Stochastic Galerkin Methods without Uniform Ellipticity€¦ · without uniform ellipticity,Juan...

Stochastic Galerkin Methodswithout Uniform Ellipticity

Marcus Sarkis (WPI/IMPA)

Collaborator: Juan Galvis (Texas A & M)

WPI/IMPA

RICAM MS & AEE-Workshop4, Dec13/2011

Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 1 / 28

Problem of interest

Consider the Darcy’s equation−∇x . (κ(x , ω)∇xu(x , ω)) = f (x , ω), for x ∈ D ⊂ Rd

u(x , ω) = 0, on ∂D

κ(x , ω) = eW (x ,ω)

I W (x , ω) =∑∞

k=1 ak(x)ξk(ω)

I ξk are iid standard normal random variables

I eW (x,ω) ∈ (0,∞) not bounded, not uniformly elliptic

f (x , ω)

I Random forcing term


Outline

One-dimensional log-normal noise

White noise framework

Countable infinite-dimensional log-normal noise

Galerkin spectral method

Discretization, well-posedness, a priori error

Numerical results

Conclusions


Breeding analysis on Log-Normal without ellipticity

November 2005: I. Babuska, F. Nobile and R. Tempone: A stochasticcollocation method for elliptic partial differential equations withrandom input data.

March 2008: J. Galvis and S., Approximating infinity-dimensionalstochastic Darcy’s equations without uniform ellipticity.

March 2009: X. Wan, B. Rozovskii and G. E. Karniadakis, Astochastic modeling methodology based on weighted Wiener chaosand Malliavin calculus.

May 2009: C.J. Gittelson, Stochastic Galerkin discretization of thelognormal isotropic diffusion problem.

June 2010: J. Charrier, Strong and weak error estimates for thesolutions of elliptic partial differential equations with randomcoefficients.

January 2011: A. Mugler and H.-J. Starkloff, On elliptic partialdifferential equations with random coefficients.


References

Approximating infinity-dimensional stochastic Darcy’s equationswithout uniform ellipticity, Juan Galvis and Marcus Sarkis. SIAM J.Numer. Anal., Vol. 47(5), pp. 3624-3651, 2009.

Regularity results for the ordinary product stochastic pressureequation, Juan Galvis and Marcus Sarkis. Submitted. Preprint serieIMPA A 692, 2011.

An introduction to infinite-dimensional analysis, Giuseppe Da Prato.Universitext, Springer-Verlag, Berlin, 2006.

Stochastic analysis, Ichiro Shigekawa. Translations of MathematicalMonographs, Vol. 224, AMS, Providence, RI, 2004.


A simple example: one-dimensional log-normal noise

−(ea1(x)ξ1ux)x = f (x), in D = (0, 1)

Young Inequality: Let C1 := maxx∈D |a1(x)| and ε > 0

e−C21

2ε e−ε2ξ21 ≤ ea1(x)ξ1 ≤ e

C21

2ε eε2ξ21

Idea: Use weights of the type esξ21 (easy to integrate)∫

G (ξ1)esξ21dξ1 =

1√2π

∫ ∞−∞

G (y)esy2− 1

2y2dy

Lax Milgram:

‖ux(·, ξ1)‖2L2(0,1) ≤ C 2Pe

C21ε ‖f ‖2H−1(0,1)e

εξ21



‖ux(·, ξ1)‖2L2(D) ≤ C 2Pe

C21ε ‖f ‖2H−1(D)e

εξ21

Integrate over ξ1, (0 < ε < 1/2)

u ∈ H10 (D)⊗ (L2)(dξ1) bounded by f ∈ H−1(D)⊗ (L2)ε(dξ1)

Integrate over ξ1, (ε > 0 and s ∈ R such that s + ε < 1/2)∫||ux(·, ξ1)||2L2(D)e

sξ21dξ1 =1√2π

∫ ∞−∞‖ux(·, y)‖2L2(D)e

sy2− 12y2dy

≤ C 2Pe

C21ε ‖f ‖2H−1

1√2π

∫ ∞−∞

e(s+ε)y2− 1

2y2dy

= C 2Pe

C21ε (1− 2(s + ε))−

12 ‖f ‖2H−1(D)

For s 6= 0, ux ∈ L2(D)⊗ L2s (dξ1) important when f (x , ξ1)



RHS f (x , ξ1) with lack (s < 0) extra (s > 0 ) decay in ξ1

∫‖ux(·, ξ1)‖2L2(D)e

sξ21dξ1 ≤ C 2Pe

C21ε

∫‖f (·, ξ1)‖2H−1(D)e

(s+ε)ξ21dξ1

Note: u → s, f → s + ε, and v → −(s + ε)

Given f ∈ H−1(D)⊗ L2s+ε(dξ1), find u ∈ H10 (D)⊗ L2s (dξ1) such that

a(u, v) = f (v), ∀v ∈ H10 (D)⊗ L2−(s+ε)(dξ1)

Existence and uniqueness (inf-sup condition) Galvis and S. (09’)


We need a theoretical framework to deal:

Infinite-dimensional case (Gaussian measure)

Generalized Wiener-chaos expansions (on Hilbert spaces)

Galerkin spectral methods (Explicit computations)

A priori error estimates (Natural to establish)

Regularity theory (Derivatives in ω and x)

Gaussian Sobolev (Hilbert) spaces

Constants that depend on few quantities


Probability space: constructed from a pair (H ,A)

Using KL:

I H := L2(D)

I Correlation operator: C v(x) :=∫Dk(x , x)v(x)dx

I C →: Eigenfunctions qk , eigenvalues µk

I A = C−1

I A→: Eigenfunctions qk , eigenvalues λk = 1/µk

Using convolution (1D- Smoothed white noise)

I H := L2(R)

I A := − d2

dx2 + x2 + 1

I A→: Hermite functions qk , eigenvalues λk = 2k


White noise framework

Hilbert space H, operator A, and H-orthonormal basis qk∞k=1:

I Aqk = λkqk , k = 1, 2, . . .

I 1 < λ1 ≤ λ2 ≤ · · ·

I∑∞

k=1 λ−2θk <∞ for some constant θ > 0

p ≥ 0, ξ ∈ H, ‖ξ‖2p := ‖Apξ‖2H =∑∞

k=1 λ2pk (ξ, qk)2H

Sp := ξ ∈ H; ‖ξ‖p <∞ S := ∩p≥0Sp S ′

Probability measure µ (Bochner-Minlos theorem) characterized by

Eµei〈·,ξ〉 :=

∫S′e i〈ω,ξ〉dµ(ω) = e−

12‖ξ‖2H , for all ξ ∈ S

Probability space (Ω,F ,P) = (S ′,B(S ′), µ)


Remarks: Fernique and change of variables

µ(S−θ) = 1 where S−θ = ω ∈ S ′ : ‖ω‖−θ <∞

Eµei〈·,ξ〉 :=

∫S−θ

e i〈ω,ξ〉dµ(ω) = e−12‖ξ‖2H for all ξ ∈ S

(S−θ,B(S−θ), µ): normally distributed RV Xξ(ω) = 〈ω, ξ〉Equivalent formulation:

Eµei〈·,ξ〉 :=

∫He i〈h,ξ〉d µ(h) = e

− 12‖ξ‖2

A−2θ for all ξ ∈ S

µ Gaussian measure with covariance A−2θ

(H,B(H), µ): normally distributed RV Yξ(h) = 〈h,Aθξ〉See Da Prato (06’) for θ = 1/2 and Galvis and S. (11’)


Gaussian Field W (x , ω) = 〈ω, φx〉

In KL:

I φx(x) ≡ φ(x , x) =∑∞

k=1 λ− 1

2

k qk(x)qk(x)

I W (x , ω) = 〈ω, φx〉 =∑∞

k=1 ak(x)〈ω, qk〉

I ak(x) = λ− 1

2

k qk(x)

I ξk(ω) = 〈ω, qk〉 i.i. normally distributed

In convolution:

I φx(x) ≡ φ(x − x) (smoothed window)

I W (x , ω) = 〈ω, φx〉 =∑∞

k=1 ak(x)〈ω, qk〉

I ak(x) =∫Rφ(x − x)qk(x)dx

I ξk(ω) = 〈ω, qk〉 i.i. normally distributed


Countable independent normals

W (x , ω) = 〈ω, φx〉 =∑∞

k=1 ak(x)ξk(ω)

−λ2θk a2k(x)

2ε−ελ−2θk ξk(ω)2

2≤ ak(x)ξk(ω) ≤

λ2θk a2k(x)

2ε+ελ−2θk ξk(ω)2

2

−|||φ|||2θ

2ε−ε‖ω‖2−θ

2≤ 〈ω, φx〉 ≤

|||φ|||2θ2ε

+ε‖ω‖2−θ

2

|||φ|||2θ := supx∈D ‖φx‖2θ

‖φx‖2θ =∞∑k=1

λ2θk (φx , qk)2H =∞∑k=1

λ2θk a2k(x) <∞

‖ω‖2−θ :=∑∞

k=1 λ−2θk 〈ω, qk〉2∫S′‖ω‖2−θdµ(ω) =

∞∑k=1

λ−2θk <∞


Two examples

KL:

I∑∞

k=1

√µk <∞

I supk ‖qk‖L∞(D) <∞

I Take θ = 1/4, both conditions are satisfied

Smoothed white noise:

I λk = 2k (note that∑∞

k=1 λ−1k =∞)

I φ(x − x) a smooth window (the ak(x) decay fast)

I Take θ > 1, both conditions are satisfied


Generalization to infinite-dimension

Ellipticity

κmin(ω) := e−|||φ|||2θ

2ε−ε‖ω‖2−θ

2 ≤ e〈ω,φx 〉

Lax-Milgram (fixed ω)

|u(·, ω)|2H1(D) ≤C 2P

κmin(ω)2‖f (·, ω)‖2H−1(D)

For ε > 0 and s ∈ <

|u(·, ω)|2H10 (D)e

s‖ω‖2−θ ≤ C 2Pe|||φ|||2θε ‖f (·, ω)‖2H−1(D)e

(s+ε)‖ω‖2−θ

Solution space and test space. Stability

|u|2H1(D)×(L2)s ≤ C 2Pe|||φ|||2θε ‖f ‖2H−1(D)×(L2)s+ε


Existence and uniqueness

Given f ∈ H−1 × (L2)s+ε

Find u ∈ H10 × (L2)s such that for all v ∈ H1

0 × (L2)−s−ε∫S′×D

e〈ω,φx 〉∇u(x , ω)∇v(x , ω)dxdµ =

∫S′×D

f (x , ω)v(x , ω)dxdµ

Existence and uniqueness (inf-sup condition)

Details in Galvis and S’ (09’)


Generalized Hermite polynomials in (L)s norm

Multi-indices J : α = (α1, α2, . . . ) ∈ (NN0 )c

Order of α : d(α) := max k : αk 6= 0Length of α : |α| := α1 + α2 + · · ·+ αd(α)

Note esλ−2θk y2− y2

2 = e− 1

2σ2k

y2

if σk(s) :=(

1− 2sλ2θk

)− 12

σ∗(s) :=

∫S′es‖ω‖

2−θdµ(ω) =

∏∞k=1 σk(s) s <

λ2θ12

+∞ s ≥ λ2θ12

σk -Hermite polynomials hσ2k ,αk

, orthogonal in L2(R, e− 1

2σ2k

y2

dy)

s <λ2θ12 , α = (α1, α2, . . . ) ∈ J and σ(s) = (σ1(s), σ2(s), . . . ), define

Hσ2(s),α(ω) :=1√σ∗(s)

d(α)∏k=1

hσ2k (s),αk

(〈ω, qk〉); ω ∈ S ′.


Wiener-chaos expansion in the Hm × (L)s norm

Wiener-chaos basis (for s <λ2θ12 )

‖Hσ2(s),α‖2(L)s = α!σ(s)2α

α! = α1!α2! · · ·αd(α)! σ(s)2α = σ1(s)2α1σ2(s)2α2 · · ·σd(α)(s)2αd(α)

z ∈ (L)s represented by a Wiener-chaos expansion

z =∑α∈J

zα,sHσ(s)2,α with ‖z‖2(L)s =∑α∈J

α!σ(s)2αz2α,s

u ∈ Hm × (L)s

‖u‖2Hm×(L)s =∑α∈J

α!σ(s)2α‖uα,s‖2Hm(D)


Weighted chaos norms

z ∈ (L)s represented by a Wiener-Chaos expansion

z =∑α∈J

zα,sHσ(s)2,α with ‖z‖2(L)s =∑α∈J

α!σ(s)2αz2α,s

z ∈ (L)p;s and weighted chaos norms

‖z‖2p;s :=∑α∈J

(1 + 〈α, λ〉2p) α!σ(s)2αz2α,s ,

〈α, λ〉 = α1λ1 + α2λ2 + · · ·+ αd(α)λd(α)

Measure how fast the chaos coefficients decay

u ∈ Hm × (L)p;s

‖u‖2Hm×(L)p;s =∑α∈J

(1 + 〈α, λ〉2p) α!σ(s)2α‖uα,s‖2Hm(D)


Analysis

Isomorphism between weighted chaos norms and Sobolev stochasticderivatives. Galvis and S. (11’)

Weighted chaos norms: easy for establishing a priori error estimates.Galvis and S. (09’) and (11’)

Stochastic derivatives: easier for establishing regularity theory. Galvisand S. (11’)


Finite dimensional discretization

FEM spatial discretization X h0 (D) ⊂ H1

0 (D)

Let N,K ∈ N0 and define

J N,K := α ∈ J : d(α) ≤ K , and, |α| ≤ N

Polynomials in 〈ω, q1〉, . . . , 〈ω, qK 〉 of total degree at most N

PN,K := spanHσ(s)2,α : α ∈ J N,K

QK is the (H-orthogonal) projection on the spanq1, . . . , qK

QKω :=K∑

k=1

〈ω, qk〉qk , for all ω ∈ S ′.


Finite dimensional formulation

Solution space:XN,K ,h := X h

0 (D)× PN,K

Test space:

YN,K ,hs :=

v : v(x , ω) = v(x , ω)e(s+

ε2)‖PKω‖2−θ , v ∈ XN,K ,h

Find uN,K ,h ∈ XN,K ,h such that

a(uN,K ,h, v) = 〈f , v〉 for all v ∈ YN,K ,hs

Discrete inf-sup conditions, existence and uniqueness


A priori error estimate

U1s := H1(D)× (L)s

Let ε > 0, s + 2ε <λ2θ12 and −s − ε < λ2θK+1

2 . Then

|u− uN,K ,h|U1s≤

(1 + e

|||φ|||2θε

∞∏k=K+1

σk(−s − ε)

)inf

z∈XN,K ,h|u− z |U1

s+2ε

infz∈XN,K ,h |u − z |U1s+2ε

bounded by

max 1

1 + (N + 1)λ1,

1

1 + λK+1

p|u|U1

p;s+2ε+ Ch|u|U2

s+2ε

U1p;s+2ε := H1(D)× (L)p;s+2ε U2

s+2ε := H2(D)× (L)s+2ε


Numerical experiments 1D

Smoothed white noise (convolution method)

Window φ : R → R

D = [0, 1] and φ(x) = e−12x2

T = − d2

dx2+ x2 + 1, Tqk = (2k)qk and λk = 2k

Hermite functions orthonormal in L2(R):

qk(x) :=1√√

π(k − 1)!e−

12x2hk−1(

√2x), k = 1, 2, . . .

ak(x) =∫R φ(x − x)qk(x)dx

Exact solution

u =x(1− x)

2e∑∞

k=1 ak (x)yk


Numerical experiments

k 0 1 2 3 4 5(K+NK

)1 2 6 20 70 252

|u − QN,Ku|U10

1.6284 1.3761 0.9767 0.6162 0.3570 0.1920

(1.18) (1.41) (1.59) (1.73) (1.86)

|u − uN,K ,h|U10

1.7292 1.6157 1.3590 1.0375 0.7281 0.4626

1.7291 1.6153 1.3575 1.0340 0.7214 0.4659(1.07) (1.18) (1.31) (1.43) (1.55)

|u − uN,K ,h|κ 0.4319 0.3691 0.2598 0.1573 0.0836 0.04540.4318 0.3688 0.2589 0.1552 0.0790 0.0279

(1.17) (1.42) (1.67) (1.96) (2.83)

Errors for K = N = k, h = 1/16, 1/32 and ε = 12 , s = 0. For

h = 1/32 we have added in parenthesis the reduction factor, whenpassing to next value of k, corresponding to the projection and finiteelement error in the seminorm | · |U1

0and the finite element error in

the κ-energy norm.


Numerical experiments

Approximation of u(0,0,0,... ) for K = N = 3, h = 110 and ε = 0, 1, 2


Conclusions

Ellipticity treatment

Unified framework for KL and smoothed white noise

More general f and infinite-dimensional case

Weighted norms, well-posedness, a priori error estimates

Framework for establishing regularity theory


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