Stochastic Galerkin approximations of elliptic PDEs driven by - - PowerPoint PPT Presentation

Stochastic Galerkin approximations

• f elliptic PDEs driven by spatial

white noise

Xiaoliang Wan Division of Applied Mathematics, Brown University Sea Grant College Program, MIT

(with G. E. Karniadakis, B. Rozovsky)

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 1

Two stochastic elliptic models

Model I: −∇ · ((E[a](x) + ǫ(x, ω))∇u) = f

ǫ(x, ω) is a colored noise with a known correlation function.

Model II: −∇ · ((E[a](x) + ˙ W(x, ω)) ⋄ ∇u) = f

˙ W(x, ω) is spatial white noise on L2(D). ‘⋄’ indicates the Wick product corresponding to Itô-Skorokhod integral.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 2

Outline

A brief overview of numerical methods for Model I

Karhunen-Loève expansion of the noise. Polynomial chaos methods and variants.

A stochastic finite element method for Model II

Spectral expansion of the white noise. Weighted Wiener chaos space. A stochastic FEM method.

A simple comparison between Model I and II.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 3

Karhunen-Loève expansion of the colored noise

Let ǫ(x, ω) be a second-order random process with zero mean and unit variance, i.e., E[ǫ](x) = 0, E[ǫ2](x) = 1. If the correlation function R(x, y) = E[ǫ(x, ω)ǫ(y, ω)] is known, the noise can be expressed as ǫ(x, ω) =

∞

• i=1
• λihi(x)ξi

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 4

Karhunen-Loève expansion of the colored noise

Let ǫ(x, ω) be a second-order random process with zero mean and unit variance, i.e., E[ǫ](x) = 0, E[ǫ2](x) = 1. If the correlation function R(x, y) = E[ǫ(x, ω)ǫ(y, ω)] is known, the noise can be expressed as ǫ(x, ω) =

∞

• i=1
• λihi(x)ξi

{(√λi, hi(x))}∞

i=1 are eigen-pairs of R(x, y), where

• D

R(x, y)hi(y)dy = λihi(x),

• D

hi(x)hj(x)dx = δij.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 4

Karhunen-Loève expansion of the colored noise

Let ǫ(x, ω) be a second-order random process with zero mean and unit variance, i.e., E[ǫ](x) = 0, E[ǫ2](x) = 1. If the correlation function R(x, y) = E[ǫ(x, ω)ǫ(y, ω)] is known, the noise can be expressed as ǫ(x, ω) =

∞

• i=1
• λihi(x)ξi

{(√λi, hi(x))}∞

i=1 are eigen-pairs of R(x, y), where

• D

R(x, y)hi(y)dy = λihi(x),

• D

hi(x)hj(x)dx = δij. {ξi} is a set of mutually uncorrelated random variables with zero mean and unit variance.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 4

Karhunen-Loève expansion of the colored noise

Let ǫ(x, ω) be a second-order random process with zero mean and unit variance, i.e., E[ǫ](x) = 0, E[ǫ2](x) = 1. If the correlation function R(x, y) = E[ǫ(x, ω)ǫ(y, ω)] is known, the noise can be expressed as ǫ(x, ω) =

∞

• i=1
• λihi(x)ξi

{(√λi, hi(x))}∞

i=1 are eigen-pairs of R(x, y), where

• D

R(x, y)hi(y)dy = λihi(x),

• D

hi(x)hj(x)dx = δij. {ξi} is a set of mutually uncorrelated random variables with zero mean and unit variance. The convergence of K-L expansion is optimal in the L2 sense.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 4

Approximate the problem using a series of random variables

   −∇ · (a(x, ω)∇u(x, ω)) = f(x) on D, u(x, ω) = 0 on ∂D.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 5

Approximate the problem using a series of random variables

   −∇ · (a(x, ω)∇u(x, ω)) = f(x) on D, u(x, ω) = 0 on ∂D.    −∇ · (aM(x, ξ)∇u(x, ξ)) = f(x) on D, u(x, ξ) = 0 on ∂D. aM(x, ξ) = E[a](x) + σ M

i=1

√λihi(x)ξi. ξ = (ξ1, . . . , ξM), and σ indicates the degree of perturbation.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 5

Approximate the problem using a series of random variables

   −∇ · (a(x, ω)∇u(x, ω)) = f(x) on D, u(x, ω) = 0 on ∂D.    −∇ · (aM(x, ξ)∇u(x, ξ)) = f(x) on D, u(x, ξ) = 0 on ∂D. aM(x, ξ) = E[a](x) + σ M

i=1

√λihi(x)ξi. ξ = (ξ1, . . . , ξM), and σ indicates the degree of perturbation. Need to approximate the random function u(x, ξ) efficiently!

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 5

(Generalized) polynomial chaos (gPC)

Polynomial chaos: [Wiener, 38; Cameron and Martin, 47]

Let C be the space of continuous functions induced by the Wiener process {Wt, 0 < t < 1}. If F is a functional of L2(C), i.e., E[F 2] < ∞, then the Fourier-Hermite expansion F =

∞

• |α|=0

fαφα(ξ) =

∞

• |α|=0

fα

∞

• j=1

Hj,αj(ξj), ξj = 1 bj(t)dWt converges in the L2(C) sense, where αi ∈ N0, |α| = ∞

i=1 αi,

{bi(t)} is a complete orthonormal set of real functions in L2(0, 1).

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 6

(Generalized) polynomial chaos (gPC)

Polynomial chaos: [Wiener, 38; Cameron and Martin, 47]

Let C be the space of continuous functions induced by the Wiener process {Wt, 0 < t < 1}. If F is a functional of L2(C), i.e., E[F 2] < ∞, then the Fourier-Hermite expansion F =

∞

• |α|=0

fαφα(ξ) =

∞

• |α|=0

fα

∞

• j=1

Hj,αj(ξj), ξj = 1 bj(t)dWt converges in the L2(C) sense, where αi ∈ N0, |α| = ∞

i=1 αi,

{bi(t)} is a complete orthonormal set of real functions in L2(0, 1).

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 6

(Generalized) polynomial chaos (gPC) - cont’d. Generalized polynomial chaos:[Xiu and Karniadakis, 02]

u(x; ξ) =

∞

• |α|=0

uα(x) φα(ξ) =

∞

• |α|=0

uα(x) M

j=1 φj,αj(ξj),

α ∈ NM

0 , ξ ∈ RM, |α| = M i=1 αi.

ξi are independent; E[φαφβ] = δαβ w.r.t. the PDF f(ξ) Correspondence between PDF and classical orthogonal polynomials: uniform - Legendre; Gaussian -Hermite, etc. L2 completeness of orthogonal polynomials.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 7

Apply gPC to the stochastic elliptic problem

L(x, u; ξ) = f(x) Galerkin projection:

PC expansions: u = Pp

|α|=0 uαφα.

Residual: R(ξ) = L(x, Pp

|α|=0 uαφα) − f(x).

Deterministic system of uα: E ˆR(ξ)φβ(ξ)˜ = 0, |β| = 0, . . . , p. # of φα: (M+p)!

M!p!

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 8

Apply gPC to the stochastic elliptic problem

L(x, u; ξ) = f(x) Galerkin projection:

PC expansions: u = Pp

|α|=0 uαφα.

Residual: R(ξ) = L(x, Pp

|α|=0 uαφα) − f(x).

Deterministic system of uα: E ˆR(ξ)φβ(ξ)˜ = 0, |β| = 0, . . . , p. # of φα: (M+p)!

M!p!

Collocation projection:

Interpolation operator: {ξ(j)}Ng

j=1: a set of grid points in the parametric space.

Deterministic system on grid points: L(x, u; ξ(j)) = f(x). Choices of {ξ(j)}: full tensor-products of Gauss quadrature points - O(NM ), sparse grids - O(N log(N)M−1)

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 8

Comments on (generalized) polynomial chaos

Advantages of gPC: Fast convergence due to spectral expansion. Efficiency due to orthogonality.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 9

Comments on (generalized) polynomial chaos

Advantages of gPC: Fast convergence due to spectral expansion. Efficiency due to orthogonality. Disadvantages of gPC: Efficency decreases as the number of random dimensions increases. Inefficient for problems with low regularity in the parametric space. May diverge for long-time integrations.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 9

Variants of the polynomial chaos method Choices of global approximation bases.

Sparse polynomial chaos bases. [Schwab et al., Webster et al.]

} { : p V ≤ = α φα

} { :

i i s

V β α φα ≤ =

• ( )

( )

V Vs dim dim <

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 10

Variants of the polynomial chaos method Choices of global approximation bases.

Sparse polynomial chaos bases. [Schwab et al., Webster et al.]

} { : p V ≤ = α φα

} { :

i i s

V β α φα ≤ =

• ( )

( )

V Vs dim dim <

• ✁
• ✂
• ✄

• ✁
• ✂
• ✄

Choices of local approximation bases - ξ ∈ Γ

Piecewise finite element space. [Babuska et al.] Wavelets approximation. [Le Maitre et al.] Adaptive multi-element gPC. [Wan and Karniadakis]

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 10

Strong ellipticity and Model I

Strong ellipticity: aM = E[a](x) + M

i=1

√λihi(x)ξi > c > 0 a.s.

10

−3

10

−2

10

−1

10 10

1

10

−3

10

−2

10

−1

10 Correlation length σ

As M → ∞, strong ellipticity condition may fail.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 11

Outline

A brief overview of numerical methods for Model I

Karhunen-Loève expansion of the noise. Polynomial chaos methods and variants.

A stochastic finite element method for Model II

Spectral expansion of the white noise. Weighted Wiener chaos space. A stochastic FEM method.

A simple comparison between Model I and II.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 12

Stochastic elliptic PDE - Model II (white noise)

   Au + δ(Mu) = f(x)

• n D,

u(x) =

• n ∂D.

Au(x) := −Di(aij(x)Dju(x)) δ(Mu) = Itô-Skorokhod integral of Mu Mu(x) := Di(σij(x)Dju(x)) Note: M can be an operator up to order two.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 13

White noise on a real separable Hilbert space U

Given a complete orthonormal basis {wk}∞

k=1 in U and a zero-mean

Gaussian family ˙ W = { ˙ W(h), h ∈ U} such that E[ ˙ W(h1) ˙ W(h2)] = (h1, h2)U, ∀h1, h2 ∈ U, the formal series ˙ W =

∞

• k=1

˙ W(wk)wk is called (Gaussian) white noise on U. Note: Due to the fact that (wi, wj) = δij, ˙ W(wk) ∼ N(0, 1). Spectral expansion of ˙ W: ˙ W =

k≥1 wk(x)ξk, ξk ∼ N(0, 1).

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 14

Weighted Wiener-chaos space

Define a linear bounded operator R on L2(F = (Ω, F, P)): RHα = rαHα, R−1Hα = r−1

α Hα,

0 < rα < C.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 15

Weighted Wiener-chaos space

Define a linear bounded operator R on L2(F = (Ω, F, P)): RHα = rαHα, R−1Hα = r−1

α Hα,

0 < rα < C. RL2(F; X): the closure of L2(F; X) with respect to the norm f2

RL2(F;X) := Rf2 L2(F;X) =

• α∈J

fα2

Xα!r2 α,

where the chaos expansion of f takes the form f = ∞

|α|=0 fαHα.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 15

Weighted Wiener-chaos space

Define a linear bounded operator R on L2(F = (Ω, F, P)): RHα = rαHα, R−1Hα = r−1

α Hα,

0 < rα < C. RL2(F; X): the closure of L2(F; X) with respect to the norm f2

RL2(F;X) := Rf2 L2(F;X) =

• α∈J

fα2

Xα!r2 α,

where the chaos expansion of f takes the form f = ∞

|α|=0 fαHα.

Given f ∈ RL2(F; X) and g ∈ R−1L2(F; X), we define a scalar inner product as f, g := E[(Rf, R−1g)X].

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 15

Wick product and Itô-Skorokhod integral

Wick product with respect to Hermite polynomials Hα(ξ): Hα(ξ) ⋄ Hβ(ξ) = Hα+β(ξ), ∀α, β ∈ NN

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 16

Wick product and Itô-Skorokhod integral

Wick product with respect to Hermite polynomials Hα(ξ): Hα(ξ) ⋄ Hβ(ξ) = Hα+β(ξ), ∀α, β ∈ NN E[Hα ⋄ Hβ] = E[Hα+β] = I(α+β=0); E[HαHβ] = α!δαβ

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 16

Wick product and Itô-Skorokhod integral

Wick product with respect to Hermite polynomials Hα(ξ): Hα(ξ) ⋄ Hβ(ξ) = Hα+β(ξ), ∀α, β ∈ NN E[Hα ⋄ Hβ] = E[Hα+β] = I(α+β=0); E[HαHβ] = α!δαβ Wick product and Itô-Skorokhod integral - an example

• [0,T ]

fdWt =

• [0,T ]

f ⋄ ˙ Wdt, where Wt is a one-dimensional Wiener process.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 16

Wick product and Itô-Skorokhod integral

Wick product with respect to Hermite polynomials Hα(ξ): Hα(ξ) ⋄ Hβ(ξ) = Hα+β(ξ), ∀α, β ∈ NN E[Hα ⋄ Hβ] = E[Hα+β] = I(α+β=0); E[HαHβ] = α!δαβ Wick product and Itô-Skorokhod integral - an example

• [0,T ]

fdWt =

• [0,T ]

f ⋄ ˙ Wdt, where Wt is a one-dimensional Wiener process. Itô-Skorokhod integral of Mu: δ(Mu) = Mu ⋄ ˙ W :=

• k≥1

Mku ⋄ ξk, where Mku(x) := wk(x)Di(σij(x)Dju(x)).

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 16

Stochastic elliptic PDE - Model II (white noise)

   Au + δ(Mu) = f(x)

• n D,

u(x) =

• n ∂D,

˙ W(x) = P∞

k=1 wk(x)ξk,

δ(Mu) = Mu ⋄ ˙ W := P

k≥1 Mku ⋄ ξk

Au(x) := −Di(aij(x)Dju(x)), Mku(x) := wk(x)Di(σij(x)Dju(x))

Assumption: aij(x) and σij(x) are measurable and bounded in ¯ D. A1|y|2 ≤ aijyiyj ≤ A2|y|2, ∀x ∈ ¯ D, y ∈ Rd, where A1 and A2 are positive numbers. wk is bounded and Lipschitz continuous.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 17

Model II is unbiased

u =

• α∈NN

uαHα ⇒ E[δ(Mu)] =

• k≥1
• α∈NN

MkuαE[Hα ⋄ ξk] = 0

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 18

Model II is unbiased

u =

• α∈NN

uαHα ⇒ E[δ(Mu)] =

• k≥1
• α∈NN

MkuαE[Hα ⋄ ξk] = 0

E[Au + δ(Mu)] = E[f] AE[u] = E[f]

Mean of the Stochastic PDE is the unperturbed deterministic PDE.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 18

A variational approach

Function spaces: V := RL2(F; H1

0(D))

ˆ V := R−1L2(F; H1

0(D)).

Bilinear form: B(u, v) = Au, v + Mu ⋄ ˙ W, v. Linear form: L (v) = f, v. Find a solution u ∈ V satisfying the weak form: B(u, v) = L (v), ∀v ∈ ˆ V .

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 19

Uncertainty propagator

Let u(x, ξ) =

∞

• |α|=0

uαhα(ξ), where hα(ξ) are normalized Hermite polynomials. The Galerkin projection in probability space yields ‘uncertainty propagator’ as Auα +

• k≥1

√αkMkuα−ǫk = fα, where ǫk = (0, . . . , 1, . . . , 0): only the k-th component equal to 1. The ellipticity of A guarantees the wellposedness. uα only depends on coefficients β < α.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 20

Equivalence between the model problem and its propagator

• Theorem. There exist a unique solution u ∈ RL2(F; H1

0(D)), if the

• perator R is defined by the weights rα as

rα = qα 2|α| |α|! , with qα =

∞

• k=1

qαk

k ,

where the number qk are chosen so that

• k≥1

k2q2

kC2 k < 1.

u0H1

0(D)

≤ CAfH−1(D), A−1MkvH1

0(D)

≤ CkvH1

0(D),

∀v ∈ H1

0(D).

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 21

Continuity of A−1Mk

|wk(x)| ≤ Cw

k and |wk(x) − wk(y)| ≤ CL k |x − y|, ∀x, y ∈ ¯

D. A1ˆ v2

H1

0(D)

≤ (Aˆ v, ˆ v) = (Mkv, ˆ v) =

• i,j(wkDi(σijDjv), ˆ

v) =

i,j(σijDjv, Di(wkˆ

v)) =

• i,j[(σijDjv, ˆ

vDiwk) + (σijDjv, wkDiˆ v)] ≤ maxi,jσij[CL

k vH1

0(D)ˆ

vL2(D) + Cw

k vH1

0(D)ˆ

vH1

0(D)]

≤ maxi,jσij(CL

k Cp + Cw k )vH1

0(D)ˆ

vH1

0 (D),

Thus, ˆ vH1

0 (D) = A−1MkvH1 0(D) ≤ maxi,jσij(CL

k Cp + Cw k )/A1vH1

0 (D),

where Cp is the Poincaré constant in D.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 22

A stochastic FEM method

Vh ⊂ H1

0(D): finite element space for the physical discretization.

Truncated Wiener chaos space: Vc := {f =

• α∈NM

0 ,|α|≤p

fαhα|fα ∈ R, fRL2(F) < ∞} The dual space of Vc: V −1

c

:= {f =

• α∈NM

0 ,|α|≤p

fαhα|fα ∈ R, fR−1L2(F) < ∞} Find a solution uM,p

h

∈ Vh ⊗ Vc such that AuM,p

h

, v + M

k=1 MkuM,p h

⋄ ξk, v = f, v, ∀v ∈ Vh ⊗ V −1

c

.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 23

Convergence of the stochastic FEM

• Theorem. Assume that αij, σij, wk have proper regularity and

u ∈ RL2(F; H) ∩ RL2(F; Hm+1(D)). The approximation solution uM,p

h

given by the stochastic finite element method can be bounded as u − uM,p

h

RL2(F;H) ≤ C( hmuRL2(F;Hm+1(D)) +

ˆ qW 1−ˆ q + ˆ qp+1 1−ˆ q ),

where the constant C is independent of h. ˆ qW =

k>M k2C2 kq2 k and ˆ

q =

k≥1 k2C2 kq2 k < 1.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 24

Convergence of the stochastic FEM

• Theorem. Assume that αij, σij, wk have proper regularity and

u ∈ RL2(F; H) ∩ RL2(F; Hm+1(D)). The approximation solution uM,p

h

given by the stochastic finite element method can be bounded as u − uM,p

h

RL2(F;H) ≤ C( hmuRL2(F;Hm+1(D)) +

ˆ qW 1−ˆ q + ˆ qp+1 1−ˆ q ),

where the constant C is independent of h. ˆ qW =

k>M k2C2 kq2 k and ˆ

q =

k≥1 k2C2 kq2 k < 1.

Error from the finite element discretization int the physical space.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 24

Convergence of the stochastic FEM

• Theorem. Assume that αij, σij, wk have proper regularity and

u ∈ RL2(F; H) ∩ RL2(F; Hm+1(D)). The approximation solution uM,p

h

given by the stochastic finite element method can be bounded as u − uM,p

h

RL2(F;H) ≤ C( hmuRL2(F;Hm+1(D)) +

ˆ qW 1−ˆ q + ˆ qp+1 1−ˆ q ),

where the constant C is independent of h. ˆ qW =

k>M k2C2 kq2 k and ˆ

q =

k≥1 k2C2 kq2 k < 1.

Error from the truncation in approximation of white noise: ˆ qW =

k>M k2C2 kq2 k as M → ∞.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 24

Convergence of the stochastic FEM

• Theorem. Assume that αij, σij, wk have proper regularity and

u ∈ RL2(F; H) ∩ RL2(F; Hm+1(D)). The approximation solution uM,p

h

given by the stochastic finite element method can be bounded as u − uM,p

h

RL2(F;H) ≤ C( hmuRL2(F;Hm+1(D)) +

ˆ qW 1−ˆ q + ˆ qp+1 1−ˆ q ),

where the constant C is independent of h. ˆ qW =

k>M k2C2 kq2 k and ˆ

q =

k≥1 k2C2 kq2 k < 1.

Error from the truncation of Wiener-chaos expansion: spectral convergence with respect to the weighted norm.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 24

An numerical example

   −∇ · [(E[a](x) + ǫ ˙ W) ⋄ ∇u(x, ω)] = f(x), x ∈ D, u(x, ω) = 0, x ∈ ∂D, D = (0, 1)2, E[a](x) = 1, ǫ = 1, and f(x) = 1. Orthonormal basis on L2(D):

wm,n(x) = 8 > > > > > < > > > > > : 1, m = n = 0 √ 2 sin(mπx), n = 0 √ 2 sin(nπy), m = 0 2 sin(mπx)sin(nπy), m, n = 1, 2, . . . , ∞.

Mku = −∇ · (wk(x)∇u) and continuity of A−1Mk:

Ck = maxwk(x) ≤ ˆ Ck = 8 > > < > > : 1, m=n=0, √ 2, mn = 0, m + n > 0, 2,

• therwise.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 25

Convergence of approximation of white noise

Weights rα : rα = qα 2|α| |α|! qα =

∞

• k=1

qαk

k

qk = 1 (k + 1)k ˆ Ck

10 20 30 40 0.25 0.3 0.35 0.4 0.45 n Weighted Norm

Weighted L2 norm of approximate white noise.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 26

p-convergence of the approximate solution

1 2 3 4 10

−4

10

−3

10

−2

10

−1

p Error

Spectral convergence of uM,p

h

RL2(F;H1

0 (D)). M = 21.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 27

Generalization of Model II Itô-Skorokhod integral: δ(Mu) = Mu ⋄ ˙ W Convolution with respect to g ∈ RL2(F; X) : Mu ⋄ g = Mu ⋄

• α∈NN

gαHα

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 28

Comparison between Wick and ordinary products

− d

dx

• K(x, ω) ∗ du

dx

• = 1

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

‘∗’ indicates Wick product ‘⋄’ or ordinary product ‘·’. K(x) = ecξ− 1

2c2, ξ ∼ N(0, 1), c is constant.

E[K] = 1, Var[K] = ec2 − 1.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 29

Wellposedness of the 1D models

u(x) =

p

• i=0

ui(x)hi(ξ)    − p

i=0 E[Kξiξj] d2 dx2 ui

= δ0j, ∀j = 0, . . . , p − p

i=0 E[(K ⋄ ξi)ξj] d2 dx2 ui

= δ0j, ∀j = 0, . . . , p E[Kξiξj] is symmetric and nonnegative-definite. E[(K ⋄ ξi)ξj] is lower-triangular with E[(K ⋄ ξi)ξi] = 1, which means that ui only depends on uj with j < i. Both systems are wellposed.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 30

Comparison between coefficients of Wiener chaos expansions

0.2 0.4 0.6 0.8 1 −0.05 0.05 0.1 0.15 0.2 0.25 0.3

x ui

ud with E[K] u0 u3 u6 0.2 0.4 0.6 0.8 1 −0.05 0.05 0.1 0.15 0.2 0.25 0.3

x ui

ud with E[K] u0 u3 u6

Ordinary product Wick product

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 31

Comparison between variances

0.5 1 1.5 2 2.5 10

−4

10

−2

10 10

2

10

4

10

6

10

8

c Variance

Ordinary product Wick product

Variance versus perturbation at x = 0.5.

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 32

Summary A brief overview of polynomial chaos methods for elliptic PDEs perturbed by colored noise. Karhunen-Loève expansion of colored noise. Adaptive polynomial chaos methods. A stochastic FE method for elliptic PDEs perturbed by spatial white noise. Itô-Skorokhod integral - convolution with respect to white noise through the wick product. Weighted Wiener chaos space p-convergence under the weighted norm

Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 33