnorms are

(u, v)k =∑

|α|≤k

(Dαu,Dαv), ‖u‖k = (u, u)1/2k .



The Riesz Representation theorem says that every linearfunctional on Hk has the form (u, v)k for some fixed v in Hk.

Defining the dual space to Hk runs into subtle difficulties due toa collision with the requirements for using distribution theory.

We define the dual spaces to a subspace.

Hk0 (Ω) =

u ∈ Hk(Ω) : u =∂u

∂n= · · · =

∂k−1u

∂nk−1= 0 on ∂Ω

,

where ∂/∂n denotes the normal derivative on the boundary ∂Ω.

We let H−k(Ω) is the set of all linear functionals on Hk0 (Ω).


Adjoint operators - definition

We now explain how a linear transformation between twonormed vector spaces X and Y is naturally associated withanother linear transformation between Y ∗ and X∗.

This is the infamous adjoint operator.



Suppose L ∈ L(X,Y ) is a bounded linear transformation.

For each y∗ ∈ Y ∗,

y∗ L(x) = y∗(L(x)) =< Lx, y∗ >

assigns a number to each x ∈ X, hence defines a functionalF (x). F (x) is clearly linear.

It is also bounded since

|F (x)| = |y∗(L(x))| ≤ ‖y∗‖Y ∗‖L(x)‖Y ≤ ‖y∗‖Y ∗‖L‖ ‖x‖X .



There is an x∗ ∈ X∗ such that y∗(L(x)) = x∗(x) for all x ∈ X.

x∗ is unique.

Thus, to each y∗ ∈ Y ∗, we have assigned a unique x∗ ∈ X∗ andthus have defined a linear transformation L∗ : Y ∗ → X∗.

We can write these relations as

y∗(L(x)) = L∗y∗(x)



Using the bracket notation,

< L(x), y∗ >=< x,L∗(y∗) > x ∈ X, y∗ ∈ Y ∗.

DefinitionThis is called the bilinear identity .

It defines the adjoint operator L∗ : Y ∗ → X∗ associated to abounded linear transformation L : X → Y .

Note that we have defined the adjoint transformation viasampling by elements in the dual space.



ExampleLet X = R

m and Y = Rn, where we take the standard inner

product and norm.

By the Riesz Representation theorem, the bilinear identity forL ∈ L(Rm,Rn) reads

(Lx, y) = (x, L∗y), x ∈ Rm, y ∈ R

n.



There is a unique n×m matrix A so that if y = L(x) = Ax where

A =

a11 · · · a1m

......

an1 · · · anm

, y =

y1

...yn

, x =

x1

...xm

and

yi =m∑

j=1

aijxj , 1 ≤ i ≤ n.



For a linear functional y∗ = (y∗1, · · · , y∗n)

> ∈ Y ∗

L∗y∗(x) = y∗(L(x)) =

(y∗1, · · · , y∗n),

∑mj=1 a1jxj

...∑m

j=1 anjxj

=

m∑

j=1

y∗1a1jxj + · · ·m∑

j=1

y∗nanjxj

=m∑

j=1

(

n∑

i=1

y∗i aij)

xj



L∗(y∗) is given by the inner product with y = (y1, · · · , ym)>

where

yj =

n∑

i=1

y∗i aij .



The matrix A∗ of L∗ is

A∗ =

a∗11 · · · a∗1n...

...a∗m1 · · · a∗mn

=

a11 a21 · · · an1

......

a1m a2m · · · anm

= A>.

The bilinear identity is

y>Ax = x>A>y

using the fact that (x, y) = (y, x).



The following theorem is crucial.TheoremL∗ ∈ L(Y ∗, X∗) and ‖L∗‖ = ‖L‖.

proof

|L∗y∗(x)| ≤ ‖y∗‖Y ∗‖L‖‖x‖X .

Therefore,

‖L∗y∗‖X∗ = supx 6=0

|L∗y∗(x)|

‖x‖≤ ‖y∗‖Y ∗‖L‖,

which implies that L∗ ∈ L(Y ∗, X∗) and ‖L∗‖ ≤ ‖L‖.



To show the reverse inequality, we prove that

‖Lx‖Y ≤ ‖L∗‖ ‖x‖X , x ∈ X.

The bilinear identity implies that

|y∗(Lx)| ≤ ‖L∗y∗‖X∗‖x‖X ≤ ‖L∗‖‖y∗‖Y ∗‖x‖X

and

supy∗ 6=0

|y∗(Lx)|

‖y∗‖Y ∗

≤ ‖L∗‖ ‖x‖X , x ∈ X.



Properties of the adjoint operator:

TheoremLet X, Y , and Z be normed linear spaces. Then,

0∗ = 0

(L1 + L2)∗ = L∗

1 + L∗2, all L1, L2 ∈ L(X,Y )

(αL)∗ = αL∗, all α ∈ R, L ∈ L(X,Y )

If L2 ∈ L(X,Y ) and L1 ∈ L(Y,Z), then L1L2 ∈ L(X,Z),(L1L2)

∗ ∈ L(Z∗, X∗), and

(L1L2)∗ = L∗

2L∗1.



We are often deal with linear operators that are not defined onthe entire space.

ExampleConsider D = d/dx on X = C([0, 1]). This linear map is onlydefined on the subspace C1([0, 1]).



DefinitionLet X and Y be normed vector spaces. A map L that assigns toeach x in a subset D(L) of X a unique element y in Y is called amap or operator with domain D(L).

L is linear if (1) D(L) is a vector subspace of X and (2)L(αx1 + βx2) = αL(x1) + βL(x2) for all α, β ∈ R andx1, x2 ∈ D(L).



We define the dual of a linear operator by examining its behavioron its domain.

We wantL∗y∗(x) = y∗(Lx) all x ∈ D(L).

We say that y∗ ∈ D(L∗) if there is an x∗ ∈ X∗ such that

x∗(x) = y∗(Lx), all x ∈ D(L).

The existence of x∗ is no longer automatic.

When x∗ exists, we define L∗y∗ = x∗.



For this to work, x∗ must be unique.

In other words, x∗(x) = 0 for all x ∈ D(L) should imply x∗ = 0.

This depends on the “size” of D(L).

DefinitionA subspace A of a normed linear space X is dense if everypoint in X is either in A or the limit of a sequence of points in A.

ExampleThe rational numbers are dense in the real numbers and thepolynomials are dense in C([a, b]).



We can define L∗ for a linear operator L : X → Y provided D(L)is dense in X.

We define D(L∗) to be those y∗ ∈ Y ∗ for which there is anx∗ ∈ X∗ with x∗(x) = y∗(Lx) for all x ∈ X.

This x∗ is unique and L∗y∗ = x∗.

The Hahn-Banach theorem implies that if there is a C such that|y∗(Lx)| ≤ C‖x‖ for all x ∈ D(L), then y∗ ∈ D(L∗).


Adjoint operators - motivation

Common problem: Given normed vector spaces X and Y , anoperator L(X,Y ), and b ∈ Y , find x ∈ X such that

Lx = b.

We explain the role of the adjoint in solving this kind of problems.



DefinitionThe set of b for which there is a solution is called the range ,R(L), of L.

The set of x for which L(x) = 0 is called the null space , N (L),of L.

0 is always in N (L).

If it is the only element in N (L), then Lx = b can have at mostone solution.



TheoremN (L) is a subspace of X and R(L) is a subspace of Y .



If y ∈ R(L), there is an x with Lx = y.

For y∗ ∈ Y ∗,y∗(Lx) = y∗(y).

By the definition of the adjoint,

L∗y∗(x) = y∗(y).

If y∗ ∈ N (L∗), then y∗(y) = 0.

A necessary condition that y ∈ R(L) is that y∗(y) = 0 for ally∗ ∈ N (L∗).



Is this sufficient?

We require just one condition.

DefinitionA subset A of a normed vector space X is closed if everysequence xn in A that has a limit in X has its limit in A.

TheoremLet X and Y be normed linear spaces and L ∈ L(X,Y ). Anecessary condition that y ∈ R(L) is y∗(y) = 0 for ally∗ ∈ N (L∗). This is a sufficient condition if R is closed in Y .



ExampleSuppose that L ∈ L(X,Y ) is associated with the n×m matrixA, i.e., L(x) = Ax.

The necessary and sufficient condition for the solvability ofAx = b is that b is orthogonal to all linearly independentsolutions of AT y = 0.



ExampleIn the case X is a Hilbert space and L ∈ L(X,Y ), thennecessarily R(L∗) ⊂ N (L)⊥, where S⊥ is the subspace ofvectors that are orthogonal to a subspace S.

If R(L∗) is “large”, then N (L)⊥ must be “large” and N (L) mustbe “small”.

The existence of sufficiently many solutions of thehomogeneous adjoint equation implies there is at most onesolution of Lx = b for a given b.



ExampleThe setting of general partial differential equations:

TheoremHolmgren Uniqueness The generalized initial value problemconsisting of the equation

L(u) =∑

|α|≤m

Aα(x)Dαu = f(x), x ∈ Rn,

where Aα and f are analytic functions, together with the data

Dβu(x) = gβ(x), |β| ≤ m− 1, x ∈ S

given on an analytic noncharacteristic surface S, has at mostone solution in a neighborhood of S.



The adjoint has an important role in the solution of a generaln×m system Ax = b, where A is a n×m matrix, x ∈ R

m, andb ∈ R

n.

The reason to dwell on such problems is that differentialoperators do not tend to be “square”.

Exampley′′ = d2y/dx2 requires two boundary conditions to define aproblem for the associated differential equation that has aunique solution.

We may want to study the differential operator without anyboundary conditions, or more or less than two conditions.



ExampleDivergence,

divu =∂u1

∂x1+ · · · +

∂un∂xn

,

associates a scalar with a given vector function, and theassociated differential equation is very “under-determined”.

The gradient,

gradu =

(

∂u1

∂x1, · · · ,

∂un∂xn

)

,

associates a vector field with a given scalar function, and theassociated equations are very “over-determined”.



Consider the n×m system Ax = b, where A is a n×m matrix,x ∈ R

m, and b ∈ Rn.

We enlarge the system by adding the adjoint m× n systemA>y = c, where y and c are independent of x and b.

The new problem is an (n+m) × (n+m) symmetric problem

Sz = d,

with

z =

(

y

x

)

, d =

(

b

c

)

, S =

(

0 A

A> 0

)

.



Since S is symmetric, it is diagonalizable with eigenvaluessatisfying

Av = λu

A∗u = λvor

AA>u = λ2u

A>Av = λ2v

The eigenvalues of S are the singular values of A and last twoequations give the left and right singular vectors of A.



Consequences:• The theorem on the adjoint condition for solvability falls out

right away.• It yields a “natural” definition of a solution or gives

conditions for a solution to exist in the over-determined andunder-determined cases.

• It also gives a way of determining the condition of thesolution process.



One interesting observation is that there is a reciprocalrelationship between the degree of over/under-determination ofthe original system and the adjoint system.

The more over-determined the original system, the moreunder-determined the adjoint system, and so forth.



ExampleConsider 2x1 + x2 = 4, where L : R

2 → R. L∗ : R → R2 is given

by L∗ =

(

2

1

)

. The extended system is

0 2 1

2 0 0

1 0 0

y1

x1

x2

=

4

c1

c2

,

from which we see that 2c1 = c2 is required in order to have asolution.



ExampleOn the other hand, if the problem is

2x1 + x2 = 4

x2 = 3,

with L : R2 → R

2, then there is a unique solution. The extendedsystem is

0 0 2 1

0 0 0 1

2 0 0 0

1 1 0 0

y1

y2

x1

x2

=

c1

c2

4

3

,

where we can specify any values for c1, c2.



In the under-determined case, we can eliminate the deficiencyby posing the method of solution

AA>y = b

x = A>y.or

L(L∗(y)) = b

x = L∗(y).



This works for differential operators

ExampleConsider the under-determined problem

divF = ρ.

The adjoint to div is -grad modulo boundary conditions.

If F = gradu, where u is subject to the boundary conditionu(′′∞′′) = 0, then we obtain the “square”, well-determinedproblem

div gradu = ∆u = −ρ,

which has a unique solution because of the boundary condition.



ExampleConsider

Ax = b

A>φ = ei

where A is a n× n invertible matrix.

There are no constraints on the data for the adjoint problem.

We have specified the ith standard basis vector of Rn.

xi = (x, ei) = (x,A>φ) = (Ax, φ) = (b, φ).

y is the discrete Green’s vector associated to Ax = b.


Adjoint operators - computation

The computation of the adjoint to a general differential operatoris not easy.

A tedious, formal approach:

1. Take the problem for the linear operator, including anyboundary and/or initial conditions imposed with theoperator, and discretize it.

2. Extract the matrix from the resulting discrete system, andcompute its transpose.

3. Let the discretization parameter tend to its limit anddetermine the differential operator that is approached by thetransposed matrix.

Determination of the adjoint of a differential operator is heavilyinfluenced by the boundary and/or initial conditions, as well asthe underlying spaces.



ExampleA standard difference approximation of

−u′′(x) = f(x), 0 < x < 1,

u(0) = 0, u(1) = 0,

yields the matrix

1

h2

2 −1 0

−1 2 −1. . . . . . . . .

−1 2 −1

0 −1 2

,

The differential operator is self-adjoint.



ExampleIf we change the boundary conditions to

−u′′(x) = f(x), 0 < x < 1,

u(0) = 0, u′(0) = 0,

we get a triangular matrix after discretization.

The adjoint corresponds to a problem

−v′′(x) = g(x), 0 < x < 1,

v(1) = 0, v′(1) = 0,



Example

u′ = f(x), 0 < x < 1,

u(0) = 0⇒

−v′ = g(x), 1 > x > 0,

v(1) = 0,

Example

−u′′ = f(x), 0 < x < 1,

no boundary conditions

⇒

−v′′ = g(x), 0 < x < 1,

v(0) = v′(0) = v(1) = v′(1) = 0.



In the case of L2, we can use the bilinear identity

< Lu, v∗ >=< u,L∗v∗ > all u ∈ X, v∗ ∈ Y ∗,

which is

(Lu, v) = (u, L∗v) all suitable u, v ∈ L2(Ω).

DefinitionWe say that we are evaluating the bilinear identity when wecompute

< Lu, v∗ > − < u,L∗v∗ >= (Lu, v) − (u, L∗v)

for some suitable functions u and v.



• Since we are considering differential operators, these willnot be defined on all of L2(Ω), but only a subset ofsufficiently smooth functions.

• The L2 inner product involves integration over the domain,and we can interpret the process producing the bilinearidentity as a succession of integration by parts. This yieldsterms that involve the integrals over the boundary of Ω ofthe functions involved as well as certain derivatives.



Computing the adjoint using the bilinear identity proceeds in twostages.

First, we compute a formal adjoint neglecting all boundary termsby assuming that the functions involved have compact supportinside Ω.

DefinitionA function on a domain Ω has compact support in Ω if itvanishes identically outside a bounded set contained inside Ω.



1. Take the differential operator applied to a smooth functionwith compact support and multiply by a smooth test functionwith compact support

2. Integrate over the domain

3. Integrate by parts to move all derivatives onto the testfunction while ignoring boundary terms.

Functions that have compact support are identically zeroanywhere near the boundary and any boundary terms arisingfrom integration by parts will vanish.



DefinitionLet L be a differential operator. The formal adjoint L∗ is thedifferential operator that satisfies

(Lu, v) = (u, L∗v)

(∫

ΩLu · v dx =

∫

Ωu · L∗v dx

)

for all sufficiently smooth u and v with compact support in Ω.



ExampleConsider

Lu(x) = −d

dx

(

a(x)d

dxu(x)

)

+d

dx(b(x)u(x))

on [0, 1]. Integration by parts neglecting boundary terms gives

−

∫ 1

0

d

dx

(

a(x)d

dxu(x)

)

v(x) dx

=

∫ 1

0a(x)

d

dxu(x)

d

dxv(x) dx− a(x)

d

dxu(x)v(x)

∣

∣

∣

∣

1

0

= −

∫ 1

0u(x)

d

dx

(

a(x)d

dxv(x)

)

dx+ u(x)a(x)d

dxv(x)

∣

∣

∣

∣

1

0

,



∫ 1

0

d

dx(b(x)u(x))v(x) dx = −

∫ 1

0u(x)b(x)

d

dxv(x) dx+b(x)u(x)v(x)

∣

∣

∣

∣

1

0

,

where all of the boundary terms vanish.

Therefore,

L∗v = −d

dx

(

a(x)d

dxv(x)

)

− b(x)d

dx(v(x)).



In higher space dimensions, we use the divergence theorem.

ExampleA general linear second order differential operator L in R

n canbe written

L(u) =n∑

i=1

n∑

j=1

aij∂2u

∂xi∂xj+

n∑

i=1

bi∂u

∂xi+ cu,

where aij, bi, and c are functions of x1, x2, · · · , xn. Then,

L∗(u) =

n∑

i=1

n∑

j=1

∂2(aijv)

∂xi∂xj−

n∑

i=1

∂(biv)

∂xi+ cv.



It can be verified directly that

vL(u) − uL∗(v) =

n∑

i=1

∂pi∂xi

,

where

pi =

n∑

j=1

(

aijv∂u

∂xj− u

∂(aijv)

∂xj

)

+ biuv.

The divergence theorem yields∫

Ω(vL(u) − uL∗(v)) dx =

∫

∂Ωp · nds = 0,

where p = (p1, · · · , pn) and n is the outward normal in ∂Ω.



A typical term,

va11∂2u

∂x21

= va11∂

∂x1

(

∂u

∂x1

)

=∂

∂x1

(

va11∂u

∂x1

)

−∂(a11v)

∂x1

∂u

∂x1

=∂

∂x1

(

va11∂u

∂x1

)

−∂

∂x1

(

u∂(a11v)

∂x1

)

+ u∂2(a11v)

∂x21

yielding

va11∂2u

∂x21

− u∂2(a11v)

∂x21

=∂

∂x1

(

a11v∂u

∂x1− u

∂(a11v)

∂x1

)

.



ExampleLet L be a differential operator of order 2p of the form

Lu =∑

|α|,|β|≤p

(−1)|α|Dα(

aαβ(x)Dβu)

,

thenL∗v =

∑

|α|,|β|≤p

(−1)|α|Dα(

aβα(x)Dβv)

,

and L is elliptic if and only if L∗ is elliptic.



Some special cases.

grad∗ = −div

div∗ = −grad

curl∗ = curl

Lu =∑

|α|≤p

aα(x)Dαu

thenL∗v =

∑

|α|≤p

(−1)|α|Dα(aα(x)v(x)).



ExampleIf

Lu = ρutt −∇ · (a∇u) + bu

then L∗ = L.

IfLu = ut −∇ · (a∇u) + bu

thenL∗v = −vt −∇ · (a∇v) + bv

where time runs “backwards”.



This procedure also works for systems

ExampleLet

L(~u) = A~ux +B~uy + C~u,

where A,B, and C are n× n matrices, then

L∗(~v) = (−A>~v)x − (B>~v)y + C>v,

so that~v>L~u− ~u>L∗~v = ∇ · (~v>A~u,~v>B~u).



In the second stage of computing the adjoint, we deal withboundary conditions by removing the assumption that thefunctions involved have compact support.

Integration by parts yields additional terms involving integralsover the boundary of the functions involved and their derivatives.

We want to determine boundary conditions such that the bilinearidentity still holds, e.g., such that any boundary terms that arisevanish.



It turns out that the form of the boundary conditions imposed inthe problem for L are important, but the values given for theseconditions are not.

If the boundary conditions are not homogeneous, we make themso for the purpose of determining the adjoint.

DefinitionThe adjoint boundary conditions posed on the formal adjointoperator are the minimal conditions required to make theboundary terms that appear when evaluating the bilinear identityfor general smooth functions vanish.



Some of the boundary terms that appear when evaluating thebilinear identity will vanish because of the boundary conditionsimposed in the original problem.

The point of this assumption is to make the formal adjoint serveas the true adjoint by pairing it with the correct boundaryconditions.



This definition can be made completely precise. Issues involved:

• Placing conditions on the differential operator L so thatevaluating the bilinear identity for general smooth functionsresults in expressions involving only values on the boundary.

• Making precise the meaning of “minimal conditions” neededfor the adjoint problem, and proving these always exist.



ExampleConsider Newton’s equation of motion s′′(x) = f(x) with x =“time”, normalized with mass 1. First, suppose we assumes(0) = s′(0) = 0, and 0 < x < 1. We have

s′′v − sv′′ =d

dx(vs′ − sv′)

and∫ 1

0(s′′v − sv′′) dx = (vs′ − sv′)

∣

∣

1

0.



The boundary conditions imply the contributions at x = 0 vanish,while at x = 1 we have

v(1)s′(1) − v′(1)s(1).

To insure this vanishes, we must have v(1) = v′(1) = 0. Theseare the adjoint boundary conditions.



ExampleSuppose instead we impose conditions such that at the massreturns to the origin with zero speed at x = 1.

This gives four boundary conditionss(0) = s′(0) = s(1) = s′(1) = 0 on the original problem.

In this situation, all of the boundary terms are zero and noboundary conditions will be imposed on the adjoint.



Solving the over-determined problem.

ExampleBased on the discussion above, we require the data f to beorthogonal to the solution of the adjoint problem v′′ = 0, which isv = a+ bx.

Hence, f must be orthogonal in L2(0, 1) to 1 and x.

Assume for example that f(x) = a+ bx+ cx2. It is easy to seethat (f, 1) = 0 and (f, x) = 0 forces a = c/6 and b = −c.

Choosing c = 1 for example, means that f(x) = 1/6 − x+ x2.



We solve the forward problem by integrating twice and using theboundary conditions at x = 0 to get a formula for the solution.

The resulting solution satisfies the conditions at x = 1 as well.



ExampleSince

∫

Ω(u∆v − v∆u) dx =

∫

∂Ω

(

u∂v

∂n− v

∂u

∂n

)

ds,

the Dirichlet and Neumann boundary value problems for theLaplacian are their own adjoints.



ExampleLet Ω ⊂ R

2 be bounded with a smooth boundary and let s =arclength along the boundary. Consider

−∆u = f, x ∈ Ω,∂u∂n + ∂u

∂s = 0, x ∈ ∂Ω.

Since∫

Ω(u∆v − v∆u) dx =

∫

∂Ω

(

u

(

∂v

∂n−∂v

∂s

)

− v

(

∂u

∂n+∂u

∂s

))

ds,

the adjoint problem is

−∆v = g, x ∈ Ω,∂v∂n − ∂v

∂s = 0, x ∈ ∂Ω.


Green’s functions

For simplicity, we consider

Lu = f, x ∈ Ω,

suitable b.c. and i.v., x ∈ ∂Ω,

where L is a linear differential operator.

We specify the correct boundary and/or initial conditions sothere is a unique solution.


Green’s functions

DefinitionThe Green’s function satisfies

L∗φ(y, x) = δy(x), x ∈ Ω,

adjoint b.c. and i.v., x ∈ ∂Ω,

where L∗ is the formal adjoint of L.

We are solving the extended system,

Lu = f, x ∈ Ω,


L∗φ(y, x) = δy(x), x ∈ Ω,

adjoint b.c. and i.v., x ∈ ∂Ω.


Green’s functions

Based on the bilinear identity, we obtain a representationformula for the value of the solution of the original problem at apoint y ∈ Ω,

u(y) = (u, δy) = (y, L∗φ) = (Lu, φ) = (f, φ).

The imposition of the adjoint boundary conditions is key here.


Green’s functions

ExampleFor

−∆u = f, x ∈ Ω,

u = 0, x ∈ ∂Ω,

φ solves

−∆φ(y;x) = δy(x), x ∈ Ω,

φ(y;x) = 0, x ∈ ∂Ω,

and the bilinear identity reads

u(y) =

∫

Ωf(x)φ(y;x) dx.


Green’s functions


Green’s functions

ExampleFor

ut − ∆u = f, x ∈ Ω, 0 < t ≤ T,

u(t, x) = 0, x∂Ω, 0 < t ≤ T,

u(0, x) = 0, x ∈ Ω,

φ solves

−φt − ∆φ = δ(s,y)(t, x), x ∈ Ω, T > t ≥ 0,

φ(t, x) = 0, x∂Ω, T > t ≥ 0,

φ(T, x) = 0, x ∈ Ω,

and the bilinear identity reads

u(s, y) =

∫ T

0

∫

Ωf(t, x)φ(s, y; t, x) dxdt.


Green’s functions

Motivations:• The Green’s function is defined without reference to the

particular data f that determines a particular solution. TheGreen’s function depends on the operator and its properties.

• The Green’s function generally has special propertiesarising from the properties of the delta function, such aslocalized support and symmetry.

• The Green’s function gives a “low-dimensional” snapshot ofthe solution.


Green’s functions

ExampleThe Green’s function for the Dirichlet problem for the LaplacianL = −∆ on the ball Ω of radius r centered at the origin in R

3 is

φ(y;x) =1

4π×

|y − x|−1 − r|y|−1∣

∣

r2y|y|2 − x

∣

∣

−1, y 6= 0,

|x|−1 − r−1, y = 0,

where |x| denotes the Euclidean norm of x

The formula for the disk of radius r is

φ(y;x) =1

2π×

ln

(

|y|∣

∣r2y

|y|2−x∣

∣

r|y−x|

)

, y 6= 0,

ln(

r|x|

)

, y = 0.


Green’s functions

Important issues: Existence, uniqueness, and smoothness ofthe Green’s function.

Generally, everything goes as for the original problem exceptthat the Green’s function may not be very smooth or may evenbe undefined at a point.

The point of evaluation y must lie inside the domain Ω. TheGreen’s function often behaves badly in the limit of yapproaching the boundary.

By the way, the theory also extends to over-determinedproblems.

However, if the original problem is under-determined, thisapproach fails.


Green’s functions

Many expositions of the Green’s function avoid the adjoint.

When the original problem has a unique solution and the originaloperator is the adjoint to the adjoint operator, the Green’sfunction φ for the original problem and the Green’s function forthe adjoint problem φ∗ are related via

φ(y;x) = φ∗(x; y).

This is known as the reciprocity theorem.

This makes it possible to introduce Green’s functions for somekinds of problems without talking about the adjoint.

This depends on the dual space of the dual space of the originalspace being identifiable with the original space.


Green’s functions

Including nonhomogeneous conditions is really a minor issuethat usually “solves itself” without trouble.

We carry out the analysis using the Green’s function for thehomogeneous problem and some additional integrals involvingdata and the Green’s function will appear.


Green’s functions

ExampleSuppose the problem is

−∆u = f, x ∈ Ω,

u = g, x ∈ ∂Ω.

We define the Green’s function as for the homogeneousproblem, i.e.,

−∆φ(y;x) = δy(x), x ∈ Ω,

φ(y;x) = 0, x ∈ ∂Ω.


Green’s functions

Evaluating the bilinear identity yields

∫

Ω(u∆φ− φ∆u) dx =

∫

∂Ω

(

u∂φ

∂n− φ

∂u

∂n

)

ds =

∫

∂Ωu∂φ

∂nds.

This yields

u(y) =

∫

Ωf(x)φ(y;x) dx−

∫

∂Ωg(s)

∂φ(y; s)

∂nds.


Green’s functions

Different representations are possible.

ExampleFor

−∆u = f, x ∈ Ω,

u = 0, x ∈ ∂Ω,

We can define the Green’s function as the solution of

−∆φ(y;x) = 0, x ∈ Ω,

φ(y;x) = δy(x), x ∈ ∂Ω.


Green’s functions

Evaluating the bilinear identity yields

∫

Ω(u∆φ− φ∆u) dx =

∫

∂Ω

(

u∂φ

∂n− φ

∂u

∂n

)

ds

= −

∫

∂Ωφ∂u

∂nds = −

∫

∂Ωδy∂u

∂nds = −

∂u

∂n(y).

This gives the value of the normal derivative of u at a point y onthe boundary,

∂u

∂n(y) = −

∫

Ωφ(y;x)f(x) dx.


Green’s functions


A PosterioriError Analysis and Adaptive Error Control


A generalization of the Green’s function

The Green’s function yields

u(y) = (u, δy) = (φ(y;x), f(x)).

The value of a function at a point is an example of a linearfunctional.

There are frequently other quantities of interest that can beexpressed as functionals.

The Riesz Representation theorem suggests these functionalshave the form (u, ψ) for a suitable distribution ψ.



Some useful choices of ψ include:

• We can construct ψ = δc to get the average value∮

c e(s) ds

of the error over a curve c in Rn, n = 2, 3, and ψ = δs to get

the average value of the error over a plane surface s in R3.



This choice requires a certain smoothness, which is given by

TheoremSobolev If s > k + n/2, then there is a constant C such that forf ∈ Hs(Rn),

max|α|≤k

supx∈Rn

|Dα(x)| ≤ C‖f‖s.

This implies that the derivatives of f of order k and less arecontinuous.

The Sobolev theorem shows that if k ≤ n and s > k/2,restricting a function in Hs to a submanifold of (co)dimension kis well-defined.



Some useful choices of ψ include:

• We can construct ψ = δc to get the average value∮

c e(s) ds

of the error over a curve c in Rn, n = 2, 3, and ψ = δs to get

the average value of the error over a plane surface s in R3.

• We can obtain errors in derivatives using dipoles in a similarway.

• ψ = χω/|ω| gives the error in the average value over asubset ω ⊂ Ω, where χω is the characteristic function of ω.

• Choosing ψ to be the residual of the finite elementapproximation sometimes yields the energy norm of theerror.

• ψ = χωe/‖e‖ω, where e is the error of the finite elementdiscretization, gives the L2(ω) norm of the error.

Only some of these data ψ have spatially local support.



We consider

Lu = f, x ∈ Ω,


where L is a linear differential operator.

The boundary and/or initial conditions should insure there is aunique solution.



DefinitionThe generalized Green’s function corresponding to thequantity of interest (u, ψ) satisfies

L∗φ(y, x) = ψ(x), x ∈ Ω,

adjoint b.c. and i.v., x ∈ ∂Ω,

where L∗ is the formal adjoint of L.


Discretization by the finite element method

We concentrate on

Lu = f, x ∈ Ω,

u = 0, x ∈ ∂Ω,

where

L(D,x)u = −∇ · a(x)∇u+ b(x) · ∇u+ c(x)u(x),

with u : Rn → R, a is a n× n matrix function of x, b is a n-vector

function of x, and c is a function of x.



We assume• Ω ⊂ R

n, n = 2, 3, is a smooth or polygonal domain

• a = (aij), where ai,j are continuous in Ω for 1 ≤ i, j ≤ n andthere is a a0 > 0 such that v>av ≥ a0 for all v ∈ R

n \ 0 andx ∈ Ω

• b = (bi) where bi is continuous in Ω

• c and f are continuous in Ω



The associated variational formulation reads

Find u ∈ H10 (Ω) such that

A(u, v) = (a∇u,∇v)+(b·∇u, v)+(cu, v) = (f, v) for all v ∈ H10 (Ω).



We construct a piecewise polygonal approximation of ∂Ω whosenodes lie on ∂Ω and which is contained inside Ω.

This forms the boundary of a convex polygonal domain Ωh.

Th denotes a simplex triangulation of Ωh that is locallyquasi-uniform.

hK denotes the length of the longest edge of K ∈ Th and h isthe mesh function with h(x) = hK for x ∈ K.

We also use h to denote maxK hK .



U = 0

Th

K

hK

Discretization of a domain Ω with curved boundaries.



Vh denotes the space of functions that are• continuous on Ω

• piecewise linear on Ωh with respect to Th• zero on the boundary ∂Ωh and Ω \ Ωh

Vh ⊂ H10 (Ω), and for smooth functions, the error of interpolation

into Vh is O(h2) in ‖ ‖

DefinitionThe finite element method is:

Compute U ∈ Vh such that A(U, v) = (f, v) for all v ∈ Vh.


An a posteriori analysis for an algebraic equation

Estimate the error e = x−X of a numerical solution X of

Ax = b

where b, x ∈ Rn and A is a n× n matrix.

The computable residual error of X is

R = AX − b

Goal: Find a relation between the residual and the error.

The residual can be small even if the error is large.



Since the residual error of the true solution is zero

Ae = −R.

To obtain an estimate on a quantity of interest, we introduce thegeneralized Green’s vector solving the adjoint problem

A>φ = ψ,

where ψ is any unit vector.

This gives an a posteriori estimate

|(e, ψ)| = |(e,A>φ)| = |(Ae, φ)| = |(R,φ)|.



A posteriori error bound

|(e, ψ)| ≤ ‖φ‖ ‖R‖.

Definition‖φ‖ is called the stability factor

∣

∣

∣

∣

(

e

‖x ‖, ψ

)∣

∣

∣

∣

≤ cond ψ(A)‖R‖

‖b‖,

where the “weak” condition number is

cond ψ(A) = ‖φ‖ ‖A‖ = ‖A−>ψ‖ ‖A‖


An a posteriori analysis for a finite element method

Goal: Estimate the error in a quantity of interest computed fromthe finite element solution U .

We use a generalized Green’s function φ solving the adjointproblem corresponding to a special choice of data ψ.

The error in the desired information may not have much to dowith the error in some global norm.

It may be computationally infeasible as well as veryinefficient to attempt to control the error in a globalnorm when all that is desired is accuracy in somequantities of interest.



We assume that the information can be represented as (u, ψ).

DefinitionThe generalized Green’s function φ solves the weak adjointproblem : Find φ ∈ H1

0 (Ω) such that

A∗(v, φ) = (∇v, a∇φ)−(v,div (bφ))+(v, cφ) = (v, ψ) for all v ∈ H10 (Ω),

corresponding to the adjoint problem L∗(D,x)φ = ψ.



(e, ψ) = (∇e, a∇φ) − (e,div (bφ)) + (e, cφ)

= (a∇e,∇φ) + (b · ∇e, φ) + (ce, φ)

= (a∇u,∇φ) + (b · ∇u, φ) + (cu, φ)

− (a∇U,∇φ) − (b · ∇U, φ) − (cU, φ)

= (f, φ) − (a∇U,∇φ) − (b · ∇U, φ) − (cU, φ).



Definitionπhφ denotes an approximation of φ in Vh

TheoremThe error in the quantity of interest computed from the finiteelement solution satisfies the error representation ,

(e, ψ) = (f, φ−πhφ)−(a∇U,∇(φ−πhφ))−(b·∇U, φ−πhφ)−(cU, φ−πhφ),

where φ is the generalized Green’s function corresponding todata ψ.



We use the error representation by approximating φ using arelatively high order finite element method.

Good results are obtained using the space V 2h

DefinitionThe approximate generalized Green’s function satisfies:Compute Φ ∈ V 2

h such that

A∗(v,Φ) = (∇v, a∇Φ)−(v,div (bΦ))+(v, cΦ) = (v, ψ) for all v ∈ V 2h .

The approximate error representation is

(e, ψ) ≈ (f,Φ − πhΦ) − (a∇U,∇(Φ − πhΦ))

− (b · ∇U,Φ − πhΦ) − (cU,Φ − πhΦ).



Examples are computed using FETkLab.• Runs under MATLAB• Solves general nonlinear elliptic systems on general

domains in two space dimensions• Implements the a posteriori error estimate for 16

simultaneous adjoint data ψi• Uses bisection or red-green quadrisection to refine

elements• Only those elements whose element indicators are larger

than the mean plus one standard deviation of all of theelement indicators in that level are refined.



ExampleAn elliptic problem on the unit square Ω = (0, 1) × (0, 1)

−∆u = 200 sin(10πx) sin(10πy), (x, y) ∈ Ω,

u(x, y) = 0, (x, y) ∈ ∂Ω

The solution is highly oscillatory

u(x, y) = sin(10πx) sin(10πy).



-1

1

0u

xy

Highly oscillatory solution.



We estimate the error in the average value by choosing ψ ≡ 1.

The generalized Green’s function is approximated on the samemesh using a piecewise quadratic finite element function.

We plot the ratios error/estimate for a wide variety of meshes.



102

103

104

Number of elements

0.0

0.4

0.8

1.2

err

or/

esti

mate

0.0 0.1 1.0 10.0 100.0

percent error

0

1

2

3

erro

r/es

tim

ate

Plots of the error/estimate ratio.


Adaptive error control

Goal of adaptive error control: Generate a mesh with a relativelysmall number of elements such that for a given tolerance TOLand data ψ,

|(e, ψ)| ≤ TOL.

This cannot be verified in practice.



Rewrite the error representation elementwise:

(e, ψ) ≈∑

K∈Th

∫

K

(

(f−b·∇U−cU)(Φ−πhΦ)−a∇U ·∇(Φ−πhΦ))

dx.

DefinitionThe mesh acceptance criterion is∣

∣

∣

∣

∣

∑

K∈Th

∫

K

(

(f − b · ∇U − cU)(Φ − πhΦ) − a∇U · ∇(Φ − πhΦ))

dx

∣

∣

∣

∣

∣

≤ TOL

If this is satisfied, then the numerical solution is deemedacceptable.



When the mesh criterion is not satisfied, we have to “enrich” thediscretization.

The cancellation among the contributions from each elementmake the enrichment decision problematic.

There is currently no theory or practical method foraccommodating cancellation of errors in an adaptiveerror control in a way that achieves true optimality ofefficiency.



A standard approach is to formulate the discretization selectionproblem as an optimization problem.

This requires an estimate consisting of a sum over elements ofpositive quantities.

Inserting norms gives

|(e, ψ)| ≤∑

K∈Th

∫

K

∣

∣(f−b·∇U−cU)(Φ−πhΦ)−a∇U ·∇(Φ−πhΦ)∣

∣ dx.

If the acceptance criterion fails, the mesh is refined to achieve

∑

K∈Th

∫

K

∣

∣(f−b·∇U−cU)(Φ−πhΦ)−a∇U ·∇(Φ−πhΦ)∣

∣ dx ≤ TOL.



The calculus of variations yields“Principle of Equidistribution” The element contributions ona nearly optimal mesh are roughly equal across the elements.

DefinitionTwo element acceptance criteria for the element indicatorsare

maxK

∣

∣(f − b · ∇U − cU)(Φ − πhΦ) − a∇U · ∇(Φ − πhΦ)∣

∣ .TOL|Ω|

,

or∫

K

∣

∣(f − b · ∇U − cU)(Φ− πhΦ)− a∇U · ∇(Φ− πhΦ)∣

∣ dx .TOLM

,

where M is the number of elements in Th.



Computing a mesh using these criteria is usually performed by a“compute-estimate-mark-refine” adaptive strategy.

We start with a coarse mesh then successively refine somefraction of the elements on which the element acceptancecriterion fails.



Example

−∆u =4800π

(

1 − 400((x− .5)2 + (y − .5)2))

e−400((x−.5)2+(y−.5)2),

(x, y) ∈ Ω,

u(x, y) = 0, (x, y) ∈ ∂Ω,

f is an approximation of a delta function at (.5, .5).

We control the error in the average value using a tolerance of.05%.



0

350 0.08

0

Solution Generalized Green’s Function

-30

-20

-10

0

Mesh Element Contributions

Results for the initial mesh.



0

140

0

0.08

Solution Generalized Green’s Function

-30

-20

-10

0

Mesh Element Contributions

Results after 10 iterations.



0 50 100 150 200

percent error

0.90

0.95

1.00

1.05

1.10

erro

r/es

tim

ate

The error/estimate ratio.


Further analysis on the a posteriori error estimate

We derive a more common form of the estimate for the simpleproblem with L(u) = −∆u.

The error representation formula is

(e, ψ) =

∫

Ωf(φ− πhφ) dx−

∫

Ω∇U · ∇(φ− πhφ) dx.

We break up the second integral on the right as∫

Ω∇U · ∇(φ− πhφ) dx =

∑

K∈Th

∫

K∇U · ∇(φ− πhφ) dx.



Using Green’s formula,

∫

K∇U · ∇(φ− πhφ) dx = −

∫

K∆U(φ− πhφ) dx

+

∫

∂K∇U · n∂K(φ− πhφ) ds,

where the last term is a line integral and n∂K denotes theoutward normal to ∂K.

Summing over all of the elements, the boundary integrals givetwo contributions from each element edge, computed inopposite directions.



If K1,K2 ∈ Th share a common edge σ1 ⊂ ∂K1 = σ2 ⊂ ∂K2,then

∫

σ1

∇U |K1· nσ1

(φ− πhφ) ds+

∫

σ2

∇U |K2· nσ2

(φ− πhφ) ds

=

∫

σ1

∇U |K1· nσ1

(φ− πhφ) ds

−

∫

σ1

∇U |K2· nσ1

(φ− πhφ) ds

= −

∫

σ1

[∇U ] · nσ1(φ− πhφ) ds,

where [U ] = ∇U |K2−∇U |K1

denotes the “jump” in ∇U acrossσ1 in the direction of the normal n∂K1

.



An alternate error representation,:

(e, ψ) = −∑

K∈Th

(∫

K(∆U + f)(φ− πhφ) dx

−1

2

∫

∂K[∇U ] · n∂K(φ− πhφ) ds

)

.



DefinitionThe residual and corresponding adjoint or dual weights are

RK =

(

‖∆U + f‖K‖h−1/2[∇U ]‖∂K/2

)

, WK =

(

‖φ− πhφ‖K‖h1/2(φ− πhφ)‖∂K

)

.

TheoremThe error of the finite element approximation is bounded by

|(e, ψ)| ≤∑

K∈Th

RK · WK .



We derive a priori bounds on the residual and dual weights.

There is a constant C independent of the mesh such that

RK ≤ C|K|1/2,

where |K| denote the area of K ∈ Th.

The bound on the first component follows from‖∆U + f‖K = ‖f‖K ≤ maxΩ |f | × |K|1/2.



For the second component, consider an integral over thecommon edge σ between two elements K1 and K2,

‖[∇U ]‖σ = ‖∇U |K2−∇U |K1

‖σ ≤ ‖∇U |K2−∇u|σ‖σ+‖∇u|σ−∇U |K1

‖σ.

By a trace inequality, the standard energy norm convergenceresult, and a standard elliptic regularity result,

‖∇U |Ki−∇u|σ‖σ ≤ ‖∇U −∇u‖

1/2Ki

‖∇U −∇u‖1/21,Ki

≤ C‖hu‖1/22,Ki

‖u‖1/22,Ki

≤ C‖h1/2f‖Ki,

for i = 1, 2. The local quasi-uniformity of the mesh implies

1

2‖h−1/2[∇U ]‖∂K ≤ Cmax

Ω|f | × |K|1/2.



Clearly, the convergence of the Galerkin approximation isstrongly influenced by the dual weights φ− πhφ, i.e. by theapproximation properties of Vh and the smoothness of φ.

This reflects the importance of the cancellation of errors inherentto the Galerkin method.


The Effective Domain of Influence and SolutionDecomposition


The Effective Domain of Influence

A characteristic property of elliptic partial differential equations isa global domain of influence.

A local perturbation of data near one point affects the solutionthroughout the domain of the problem.

Elliptic problems also have the property that the strength of theeffect of a localized perturbation on a solution decayssignificantly with the distance from the support of theperturbation, at least in some directions.

It turns out that this property also has profound consequencesfor the numerical solution of elliptic problems.



ExampleConsider the Green’s function for the Dirichlet problem for theLaplacian −∆u = f on the ball Ω of radius r centered at theorigin in R

3.

If the data f is perturbed by a smooth function δf , theperturbation in the value of the solution δu(y) is given by

δu(y) =

∫

Ωφ(y;x)δf(x) dx, y ∈ Ω.



If δf has compact support supp(δf) ⊂ Ω,

|y − x| ≤

∣

∣

∣

∣

r2y

|y|2− x

∣

∣

∣

∣

, x ∈ supp(δf), y ∈ Ω \ supp(δf).

We conclude that

|δu(y)| ≤max |δf | × volume of supp(δf) ×

(

1 + r|y|

)

4π × the distance from y to supp(δf).


Poisson’s equation in a disk

Ω denotes the disk of radius r centered at the origin in R2

Consider the Dirichlet problem for the Laplacian L = −∆u = f .

Suppose that ω is a small region contained in Ω located wellaway from ∂Ω.

We wish to estimate the error e = u− U in the energy norm‖e‖1,ω in ω.



Ω

ω



We evaluate the norm weakly

‖e‖1,ω = supψ∈H−1(ω)‖ψ‖−1,ω=1

(e, ψ).

The supremum is achieved for some ψ ∈ H−1(ω).

We extend ψ to H−1(Ω) by setting it to zero in Ω \ ω.

To discuss orders of convergence, we use the a posterioribound.



G(x; y) denotes the Green’s function for the Laplacian on Ω, so

φ(x) =

∫

ΩG(x; y)ψ(y) dy =

∫

ωG(x; y)ψ(y) dy.

Recall,

G(x; y) =1

2π×

ln

(

|x|∣

∣ r2x

|x|2−y∣

∣

r|x−y|

)

, x 6= 0,

ln(

r|y|

)

, x = 0.



Case 1: For y ∈ ω, G(x; y) is a smooth function of x for x ∈ Ω \ ωand so is φ.

We assume that δ > 0 satisfies• It is small enough that ωδ = x ∈ Ω : dist (x, ω) ≤ δ is

contained in Ω

• It is large enough that for K ⊂ Ω \ ωδ, the union N (K) of Kand the elements bordering K does not intersect ω



∂Ω

∂ω

∂ωδ

For K ⊂ Ω \ ωδ, we let πh be the Lagrange nodal interpolant withrespect to Th, so that

‖φ− πhφ‖K ≤ C∑

|α|=2

‖h2Dαφ‖K .



Case 2: φ is not so smooth in ω.

For K ∩ ωδ 6= ∅, πh is the Scott-Zhang interpolant, for which

‖φ− πhφ‖K ≤ C|hφ|1,N (K),

for a mesh-independent constant C.



The second component of WK is bounded similarly after using atrace theorem,

‖h1/2(φ− πhφ)‖∂K ≤ ‖φ− πhφ‖1/2N (K)‖h(φ− πhφ)‖

1/21,N (K),

and the local quasi-uniformity of the mesh.



We conclude,TheoremFor any δ > 0 small enough that ωδ ⊂ Ω but large enough thatN (K) ∩ ω = ∅ for K ⊂ Ω \ ωδ, there is a constant C such that

‖e‖1,ω ≤∑

K⊂Ω\ωδ

∑

|α|=2

C‖h2Dαφ‖K |K|1/2+∑

K∩ωδ 6=∅

C|hφ|1,N (K)|K|1/2.



There is a constant C such that

|DαxD

βyG(x, y)| ≤

C

|x− y|2, x 6= y ∈ Ω, |α| = 2, |β| ≤ 1.

We conclude there is a constant C independent of the meshsuch that for K ⊂ Ω \ ωδ,

‖φ− πhφ‖K ≤Ch2

K

dist (K,ω)2|K|1/2.



To handle the second sum on the right, we use the basic stabilityestimate,

‖φ‖1,Ω ≤ ‖ψ‖−1,Ω = ‖ψ‖−1,ω = 1.

If we assume a uniform (small) size hK = h for elements suchthat K ∩ ωδ 6= ∅, we obtain

∑

K∩ωδ 6=∅

ChK |φ|1,N (K) ≤ Ch‖φ‖1,Ω = Ch≤C

|ωδ|

∑

K∩ωδ 6=∅

h|K|.



We concludeTheoremFor any δ > 0 small enough that ωδ ⊂ Ω but large enough thatN (K) ∩ ω = ∅ for K ⊂ Ω \ ωδ, there is a constant C such that

‖e‖1,ω ≤∑

K⊂Ω\ωδ

Ch2K

dist (K,ω)2|K| +

∑

K∩ωδ 6=∅

Ch|K|.



We apply the “Principle of Equidistribution”

The element indicators are Ch2K/dist (K,ω)2 respectively Ch,

and in an optimal mesh,

h2K

dist (K,ω)2≈ h or hK ≈ h1/2 × dist (K,ω), K ⊂ Ω \ ωδ.



The decay of influence inherent to the Laplacian on the diskmeans that away from the region ω where we estimate the norm,we can choose elements asymptotically larger than the elementsize used in ωδ.

Moreover, the elements can increase in size as the distance toωδ increases

ωδ is an the effective domain of influence for the error in theenergy norm in ω.



DefinitionAn effective domain of influence corresponding to the data ψis the region ωψ in which the corresponding elements must besignificantly smaller in size than the elements used in thecomplement Ω \ ωψ in order to satisfy the adaptive goal.

Equivalently, if Th comprises uniformly sized elements, then theeffective domain of influence comprises those elements onwhich the element indicators, are substantially larger than thosein the complement.


A decomposition of the solution

Often, the goal of solving a differential equation is to computeseveral pieces of information.

The problem of computing multiple quantities of interest alsoarises naturally when the data ψ for the adjoint problem does nothave spatially localized support (averages and norms).

We cannot expect a significant localization effect from the decayof influence when the support of the data for the adjoint problemis not spatially localized.

If the data ψ has the property that the adjoint weight φ− πhφ hasa more-or-less uniform size throughout Ω, then the degree ofnon-uniformity in an adapted mesh depends largely on thespatial variation of the residual.



We use a partition of unity to “localize” a problem in whichsupp (ψ) does not have local support.

DefinitionSuppose that Ωi

Ni=1 is a finite open cover of Ω. A Lipschitz

partition of unity subordinate to Ωi is a collection of functionspi

Ni=1 with the properties

supp (pi) ⊂ Ωi, 1 ≤ i ≤ N,N∑

i=1

pi(x) = 1, x ∈ Ω,

pi is continuous on Ω and differentiable on Ωi, 1 ≤ i ≤ N,

‖pi‖L∞(Ω) ≤ C and ‖∇pi‖L∞(Ωi) ≤ C/diam (Ωi), 1 ≤ i ≤ N,

where C is a constant and diam (Ωi) is the diameter of Ωi.



Corresponding to a partition of unity pi, ψ ≡∑N

i=1 ψpi,

DefinitionThe quantities (U,ψpi) corresponding to the data ψi = ψpiare called the localized information corresponding to thepartition of unity.

We consider the problem of estimating the error in the localizedinformation for 1 ≤ i ≤ N .



We obtain a finite element solution via:

Compute Ui ∈ Vi such that A(Ui, v) = (f, v) for all v ∈ Vi,

where Vi is a space of continuous, piecewise linear functions ona locally quasi-uniform simplex triangulation Ti of Ω obtained bylocal refinement of an initial coarse triangulation T0 of Ω.

Vi is globally defined and the “localized” problem is solvedover the entire domain

We hope that this will require a locally refined mesh because thecorresponding data has localized support.



A partition of unity approximation in the sense of Babuška andMelenk uses Ui = χiUi, 1 ≤ i ≤ N , where χi is the characteristicfunction of Ωi.

The local approximation Ui is in the local finite element spaceVi = χiVi.

DefinitionThe partition of unity approximation is defined byUp =

∑Ni=1 Uipi, which is in the partition of unity finite element

space

Vp =

N∑

i=1

Vipi =

N∑

i=1

vipi : vi ∈ Vi

.



The partition of unity approximation recovers the fullconvergence properties of an approximation of the originalsolution.

Note

Up =

N∑

i=1

Uipi =

N∑

i=1

χiUipi ≡N∑

i=1

Uipi.

The values of Ui or Ui outside of Ωi are immaterial in forming theglobal partition of unity approximation.



We use the generalized Green’s function satisfying the adjointproblem:

Find φi ∈ H10 (Ω) such that A∗(v, φi) = (v, ψi) for all v ∈ H1

0 (Ω).

We expand the global error

(u− Up, ψ) =N∑

i=1

(

(u− Ui)pi, ψ)

.

We estimate each summand on the right as

(

(u− Ui)pi, ψ)

= (u− Ui, ψi) = A∗(u− Ui, φi)

= (f, φi) − (a∇Ui,∇φi) − (b · ∇Ui, φi) − (cUi, φi).



Letting πiφi denote an approximation of φi in Vi,

TheoremThe error of the partition of unity finite element solution Upsatisfies the error representation,

(u− Up, ψ) =N∑

i=1

(

(f, φi − πiφi) − (a∇Ui,∇(φi − πiφi))

− (b · ∇Ui, φi − πiφi) − (cUi, φi − πiφi))

.



DefinitionThe approximate generalized Green’s functions solve:Compute Φi ∈ V 2

i such that

A∗(v,Φi) = (v, ψi) for all v ∈ V 2i , 1 ≤ i ≤ N,

where V 2i is the space of continuous, piecewise quadratic

functions with respect to Ti.

The approximate error representations are

(u− Ui, ψi) ≈ (f,Φi − πiΦi) − (a∇Ui,∇(Φi − πiΦi))

− (b · ∇Ui,Φi − πiΦi) − (cUi,Φi − πiΦi).



If the localized error satisfies

∣

∣

(

u− Ui, ψi)∣

∣ ≤TOLN

, 1 ≤ i ≤ N,

then |(u− Up, ψ)| ≤ TOL.

This justifies treating the N “localized” problems independentlyin terms of mesh refinement.


Computation of multiple quantities of interest

We present an algorithm for computing multiple quantities ofinterest efficiently using knowledge of the effective domains ofinfluence of the corresponding Green’s functions.

We assume that the information is specified as (U,ψi)Ni=1 for a

set of N functions ψiNi=1.

These data might arise as particular goals or via localizationthrough a partition of unity.

We assume that the goal is to compute the informationassociated to ψi so that the error is smaller than a toleranceTOLi for 1 ≤ i ≤ N .



Two approaches:

Approach 1: A Global ComputationFind one triangulation such that the corresponding finiteelement solution satisfies |(e, ψi)| ≤ TOLi, for 1 ≤ i ≤ N .

Approach 2: A Decomposed ComputationFind N independent triangulations and finite elementsolutions Ui so that the errors satisfy |(ei, ψi)| ≤ TOLi, for1 ≤ i ≤ N .

The Global Computation can be implemented with astraightforward modification of the standard adaptive strategy inwhich the N corresponding mesh acceptance criteria arechecked on each element and if any of the N criteria fail, theelement is marked for refinement.



If the correlation, i.e., overlap, between the effective domains ofinfluence associated to the N data ψi is relatively small andthe effective domains of influence are relatively small subsets ofΩ, then each individual solution in the DecomposedComputation will require significantly fewer elements than thesolution in the Global Computation to achieve the desiredaccuracy.

This can yield significant computational advantage in terms oflowering the maximum memory requirement to solve theproblem.



Decreasing the maximum memory required to solve a problemcan be significant in at least two situations.

• If the individual solutions in the Decomposed Computationare computed in parallel and they require significantly fewerelements than the Global Computation, we can expect tosee significant speedup.

• In an environment with limited memory capabilities,decomposing a Global Computation requiring a largenumber of elements into a set of significantly smallercomputations can greatly increase the accuracy that can beachieved and/or decrease the time required.



If the effective domains of influence associated to the N dataψ have relatively large intersections, then the individualsolutions in the Decomposed Computation will require roughlythe same number of elements as the solution for the GlobalComputation.

In this case, there is little to be gained in using the DecomposedComputation.



In general, some of the N effective domains of influenceassociated to data ψi in the Decomposed Computation willcorrelate significantly and the rest will have low correlation.

We can optimize the use of resources by combiningcomputations for data whose associated domains of influencehave significant correlation and treating the rest independently.



Determining the Solution Decomposition

1. Discretize Ω by an initial coarse triangulation T0 andcompute an initial finite element solution U0.

2. Estimate the error in each quantity (U0, ψi) by solving the Napproximate adjoint problems and computing the estimate.

3. Using the element indicators associated to identify theeffective domains of influence for the data ψi in terms ofthe mesh T0 and significant correlations between theeffective domains of influence.

4. Decide on the number of approximate solutions to becomputed and the subset of information to be computedfrom each solution.

5. Compute the approximate solutions independently usingadaptive error control aimed at computing the specifiedquantity or quantities of interest accurately.


Identifying significant correlations

The key issue in the proposed algorithm is identifying theeffective domains of influence and recognizing significantcorrelation, or overlap, between different effective domains ofinfluence.

Mesh refinement decisions are based on the sizes of theelement indicators

Ei|K = maxK

∣

∣(f − b · ∇Ui− cUi)(Φi− πiΦi)− a∇Ui · ∇(Φi− πiΦi)∣

∣

or

Ei|K =

∫

K

∣

∣(f−b ·∇Ui−cUi)(Φi−πiΦi)−a∇Ui ·∇(Φi−πiΦi)∣

∣ dx,



DefinitionWe let Ei(x) denote the piecewise constant element errorindicator function associated to data ψi with Ei(x) ≡ Ei|K forK ∈ T0.

Identifying the effective domain of influence means finding a setof elements on which the element error indicators aresignificantly larger than on the complement.

Identifying significant correlation between the effective domainsof influence of two data entails showing that the effectivedomains of influence have a significant number of elements incommon.



Pattern matching uses

DefinitionThe (cross-)correlation of two functions f ∈ Lp(Ω) andg ∈ Lq(Ω), defined as:

(f g)(y) =

∫

Ωf(x)g(y + x) dx,

which is an L1(Ω) function.

The correlation indicator c(f, g) is a functional of thecorrelation function (f g).

We treat the element error indicator functions Ei as signalfunctions.



Discounting translation or rotation, correlation of Ei and Ejreduces to the L2-inner-product:

(Ei Ej)(0) =

∫

ΩEi(x)Ej(x) dx = (Ei, Ej)Ω.

DefinitionThe correlation indicator c(Ei, Ej) is

c(Ei, Ej) = |(Ei Ej)(0)| = (Ei, Ej)Ω.



DefinitionWe say that the effective domain of influence associated to ψi issignificantly correlated to the domain of influence associatedto ψj if two conditions hold.




Condition 1The correlation of Ei and Ej is larger than a fixed fraction of thenorm of Ej , or

Correlation Ratio 1 =c(Ei, Ej)

‖Ej‖2≥ γ1,

for some fixed 0 ≤ γ1 ≤ 1.

This means that the projection of Ei onto Ej is sufficiently large.



Element

Ele

men

t In

dic

ator

Element

Ele

men

t In

dic

ator

Element

Ele

men

t In

dic

ator

Ei

Ej Ei

Ej Ei

Ej

Three examples of significant correlation of Ei with Ej . Plottedare the element indicator functions Ei(x), Ej(x) versus the

element number.




Condition 2The component of Ej orthogonal to Ei is smaller than a fixedfraction of the norm of Ej , or

Correlation Ratio 2 =

∥

∥

∥Ej −c(Ej ,Ei)‖Ei‖2 Ei

∥

∥

∥

‖Ej‖≤ γ2,

for some fixed 0 ≤ γ2 ≤ 1.

This corrects for the difficulties that arise when Ei is much largerthan Ej .



Element

Ele

men

t In

dic

ato

r

Ej

Ei

An example in which the second condition fails.



1 9 18 27 36

Element Number

0

1

2

3E1

E2

E3

E4

E5

E6

E7

E8

E9

Plots of nine element indicator functions Ei versus the elementnumber.



With γ1 = .9 and γ2 = .7:

E1 with E8 E4 with none E7 with noneE2 with E6, E7 E5 with E2, E6 E8 with noneE3 with E1, E8 E6 with none E9 with none

We emphasize that the initial identification of significantcorrelation between effective domains of influence ofvarious Green’s functions in a computation is carriedout on a coarse initial partition of the domain and henceis relatively inexpensive.


Examples

We solve elliptic problems using adaptive mesh refinement toachieve accuracy in a set of quantities of interest using a GlobalComputation and a Decomposed Computation.

The results suggest that the individual solutions in theDecomposed Computation require significantly fewer elementsto achieve the desired accuracy than the Global Computation ina variety of situations.


Examples

As a relatively universal measure of the gain from using theDecomposed Computation, we use

DefinitionThe Final Element Ratio is the number of elements in the finalmesh refinement level required to achieve accuracy in thespecified quantities of interest in the Global Computation to themaximum number of elements in the final mesh refinementlevels for the individual computations in the DecomposedComputation.

Generally, we expect the gain in efficiency to scalesuper-linearly with the Final Element Ratio.

We compute the Final Element Ratio using solutions that arehave roughly the same accuracy.


Examples

Example 1

We solve

− 110π2 ∆u(x) = sin(πx) sin(πy), (x, y) ∈ Ω,

u(x, y) = 0, (x, y) ∈ ∂Ω,

on the Ω = [0, 8] × [0, 8].

The solution is u(x, y) = 5 sin(πx) sin(πy).

Goal: Control the error in the average value of u by choosingψ ≡ 1/|Ω| = 1/64.


Examples

Global Computation: Mesh is adapted the error in the averagevalue of u is smaller than the error tolerance of 5%.

Initial mesh is 10 × 10 elements and after five refinement levels,we end up with 3505 elements, achieving an error of .022.


Examples

Initial Mesh Final Mesh

Initial and final meshes.


Examples

Numerical solutions.


Examples

Estimates, errors, and error/estimate ratios:

Level Elements Estimate Error Ratio1 100 .1567 .1534 .9786

2 211 .1157 .1224 1.058

3 585 .3063 .3078 1.005

4 1309 .1159 .1166 1.006

5 3505 .02163 .02148 .9975


Examples

2 4

31

1

2 3

4

5

6 7

8

13

14 15

16

9

10 11

12

Partitions for the decomposition.


Examples

The partition of unity yields four data ψ1, ψ2, ψ3, ψ4.

We compute the four localized approximations U1, · · · , U4using the same initial mesh as before.

The Correlation Ratios indicates that all four localized solutionsshould be computed independently.


Examples

We obtain acceptable results using the tolerance of 5%.

Data Level Elements Estimateψ1 3 618 .01242

ψ2 3 575 −.0009109

ψ3 3 618 .01242

ψ4 3 575 −.0009109

This yields a partition of unity solution Up with accuracy .023.

Using the Decomposed Computation yields a Final ElementRatio of ≈ 5.7.


Examples

Final Mesh for U1^

Final Mesh for U2^

Two of the final meshes.


Examples

The generalized Green’s functions for the global average errorand the localized data ψ2.


Examples

A Decomposed Computation using a partition of unity on 16equal-sized regions yields significant correlations:

E2 with E3 E5 with E8 E10 with E9 E13 with E14

E4 with E3 E7 with E8 E12 with E9 E15 with E14

This computation yields a Final Element Ratio of ≈ 2.6


Examples

Example 2

We solve

−∇ ·(

(1.1 + sin(πx) sin(πy))∇u(x, y))

= −3 cos2(πx) + 4 cos2(πx) cos2(πx)

+2.2 sin(πx) sin(πy) + 2 − 3 cos2(πy), (x, y) ∈ Ω,

u(x, y) = 0, (x, y) ∈ ∂Ω,

where Ω = [0, 2] × [0, 2].

The exact solution is u(x, y) = sin(πx) sin(πy).


Examples

We compute the average error using ψ1 ≡ 1/4 and four pointvalues using ψ2 ≈ δ(.5,.5), ψ3 ≈ δ(.5,1.5), ψ4 ≈ δ(1.5,1.5), andψ5 ≈ δ(1.5,.5).

We use

δ(cx,cy) =400

πe−400((x−cx)2+(y−cy)2)

to approximate the delta function δ(cx,cy).


Examples

Global Computation: We use a tolerance of 2% starting with an8 × 8 mesh.

ψ1 ψ2

Lev. Elt’s1 64

2 201

3 763

4 2917

Est. Err. Rat..035 .035 1.0

.0088 .0089 1.0

.0027 .0027 1.0

.00044 .00044 1.0

Est. Err. Rat..090 .29 3.3

.042 .082 1.9

.020 .020 .99

.0050 .00504 1.0


Examples

Global Computation: We use a tolerance of 2% starting with an8 × 8 mesh.

ψ3

Lev. Elt’s1 64

2 201

3 763

4 2917

Est. Err. Rat..24 .022 .091

.0024 .014 6.0

.0020 .0020 1.0

.0049 .00504 1.0


Examples


Initial and final meshes. Final mesh has 2917 elements.


Examples

Decomposed Computation: Use data ψ1, · · · , ψ5independently using tolerances of 2%.

We vary the initial meshes; using 7 × 7 for U1; 9 × 9 for U2 andU4; and 12 × 12 for U3 and U5.

Data Level Elements Estimateψ1 3 409 −.0004699

ψ2 4 1037 −.007870

ψ3 2 281 −.005571

ψ4 4 1037 −.007870

ψ5 2 281 −.005571

The Final Element Ratio is ≈ 2.8.


Examples

Final Mesh for U1^

Final Mesh for U2^

Final Mesh for U3^

Final meshes for U1, U2, U3.


Examples

Example 3

We solve

−∆u = 16(y − y2 + x− x2) (x, y) ∈ Ω,

u(x, y) = 0, (x, y) ∈ ∂Ω,

where Ω = [0, 1] × [0, 1].

The exact solution is u(x, y) = 8x(1 − x)y(1 − y).


Examples

We estimate the error in the average value of u.

Since the domain is small and the solution and the generalizedGreen’s function are very smooth, the gain from decomposingthe solution is greatly reduced.

Beginning with a 4 × 4 mesh and using a tolerance of 1%, weobtain a sufficiently accurate solution using a GlobalComputation after five refinements.

The final mesh uses 885 elements and produces an error of.0008699.


Examples

If we partition the domain using four equal regions there are nosubstantial correlations.

Computing the four solutions independently in the DecomposedComputation yields a Final Element Ratio of around 1.5.


Examples

If we use sixteen equal regions, there are a number ofsubstantial correlations.

Correlation Ratio 1 for E1 on E2 = .98,Correlation Ratio 2 for E1 on E2 = .44,Correlation Ratio 1 for E2 on E1 = .82,Correlation Ratio 2 for E2 on E1 = .44.


Examples

Computing U1 corresponding to the localized data ψ1 using atolerance of 1% requires 367 elements.

Computing U2 requires a mesh 494 elements.

Combining these two computations by using data equal to thesum of the two partition functions for the regions Ω1 and Ω2

requires 496 elements.

We gain almost nothing by computing U1 and U2 independently


Examples

Final Mesh for U1^

Final Mesh for U2^

Final Mesh for "U1+U2"^ ^

Final meshes.


Examples

We consider the error in the average value and the point valuesat (.25, .25) and (.5, .5).

We use a partition of unity decomposition for the error in theaverage to get data ψ1, · · · , ψ4. We let ψ5 ≈ δ(.25,.25) andψ6 ≈ δ(.5,.5).

We compare the correlation indicators on initial meshes rangingfrom 16 to 144 or 400 uniformly sized elements.


Examples

0 50 100 150

Number of Elements

0.0

0.5

1.0

Corr. Rat. 1: E1 on E2




0 50 100 150

Number of Elements

0.0

0.5

1.0





Correlation Ratios versus number of elements.


Examples

0 100 200 300 400

Number of Elements

0

5

10





0 100 200 300 400

Number of Elements

0

2

4







Examples

0 50 100 150

Number of Elements

0.0

0.6

1.2





0 100 200 300 400

Number of Elements

0

1

2

3Corr. Rat. 1: E3 on E5






Examples

In general, we find that all Correlation Ratios converge to a limitas the number of elements increases.

Generally, the second Correlation Ratio varies relatively little asthe mesh density increases for all data.

The first Correlation Ratio between data representing a partitionof unity decomposition also varies relatively little.

The first Correlation Ratio varies quite a bit on coarse mesheswhen one of the data is an approximate delta function.


Examples

The determination that two effective domains of influence arenot closely correlated is relatively robust with respect to thedensity of the mesh.

There is a mild tendency to combine computations that are moreefficiently treated independently if the correlation indicators arecomputed on very coarse meshes.


Examples

Example 4We solve

−∇ ·((

.05 + tanh(

10(x− 5)2 + 10(y − 1)2))

∇u)

+

(

−100

0

)

· ∇u = 1, (x, y) ∈ Ω,

u(x, y) = 0, (x, y) ∈ ∂Ω,

where Ω = [0, 10] × [0, 2].


Examples

2

1

0

10

x

y

Dif

fusi

on

Plot of the diffusion coefficient.


Examples

We use an initial mesh of 80 elements and an error tolerance ofTOL = .04%.

Level Elements Estimate1 80 −.0005919

2 193 −.001595

3 394 −.0009039

4 828 −.0003820

5 1809 −.0001070

6 3849 −.00004073

7 9380 −.00001715

8 23989 −.000007553


Examples

Final Mesh.


Examples

Pe=1000

Pe=.1

Plots of the meshes for the original problem with Pe = 1000 anda problem with Pe = .1.


Examples

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

Domains for the partition of unity.


Examples

Significant Correlations:

E3 with E4 E6 with E7 E7 with E6 E9 with E8 E10 with E8, E9

E13 with E14 E16 with E17 E17 with E16 E19 with E18 E20 with E18, E19

Note, there are no significant correlations in the “cross-wind”direction.


Examples

Solutions are symmetric across y = 1.

Data TOL Level Elements Estimateψ1 .04% 7 7334 −6.927 × 10−7

ψ2 .04% 7 8409 −5.986 × 10−7

ψ3 .04% 7 7839 −5.189 × 10−7

ψ4 .04% 7 7177 −5.306 × 10−7

ψ5 .04% 7 7301 −4.008 × 10−7

ψ6 .02% 7 6613 −2.471 × 10−7

ψ7 .02% 7 4396 −2.938 × 10−7

ψ8 .02% 7 4248 −1.656 × 10−7

ψ9 .02% 7 3506 −1.221 × 10−7

ψ10 .02% 7 1963 −5.550 × 10−8

The Final Element Ratio is ≈ 2.9.Duality, Adjoints, Green’s Functions – p. 255/304

Examples

Final Mesh for U1^

^

Final Mesh for U5^

Plots of the final meshes for the localized solutions U1 and U5.Duality, Adjoints, Green’s Functions – p. 256/304

Examples

Final Mesh for U9^

Plots of the final mesh for the localized solution U9.


Examples

2

1

0 02

46

8100

2

4

6

x 10 -4

y

x2

1

0 02

46

8100

2

4

6

x 10 -4

y

x

Plots of the generalized Green’s functions corresponding to ψ11

(left) and ψ19 (right).


Examples

The effective domains of influence may not be spatiallycompactly-shaped.


Examples

We combine some of the localized computations by solving forlocalized solutions corresponding to summing the two of thepartition of unity data.

Data TOL Level Elements Estimateψ3 + ψ4 .04% 7 8330 −9.8884 × 10−7

ψ6 + ψ7 .02% 7 5951 −5.897 × 10−7

ψ8 + ψ9 .02% 7 4406 −3.486 × 10−7

ψ9 + ψ10 .02% 7 3202 −2.243 × 10−7

The solutions for ψ3 + ψ4 and ψ8 + ψ9 use a few more elementsthan required for either of the original localized solutions.

The solutions for ψ6 + ψ7 and ψ9 + ψ10 use less than themaximum required for the individual localized solutions.


Examples

Example 5

We solve

− 1π2 ∆u = 2 + 4e−5((x−.5)2+(y−2.5)2), (x, y) ∈ Ω,

u(x, y) = 0, (x, y) ∈ ∂Ω,

where Ω is the “square annulus” Ω = [0, 3] × [0, 3] \ [1, 2] × [1, 2].

The initial mesh has 48 elements and the error tolerance is 1%.


Examples

Level Elements Estimate1 48 −5.168

2 125 −1.584

3 380 −.6879

4 894 −.3029

5 2075 −.1435


Examples


Initial and Final Meshes.


Examples

12

3 0

1

2

3

0

1

2

3

4

5

6

y

x

12

3 0

1

2

3

0

0.4

0.8

1.2

1.6

y

x

Solution and Generalized Green’s Function on the Final Mesh.


Examples

1 2 3

4

567

8

The partition of unity has 8 square subdomains.


Examples

There are no significant correlations.


ψ2 5 1101 −.01399

ψ3 5 1144 −.01540

ψ4 5 1107 −.01360


Examples

There are no significant correlations.


ψ6 5 1110 −.01587

ψ7 5 1074 −.02529

ψ8 5 1098 −.01660

Final Element Ratio ≈ 1.8.


Examples

Final Mesh for U3^

Final Mesh for U4^

Final meshes.


Examples

Final Mesh for U6^

Final Mesh for U7^

Final meshes.


Examples

0

12

3 0

1

2

3

1

y

x

12

3 0

1

2

3

0

0.2

0.6

1

1.4

y

x

Plots of some Generalized Green’s Functions.


Nonlinear Problems


A nonlinear algebraic equation

The problem is to estimate the error e = x−X of a numericalsolution X of a system of nonlinear algebraic equations,

f(x) = b

where the data b, nonlinearity f , and the solution x all have thesame dimension.

The residual error isR = f(X) − b

which immediately gives

f(x) − f(X) = −R.

The error is related to the residual through a nonlinear equation.



The mean value theorem for integrals:

f(x) − f(X) =

∫ 1

0f ′(sx+ (1 − s)X) (x−X) ds

where f ′ is the Jacobian matrix of f .

We get

Ae = −R

with

A =

∫ 1

0f ′(sx+ (1 − s)X) ds.

This is the linear problem obtained by linearizing f around anaverage of x and −X.



We obtain

|(e, ψ)| = |(e, A>φ)| = |(Ae, φ)| = |(R,φ)|.

To obtain a computable adjoint problem, we can try to replace xby X in the definition of A,

A→ A =

∫ 1

0f ′(sX + (1 − s)X) ds = f ′(X),

But, what is the effect on the estimate?


Defining the adjoint to a nonlinear operator

We assume that the Banach spaces X and Y are Sobolevspaces and use ( , ) for the L2 inner product, and so forth.

We define the adjoint for a specific kind of nonlinear operator.

We assume f is a nonlinear map from X into Y , where thedomain D(f) is a convex set.

DefinitionA subset A of a vector space is convex if for any a, b ∈ A, theset of points on the “line segment” joining a and b, i.e.,sa+ (1 − s)b| 0 ≤ s ≤ 1 is contained in A.



We choose u inside D(f) and define

F (e) = f(u+ e) − f(u),

where we think of e as representing an “error”, i.e., e = U − u.

The domain of F is

D(F ) = v ∈ X| v + u ∈ D(f).

We assume that D(F ) is independent of e and dense in X.

Note that 0 ∈ D(F ) and F (0) = 0.

We also assume that D(F ) is a vector subspace of X.



Two reasons to work with functions of this form:• This is the kind of nonlinearity that arises when estimating

the error of a numerical solution or studying the effects ofperturbations.

• Nonlinear problems typically do not enjoy the globalsolvability that characterizes linear problems, only a localsolvability.



The first definition is based on the bilinear identity.

DefinitionAn operator A∗(e) with domain D(A∗) ⊂ Y ∗ and range in X∗ isan adjoint operator corresponding to F if

(F (e), w) = (e,A∗(e)w) for all e ∈ D(F ), w ∈ D(A∗).

This is an adjoint operator associated with F , not the adjointoperator to F .



ExampleSuppose that F can be represented as F (e) = A(e)e, whereA(e) is a linear operator with D(F ) ⊂ D(A).

For a fixed e ∈ D(F ), define the adjoint of A satisfying

(A(e)w, v) = (w,A∗(e)v) for all w ∈ D(A), v ∈ D(A∗).

Substituting w = e shows this defines an adjoint of F as well.

If there are several such linear operators A, then there will beseveral different possible adjoints.



ExampleLet (t, x) ∈ Ω = (0, 1) × (0, 1), with X = X∗ = Y = Y ∗ = L2

denoting the space of periodic functions in t and x, with periodequal to 1.

Consider a periodic problem

F (e) =∂e

∂t+ e

∂e

∂x+ ae = f

where a > 0 is a constant and the domain of F is the set ofcontinuously differentiable functions.



We can write F (e) = Ai(e)e where

A1(e)v =∂v

∂t+ e

∂v

∂x+ av

A2(e)v =∂v

∂t+

(

a+∂e

∂x

)

v

A3(e)v =∂v

∂t+

1

2

∂(ev)

∂x+ av.



The adjoints are

A∗1(e)w = −

∂w

∂t−∂(ew)

∂x+ aw

A∗2(e)w = −

∂w

∂t+

(

a+∂e

∂x

)

w

A∗3(e)w = −

∂w

∂t−e

2

∂w

∂x+ aw.



We base the second definition of an adjoint on the integral meanvalue theorem.

We assume that the original nonlinearity is Frechetdifferentiable.

The integral mean value theorem states

f(U) = f(u) +

∫ 1

0f ′(u+ se) ds e

where e = U − u and f ′ is the Frechet derivative of f .



We rewrite this as

F (e) = f(U) − f(u) = A(e)e

with

A(e) =

∫ 1

0f ′(u+ se) ds.

Note that we can apply the integral mean value theorem to F :

A(e) =

∫ 1

0F ′(se) ds.

To be precise, we should discuss the smoothness of F .



DefinitionFor a fixed e, the adjoint operator A∗(e), defined in the usual wayfor the linear operator A(e), is said to be an adjoint for F .

ExampleContinuing the previous example,

F ′(e)v =∂v

∂t+ e

∂v

∂x+

(

a+∂e

∂x

)

v.

After some technical analysis of the domains of the operatorsinvolved,

A∗(e)w = −∂w

∂t−e

2

∂w

∂x+ aw.

This coincides with the third adjoint computed above.


Analysis of a space-time finite element method

We study a system of D reaction-diffusion equations consistingof d parabolic equations and D − d ordinary equations for theRD valued function u = (ui):

ui −∇ · (εi(u, x, t)∇ui) = fi(u, x, t), (x, t) ∈ Ω × R+, 1 ≤ i ≤ D,

ui(x, t) = 0, (x, t) ∈ ∂Ω × R+, 1 ≤ i ≤ d,

u(x, 0) = u0(x), x ∈ Ω,

where there is a constant ε > 0 such that

εi(u, x, t) ≥ ε for 1 ≤ i ≤ d and εi(u, x, t) ≡ 0 for the rest.



Assumptions:

• Ω is a convex polygonal domain in R2 with boundary ∂Ω

• ε = (εi) and f = (fi) have smooth second derivatives

ui denotes the time derivative of ui

We write εi(u, x, t) = εi(u) and f(u, x, t) = f(u).



We use up and uo to denote the parts of u associated to theparabolic and ordinary differential equations respectively.

upi = ui for 1 ≤ i ≤ d and upi = 0 for d < i ≤ D

uo = u− up

The presence of ordinary differential equations in the systemhas strong consequences for the smoothness of solutions.



Discretization of the space-time domain:

We partition [0,∞) as 0 = t0 < t1 < t2 < · · · < tn < . . . , denotingeach time interval by In = (tn−1, tn] and time step bykn = tn − tn−1.

To each interval In, we associate a triangulation Tn of Ωarranged so the union of the elements in Tn is Ω.



Space

Tim

e

Ω

A space-time discretization.



As usual, we assume the intersection of any two elements in aspace triangulation is either a common edge, node, or is empty.

We also assume that the smallest angle of any triangle in atriangulation is bounded below by a fixed constant

hn denotes the piecewise constant mesh functionhn|K = diam(K) for K ∈ Tn and h denotes the global meshfunction, where h|In

= hn.

k denotes the piecewise constant function that is kn on In.



The approximations are polynomials in time and piecewisepolynomials in space on each space-time “slab” Sn = Ω × In.

Vn ⊂ (H10 (Ω))d × (H1(Ω))D−d denotes the space of piecewise

linear continuous vector-valued functions v(x) ∈ RD defined on

Tn, where the first d components of v are zero on ∂Ω.

We define

W qn =

w(x, t) : w(x, t) =

q∑

j=0

tjvj(x), vj ∈ Vn, (x, t) ∈ Sn

.

W q denotes the space of functions defined on the space-timedomain Ω × R

+ such that v|Sn∈W q

n for n ≥ 1.



Functions in W q are generally discontinuous across the discretetime levels.

[w]n = w+n − w−

n , where w±n = lims→tn± w(s), denotes the jump

across tn

Pn is the L2 projection onto Vn, i.e., Pn : L2(Ω) → Vn is definedby (Pnv, w) = (v, w) for all w ∈ Vn. P is defined by P = Pn onSn.

πn : L2(In) → Pq(In) denotes the L2 projection onto thepiecewise polynomial functions in time, where Pq(In) is thespace of polynomials of degree q or less on In. π is defined byπ = πn on Sn.



DefinitionThe continuous Galerkin cG(q) approximation U ∈W q

satisfies U−0 = P0u0 and for n ≥ 1, the Galerkin orthogonality

relation

∫ tn

tn−1

(

(Ui, vi) + (εi(U)∇Ui,∇vi))

dt =

∫ tn

tn−1

(fi(U), vi) dt

for all v ∈W q−1n , 1 ≤ i ≤ D,

U+n−1 = PnU

−n−1.

U is continuous across time nodes over which there is no meshchange.



DefinitionThe discontinuous Galerkin dG(q) approximation U ∈W q

satisfies U−0 = P0u0 and for n ≥ 1,

∫ tn

tn−1

(

(Ui, vi) + (εi(U)∇Ui,∇vi))

dt+(

[Ui]n−1, v+i

)

=

∫ tn

tn−1

(fi(U), vi) dt

for all v ∈W qn , 1 ≤ i ≤ D.



ExampleWe discretize the scalar problem

u− ∆u = f(u), (x, t) ∈ Ω × R+,

u(x, t) = 0, (x, t) ∈ ∂Ω × R+,

u(x, 0) = u0(x), x ∈ Ω,

using the dG(0) method.

We let ~Un denote the Mn vector of nodal values with respect tothe nodal basis ηn,i

Mn

i=1 for Vn on In.



Bn :(

Bn)

ij=(

ηn,i, ηn,j)

for 1 ≤ i, j ≤Mn and

Bn,n−1 :(

Bn,n−1

)

ij=(

ηn,i, ηn−1,j

)

for 1 ≤ i ≤Mn,

1 ≤ j ≤Mn−1 denotes the mass matrices

An :(

An)

ij=(

∇ηn,i,∇ηn,j)

denotes the stiffness matrices.

Un satisfies

(

Bn + knAn)

~Un − ~F (U−n )kn = Bn,n−1

~Un−1, n ≥ 1,

where (~F (U−n ))i = (f(U−

n ), ηn,i).



Use of quadrature yields standard difference schemes

ExampleIf Mn is constant and the lumped mass quadrature is used toevaluate the coefficients of Bn and Bn,n−1 = Bn, then theresulting set of equations for the dG(0) approximation is thesame as the equations for the nodal values of the backwardEuler difference scheme.

The dG(0) method is related to the backward Euler method, thecG(1) method is related to the Crank-Nicolson scheme, and thedG(1) method is related to the third order sub-diagonal Padédifference scheme.



Under general assumptions, the cG(q) and dG(q) have order ofaccuracy q + 1 in time and 2 in space at any point.

In addition, they enjoy a superconvergence property in time attime nodes. The dG(q) method will have order of accuracy2q + 1 in time and the cG(q) method will have order 2q in time attime nodes for sufficiently smooth solutions.

The discontinuous Galerkin method has stability properties thatmake it well suited for the solution of dissipative problems.

The continuous Galerkin method is “energy” preserving and iswell-suited for problems with a conserved quantity.



We use the second definition of the adjoint, so use

εi = εi(u, U) =

∫ 1

0εi(

us+ U(1 − s))

ds,

βij = βij(u, U) =

∫ 1

0

∂εj∂ui

(

us+ U(1 − s))∇(uis+ Ui(1 − s))

ds,

fij = fij(u, U) =

∫ 1

0

∂fj∂ui

(us+ U(1 − s))

ds.

Typically, ε and f are piecewise continuous with respect to t andcontinuous, H1 functions in space while β is discontinuous intime and space.



The adjoint problem is

−φi −∇ ·(

εi∇φi)

+∑D

j=1 βji · ∇φj −∑D

j=1 fijφj = ψi,

(x, t) ∈ Ω × (tn, 0], 1 ≤ i ≤ D,

φi (x, t) = 0, (x, t) ∈ ∂Ω × (tn, 0], 1 ≤ i ≤ d,

φ(x, tn) = 0, x ∈ Ω,

ExampleIn the case of the scalar problem with constant diffusion, theadjoint problem is

−φ− ε∆φ− fφ = ψ, (x, t) ∈ Ω × (tn, 0],

φ(x, t) = 0, (x, t) ∈ ∂Ω × (tn, 0],

φ(x, tn) = 0, x ∈ Ω.



ExampleIn the case of one parabolic equation with nonlinear diffusioncoupled to one ordinary differential equation, the dual problem is

−φ1 −∇ · ε1∇φ1 + β11∇φ1 − f11φ1 − f12φ2 = ψ1,

−φ2 + β12∇φ1 − f21φ1 − f22φ2 = ψ2,

φ1(x, t) = 0,

φ(x, tn) = 0.



For the cG method, we obtain

∫ tn

0(e, ψ) dt = (e+(0), φ(0))

+

∫ tn

0

(

(U , πPφ−φ)+(ε(U)∇U,∇(πPφ−φ))−(f(U), πPφ−φ))

dt.

For the dG method, we obtain

∫ tn

0(e, ψ) dt = (e−(0), φ(0)) +

n∑

j=1

(

[U ]j−1, (πPφ− φ)+j−1

)

+

∫ tn

0

(

(U , πPφ−φ)+(ε(U)∇U,∇(πPφ−φ))−(f(U), πPφ−φ))

dt.


Collaborators

Sean Eastman, Colorado State UniversityMike Holst, UCSDClaes Johnson, ChalmersMats Larson, Umea UniversityDuane Mikulencak, Georgia TechDavid Neckels, Colorado State UniversityTim Wildey, Colorado State UniversityRoy Williams, Caltech


A Short Course on Duality, Adjoint Operators, Green’s Functions,...

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