Adaptive Methods for Elliptic PDE withRandom Operators

Claude Jeffrey Gittelson

Purdue University

SIAM Conference on Uncertainty QuantificationRaleigh, North Carolina

April 4, 2012

Outline

1 Random Elliptic PDEA Model ProblemLegendre Chaos ExpansionStructure of the Operator

2 Adaptive Stochastic GalerkinGalerkin ProjectionAn Idealized Adaptive Algorithm

3 The Adaptive Wavelet ApproachComputation of the ResidualAn Adaptive AlgorithmSpatial Discretization

4 The Adaptive Finite Element ApproachA Residual-based Error Estimator

A Model ProblemElliptic boundary value problem on a domain D ⊂ Rd,

−∇ · (a∇u) = f in D,u = 0 on ∂D.

Karhunen–Loeve expansion of the random field a,

a(y, x) B a(x) +∞∑

m=1

ymam(x) , y = (ym)∞m=1∈ [−1, 1]∞ .

Assume the random variables ym are independent anduniformly distributed on [−1, 1], and

∞∑m=1

∥∥∥∥∥am

a

∥∥∥∥∥L∞(D)

< 1 , a > 0 , a, 1/a ∈ L∞(D) .

Legendre Chaos ExpansionExpand the solution u(y, x) as

u(y, x) =∑µ∈F

uµ(x)Pµ(y)

for the tensorized Legendre polynomials

Pµ(y) B∞∏

m=1

Pµm(ym) =∏

m∈supp µ

Pµm(ym) ,

µ ∈ F B {µ ∈ N∞0

; # supp µ < ∞} .

The coefficients u B (uµ)µ∈F in H10(D) satisfy an equation

Au = f ,

where A represents −∇ · (a∇ ) in the basis (Pµ)µ∈F .

Structure of the OperatorDeterministic differential operators from coefficientsa(y) = a +

∑m ymam,

Av B −∇ · (a∇v) , Amv B −∇ · (am∇v) .

Action of A on u0 = u0P0(y),

A(u0P0(y)) = (Au0)P0(y) +∞∑

m=1

1√

3(Amu0)Pεm(y) .

Similar for the other terms in u(y) =∑µ∈F uµPµ(y).

Au is an infinite sequence even if u is finitely supported.

Truncated operators A[M]: truncate series after M terms.support increases at most by a factor 2M + 1.

Structure of the OperatorDeterministic differential operators from coefficientsa(y) = a +

∑m ymam,

Av B −∇ · (a∇v) , Amv B −∇ · (am∇v) .

Action of A[M] on u0 = u0P0(y),

A[M](u0P0(y)) = (Au0)P0(y) +M∑

m=1

1√

3(Amu0)Pεm(y) .

Similar for the other terms in u(y) =∑µ∈F uµPµ(y).

Au is an infinite sequence even if u is finitely supported.

Truncated operators A[M]: truncate series after M terms.support increases at most by a factor 2M + 1.

Outline

1 Random Elliptic PDEA Model ProblemLegendre Chaos ExpansionStructure of the Operator

2 Adaptive Stochastic GalerkinGalerkin ProjectionAn Idealized Adaptive Algorithm

3 The Adaptive Wavelet ApproachComputation of the ResidualAn Adaptive AlgorithmSpatial Discretization

4 The Adaptive Finite Element ApproachA Residual-based Error Estimator

Galerkin ProjectionGalerkin projection u = (uµ)µ∈Λ, u(y) =

∑µ∈Λ uµPµ(y), Λ ⊂ F ,

〈Au, v〉 = 〈f, v〉

for all v in the same subspace as u, i.e. orthogonal projection ofu w.r.t. the energy norm ‖v‖2 = 〈Av, v〉 on `2(F ; H1

0(D)).

Residual r = (rµ)µ∈F = f − Au = A(u − u) ∈ `2(F ; H−1(D)).

‖u − u‖ h ‖u − u‖ h ‖r‖

Given Λ ⊂ F , finite element spaces for uµ, compute u byconjugate gradient iteration,preconditioner A = −∇ · (a∇ ) applies independently toeach uµ.

An Idealized Adaptive AlgorithmInitialize Λ0 B ∅, u0 = 0.

1 Compute ri = (riµ)µ∈F B f − Aui.

2 Refine Λi to Λi+1 based on ri.3 Compute the Galerkin projection ui+1 = (ui+1

µ )µ∈Λi+1

If ‖ri|Λi+1‖ ≥ ϑ‖ri‖ with 0 < ϑ < 1, then∥∥∥u − ui+1∥∥∥ ≤ √

1 − κ−1ϑ2∥∥∥u − ui

∥∥∥ .

If ϑ < κ−1/2 and #Λi+1 is minimal with ‖ri|Λi+1‖ ≥ ϑ‖ri‖, then

‖u − πN(u)‖ . N−s =⇒∥∥∥u − ui

∥∥∥ . (#Λi)−s ,

where πN(u) is a best N-term approximation of u in `2(F ; H10(D)).

Outline

1 Random Elliptic PDEA Model ProblemLegendre Chaos ExpansionStructure of the Operator

2 Adaptive Stochastic GalerkinGalerkin ProjectionAn Idealized Adaptive Algorithm

3 The Adaptive Wavelet ApproachComputation of the ResidualAn Adaptive AlgorithmSpatial Discretization

4 The Adaptive Finite Element ApproachA Residual-based Error Estimator

The Adaptive Wavelet Approachcompute the residual to sufficient accuracy

The adaptive algorithmsCohen, Dahmen and DeVore, Adaptive wavelet methods for ellipticoperator equations: convergence rates, Math. Comp., 2001.

Gantumur, Harbrecht and Stevenson, An optimal adaptive waveletmethod without coarsening of the iterands, Math. Comp., 2007.

are formulated for abstract Riesz bases.

Apply to the basis (Pµ)µ∈F in place of wavelets.

Gittelson, An adaptive stochastic Galerkin method for random ellipticoperators, Math. Comp., accepted.

Gittelson, Adaptive Galerkin methods for parametric and stochasticoperator equations, Ph.D. Thesis, ETH Zurich, 2011, supervised by Ch.Schwab.

Computation of the ResidualConstruct approximation q = (qµ)µ∈Ξ of A−1r = A−1(f − Au).

1 Approximate Audecompose u = (uµ)µ∈Λ into u = u{1} + · · · + u{J} accordingto ‖uµ‖,truncations Mj determine Ξ ⊂ F through

g = (gµ)µ∈Ξ B A[M1]u{1} + · · · + A[MJ]u{J} ≈ Au .

2 Compute q B A−1(f − g)independent solve of Aqµ = fµ − gµ for each µ ∈ Ξ.

3 Relative accuracy ‖q − A−1r‖ ≤ ω‖q‖ ensures

‖r|Ξ‖ ≥ ϑ‖r‖ , ϑ =1 − ω1 + ω

.

An Adaptive AlgorithmInitialize Λ0 B ∅, u0 = 0.

1 Compute qi = (qiµ)µ∈Ξi with∥∥∥qi

− A−1ri∥∥∥ ≤ ω ∥∥∥qi

∥∥∥ .

2 Construct a minimal set Λi ⊂ Λi+1 ⊂ Λi ∪ Ξi with∥∥∥qi|Λi+1 − A−1ri

∥∥∥ ≤ ω ∥∥∥qi|Λi+1

∥∥∥ .

3 Compute the Galerkin projection ui+1 = (ui+1µ )µ∈Λi+1 .

Ensured convergence in the energy norm,∥∥∥u − ui+1∥∥∥ ≤ √

1 − κ−1ϑ2∥∥∥u − ui

∥∥∥ , ϑ =1 − ω1 + ω

.

ExampleElliptic two-point boundary value problem

−∇ · (a∇u) = f in (0, 1), u(0) = u(1) = 0 .

Model problemf (x) = xa(x) = 1, i.e. A = −∆

Fast decayam(x) = cm−4 sin(mπx), scaled such that c

∑∞

m=1 m−4 = 5/6

Slow decayam(x) = cm−2 sin(mπx), scaled such that c

∑∞

m=1 m−2 = 1/2

Spatial discretizationlinear finite elementsuniform mesh with 1024 elements

Example (fast decay)

0 0.2 0.4 0.6 0.8 1

0.5

1

1.5

a(x)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10·10−2 u(x)

Example (fast decay)

100 101 102 103

10−5

10−4

10−3

10−2

10−1

100

#Λ, #Ξ

erro

rbou

nd

solutionresidual

Example (slow decay)

0 0.2 0.4 0.6 0.8 1

0.8

1

1.2

a(x)

0 0.2 0.4 0.6 0.8 10

2

4

6

8·10−2 u(x)

Example (slow decay)

100 101 102 103 104

10−3

10−2

10−1

#Λ, #Ξ

erro

rbou

nd

solutionresidual

Spatial DiscretizationApproximate solution

u = (uµ)µ∈Λ , u(y) =∑µ∈Λ

uµPµ(y) ,

with uµ in in separate finite element spaces (Vµ)µ∈Λ.

Computation of the residual1 Approximate Au,

refinement of Λ to Ξ.2 Compute an approximation q of A−1(f − g),

independent solves of Aqµ = fµ − gµ for all µ ∈ Ξ,a posteriori error estimator ensures accuracy,finite element spaces (Wµ)µ∈Ξ for (qµ)µ∈Ξ.

Refinement of Vµ chosen as subspace of Vµ +Wµ, µ ∈ Λ ∪ Ξ.

ExampleElliptic two-point boundary value problem

−∇ · (a∇u) = f in (0, 1), u(0) = u(1) = 0 .

Model problemf (x) = xa(x) = 1, i.e. A = −∆

Fast decayam(x) = cm−4 sin(mπx), scaled such that c

∑∞

m=1 m−4 = 5/6

Slow decayam(x) = cm−2 sin(mπx), scaled such that c

∑∞

m=1 m−2 = 1/2

Spatial discretizationlinear finite elements on uniform dyadic meshesresidual-based a posteriori error estimator

Example (fast decay)

100 101 102 103 104 105 106

10−5

10−4

10−3

10−2

10−1

100

degrees of freedom

erro

rbou

nd

solutionresidual

Example (slow decay)

100 101 102 103 104 105 106 107

10−3

10−2

10−1

degrees of freedom

erro

rbou

ndsolutionresidual

Outline

1 Random Elliptic PDEA Model ProblemLegendre Chaos ExpansionStructure of the Operator

2 Adaptive Stochastic GalerkinGalerkin ProjectionAn Idealized Adaptive Algorithm

3 The Adaptive Wavelet ApproachComputation of the ResidualAn Adaptive AlgorithmSpatial Discretization

4 The Adaptive Finite Element ApproachA Residual-based Error Estimator

The Adaptive Finite Element Approachcompute an upper bound for the residual

an a posteriori error estimator provides a bound for theerror and guides the refinement,no extra refinement just to compute the residual.

joint work withRoman Andreev, Martin Eigel, Christoph Schwab, Elmar

Zander

A Residual-based Error EstimatorThe residual r = (rν)ν∈F of u = (uµ)µ∈Λ has the form

rν = fν + ∇ · (a∇uν) +∑µ∈Λ

∇ · (aνµ∇uµ) .

Standard error estimators cannot be applied if uµ is piecewisesmooth on a finer mesh than uν. Projection Πν onto Vν,

rν = fν + ∇ · (a∇uν) +∑µ∈Λ

∇ · (aνµ∇Πνuµ)

︸ ︷︷ ︸a posteriori error estimator

+∑µ∈Λ

∇ · (aνµ∇(uµ − Πνuµ))

︸ ︷︷ ︸triangle inequality

for ν ∈ Λ, refinement of Vν based on both terms.activation of indices ν ∈ F \ Λ based on the second term.

Computations for ν ∈ F \ Λ involve only scalars such as ‖uµ‖.

SummaryThe adaptive Galerkin paradigm applies to equations withrandom operators.The parameter dependence is removed by expanding w.r.t.a tensorized polynomial basis on the parameter domain.Adaptive wavelet-type methods can ensure convergence,but computation of the residual is costly.A posteriori error estimators can be extended to bound theerror and guide refinements in an adaptive finiteelement-type method.

Thank you for your attention

SummaryThe adaptive Galerkin paradigm applies to equations withrandom operators.The parameter dependence is removed by expanding w.r.t.a tensorized polynomial basis on the parameter domain.Adaptive wavelet-type methods can ensure convergence,but computation of the residual is costly.A posteriori error estimators can be extended to bound theerror and guide refinements in an adaptive finiteelement-type method.

Thank you for your attention

ReferencesGittelson, Adaptive Galerkin methods for parametric and stochasticoperator equations, Ph.D. Thesis, ETH Zurich, 2011, supervised by Ch.Schwab.

Gittelson, An adaptive stochastic Galerkin method for random ellipticoperators, Math. Comp., accepted.

Gittelson, Uniformly convergent adaptive methods for a class ofparametric operator equations, M2AN, accepted.

Chkifa, Cohen, DeVore and Schwab, Sparse adaptive Taylorapproximation algorithms for parametric and stochastic elliptic PDEs,SAM Report, 2011-44.

Cohen, DeVore and Schwab, Convergence rates of best N-termGalerkin approximations for a class of elliptic sPDEs, FoCM, 2010.

Cohen, DeVore and Schwab, Analytic regularity and polynomialapproximation of parametric and stochastic elliptic PDE’s, Anal. Appl.,2011.

Gittelson, Convergence rates of multilevel and sparse tensorapproximations for a random elliptic PDE, SINUM, submitted.