stochastic galerkin methods without uniform ellipticity€¦ · without uniform ellipticity,juan...
TRANSCRIPT
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
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 2 / 28
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
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 3 / 28
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.
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 4 / 28
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.
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 5 / 28
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
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 6 / 28
A simple example: one-dimensional log-normal noise
‖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)
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 7 / 28
A simple example: one-dimensional log-normal noise
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’)
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 8 / 28
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
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 9 / 28
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
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 10 / 28
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)
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 11 / 28
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’)
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 12 / 28
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
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 13 / 28
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 <∞
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 14 / 28
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
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 15 / 28
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
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 16 / 28
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’)
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 17 / 28
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 ′.
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 18 / 28
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)
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 19 / 28
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)
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 20 / 28
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’)
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 21 / 28
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 ′.
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 22 / 28
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
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 23 / 28
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
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 24 / 28
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
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 25 / 28
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.
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 26 / 28
Numerical experiments
Approximation of u(0,0,0,... ) for K = N = 3, h = 110 and ε = 0, 1, 2
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 27 / 28
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
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 28 / 28