Sparse Tensor Methods for PDEs with Stochastic Data
I: MLMC and FoSM
Ch. SchwabSeminar fur Angewandte Mathematik
ETH Zurich, Switzerland
Workshop Tutorial Computation with UncertaintiesIMA, Minneapolis, MN, USA, October 16 & 17, 2010
ERC Project 247277 STAHDPDE
Swiss National Science Foundation
Numerical models in engineering can be solved with high accuracy
if input data are known exactly.
Often, however,
input data are not known exactly
and
accurate numerical solutions are of limited use.
•Mathematical description of uncertainty in input data and solution?
• How to propagate data uncertainty through an engineering FEM simulation?
• How to process statistical information in FEM?
Goal:
given statistics of input data, compute (deterministic) solution statistics.
Tool:
Formulation and solution of Stochastic Partial Differential Equation (SPDE)
1. Deterministic operator w. stochastic data:
u : Ω ∋ ω → X such that
Au = f(·, ω), f : Ω ∋ ω → Y ′
2. Stochastic operator A(ω) ∈ L(X, Y ′) and deterministic data f ∈ Y ′:u : Ω ∋ ω → X such that
A(ω)u = f .
Choices of X and Y : elliptic PDEs X = Y , parabolic/hyperbolic PDEs: X 6= Y .
examples
Examples
(Ω,Σ,P) P.-space, ω ∈ Ω, x ∈ D ⊂ Rd bounded Lipschitz domain
1. Diffusion (random medium, random source term) (Dettinger & Wilson (1985), ...)
−∇ · (a(x;ω)∇u(x;ω)) = f(x;ω) in D, u(·;ω)|∂D = 0 .
2. Random Eigenvalue Problem (R. Andreev & CS (2010))
−∇ · (a(x;ω)∇w(x;ω)) = λ(ω)w(x;ω) in D, w(·;ω)|∂D = 0 .
3. Diffusion in Random Medium
ρ(x, t;ω)∂tu−∇ · (a(x, t;ω)∇u(x, t;ω)) = f(x, t;ω) in D, u(·, ·;ω)|∂D = 0 ,
u(x, 0;ω) = u0(x;ω) in D .
4. Wave Propagation in Random Medium
ρ(x, t;ω)∂2ttu−∇ · (a(x, t;ω)∇u(x, t;ω)) = f(x, t;ω) in D, u(·, ·;ω)|∂D = 0 ,
u(x, 0;ω) = u0(x;ω) , ∂tu(x, 0;ω) = u1(x;ω) in D .
literature
References
Monte Carlo FEM
Sampling Methods w. Sparse Tensor Estimation of Correlations (CS and von Petersdorff (2006)) ,
Multilevel Monte Carlo: (Heinrichs (2000), Giles (2006))
Diffusion (Barth, CS & Zollinger (2010)) , SCL (Mishra & CS (2010)) .
Perturbation Methods “First Order Second Moment” (FOSM)
Asymptotics: J. B. Keller (1964), L. Borcea, G. Papanicolau et al., M. Kleiber, T.D. Hien (1992)
Sparse Tensor FEM: CS and R.A. Todor (2003), CS and T. von Petersdorff (2006) , CS and H. Harbrecht,
CS and A. Chernov (2009)
Stochastic Collocation and Galerkin Wiener Polynomial Chaos (WPC), Karhunen-Loeve (KL), gPCR. G. Ghanem, P. D. Spanos (1991)I. Babuska , R. Tempone etal. SINUM (2003- 2005)G. E. Karniadakis, D. Xiu et al. SIAM J. Sci. Comp. (2002)H. Matthies etal. CMAME (2005)
Outline
1 Random fields, statistics
2 Example 1: Time harmonic scattering of random incident wave
3 Example 2: SCL with random initial data
4 Monte Carlo FEM (MCFEM)
5 Sparse Tensor FEM
6 Multi Level Monte Carlo FVM (MLMCFVM) for Conservation Laws
7 Example 3: Sparse Tensor FoSM Analysis in Random Domains
8 Conclusions
random fields
Random fields, statistics
D ⊂ Rd bounded domain, Γ = ∂D = Γ0 ∪ Γ1 Lipschitz,
(Ω,Σ,P) probability space
Random fields on Γ, D:
X separable Hilbert space. u(x, ω) random field iff
u ∈ L0(Ω, X) := u(x, ω) : Ω → X| Ω ∋ ω → ‖u(·, ω)‖X is P-measurable
A random field u : Ω → X is in L1(Ω, X) if ω 7→ ‖u(ω)‖X is integrable so that
‖u‖L1(Ω,X) :=
ˆ
Ω
‖u(ω)‖X dP(ω) <∞
In this case the Bochner integral
Eu :=
ˆ
Ω
u(ω)dP(ω) ∈ X
exists and we have
‖Eu‖X ≤ ‖u‖L1(Ω,X) . (1)
B : X → Y continuous, linear.
u ∈ Lk(Ω, X) random field in X =⇒ v(ω) = Bu(ω) ∈ Lk(Ω, Y )
‖Bu‖Lk(Ω,Y ) ≤ C ‖u‖Lk(Ω,X)
and
B
ˆ
Ω
u dP(ω) =
ˆ
Ω
BudP(ω).
Statistical moments of u: for any k ∈ N need k-fold tensor product spaces
X (k) = X ⊗ · · · ⊗X︸ ︷︷ ︸k-times
,
equipped with a cross norm ‖ ‖X(k) (Schatten (TAMS 1943), Grothendieck (MAMS 1955))
∀u1, . . . , uk ∈ X ‖u1 ⊗ ...⊗ uk‖X(k) = ‖u1‖X ...‖uk‖XFor u ∈ Lk(Ω, X) consider random field
u(k) = u(ω)⊗ · · · ⊗ u(ω) ∈ L1(Ω, X (k))
and∥∥∥u(k)
∥∥∥L1(Ω,X(k))
=
ˆ
Ω
‖u(ω)⊗ · · · ⊗ u(ω)‖X(k) dP(ω)
=
ˆ
Ω
‖u(ω)‖X · · · ‖u(ω)‖X dP(ω) = ‖u‖kLk(Ω,X) (2)
Define k-th moment (k-point correlation function) Mku as expectation of u⊗ · · · ⊗ u:
Definition 1
For u ∈ Lk(Ω, X) for some integer k ≥ 1, the k-th moment of u(ω) is defined by
Mku = E[u⊗ ...⊗ u︸ ︷︷ ︸k−times
] =
ˆ
ω∈Ω
u(ω)⊗ ...⊗ u(ω)︸ ︷︷ ︸k−times
dP(ω) ∈ X (k) (3)
Application: Covariance of u ∈ L2(Ω, V ), V separable and reflexive (e.g. V = H10 (D))
C[u] = E [(u− Eu)⊗ (u− Eu)] ∈ V ⊗ V
If u “sufficiently regular”:
Covariance function:
C[u](x, x′) =
ˆ
Ω
(u(x, ω)− Eu(x))(u(x′, ω)− Eu(x′))dP(ω), x, x′ ∈ D.
k-th Moment (k-point correlation function): if u ∈ Lk(Ω, V ), then
M(k)u = E[u⊗ ...⊗ u] ∈ V (k) := V ⊗ ...⊗ V :
M(k)u(x1, ..., xk) :=
ˆ
Ω
u(x1, ω)⊗ ...⊗ u(xk, ω) dP(ω)
stochastic operator equation
Linear Operator Equation with Stochastic Data
Given A : V → V ′ linear, bounded, f ∈ L1(Ω, V ′), find u ∈ L1(Ω, V ):
Au = f
Assume ex. α > 0 and T : V → V ′ compact such that
∀v ∈ V :⟨(A + T ) v, v
⟩≥ α ‖v‖2V (4)
and
kerA = 0 (5)
Proposition 2
Assume (4) and (5). Then
• for every f ∈ L0(Ω, V ′) exists a unique u ∈ L0(Ω, V ) solution of Au = f ,
• for every f ∈ Lk(Ω, V ′) holds u ∈ Lk(Ω, V ).
Example
Example: Time harmonic waves w. random incident field
D ⊂ R3 bounded, Γ = ∂D Lipschitz, κ ∈ R wavenumber
−∆U − κ2U = 0 in Dc := R3\D
subject to Dirichlet boundary conditions
γ0U = U |Γ = uinc(·;ω) on Γ, Radiation Condition at ∞ .
Given random incident field
uinc ∈ Lk(Ω, H12(Γ)), k ≥ 0,
ex. unique solution (scattered wave)
U(x, ω) ∈ Lk(Ω, H1(D)) (Sch. & Todor 2003).
Example
Time harmonic waves w. random incident field: BEM (Harbrecht & CS)
U(x, ω) = (SLκσ)(x;ω) :=
ˆ
Γ
e(κ; x, y) σ(y;ω)dsy.
V = H−1/2(Γ), σ(x;ω) : Ω → H−1/2(Γ) random flux
Fubini: SLκ and M(1) commute. Hence
E[U ] = M(1)[U ] = M(1)[SLκσ] = SLκ
[M(1)[σ]
]= SLκ [E[σ]]
where the mean field E[σ] = M(1)[σ] ∈ H−12 (Γ) satisfies first kind deterministic integral equation
SκE[σ] = E[uinc] ∈ H12(Γ) . (6)
Unique Solvability (Nedelec and Planchard (1973)): under assumption of nonresonance,
κ2 6∈ Σ .
w. Σ set of eigenfreq. of int. Dirichlet problem, ex. cS > 0 such that
∀σ ∈ H−1/2(Γ) : 〈σ, Sκσ〉 ≥ cS ‖σ‖2H−1/2(Γ)
!! cS ≃ dist(κ2,Σ)
Example
Time harmonic waves w. random incident field: BEM (Harbrecht & CS)
If in the stochastic Dirichlet problem u ∈ L2(Ω, H
12(Γ)
)and E[u] = 0, then U ∈ L2
(Ω, H1(D)
)and
C[U ] = M(2)U = M(2)(SLκσ) = (SLκ ⊗ SLκ)M(2)σ =
ˆ
Γ
ˆ
Γ
e(κ; x, z) e(κ; y, w)C[σ](z, w)dsz dsw ,
where
C[σ] ∈ H−12 ,−1
2(Γ× Γ) := H−12(Γ)⊗H−1
2(Γ)
satisfies the first kind BIE
(Sκ ⊗ Sκ)C[σ] = C[uinc] ∈ H12 ,
12(Γ× Γ) .
Solvability:
∀C[σ] ∈ H−12 ,−1
2 (Γ× Γ) : 〈(Sκ ⊗ Sκ)C[σ], C[σ]〉 ≥ c2S‖C[σ]‖2H−1
2 ,−12 (Γ×Γ)
Stability: Condition of second (k-th) moment equation is c2S (ckS) !!!!
Example
Example: SCL in Rd (Mishra & CS (2010))
∂u
∂t+
d∑
j=1
∂
∂xj(fj(u)) = 0, x = (x1, . . . , xd) ∈ Rd, t > 0 u(x, t;ω)|t=0 = u0(x;ω) . (7)
For every 0 < T <∞, ex. unique random entropy solution u : Ω ∋ ω 7→ Cb((0, T );L1(Rd)) given by
u(·, t;ω) = S(t)u0(·, ω) , t > 0, ω ∈ Ω (8)
such that for every k ≥ m ≥ 1 and for every 0 ≤ t ≤ T <∞ holds P-a.s.
‖u‖Lk(Ω;C(0,T ;L1(Rd))) ≤ ‖u0‖Lk(Ω;L1(Rd)) , (9)
‖S(t) u0(·, ω)‖(L1∩L∞)(Rd) ≤ ‖u0(·, ω)‖(L1∩L∞)(Rd) (10)
Example
Example: SCL in Rd (Mishra & CS (2010))
Assume further that for some k ∈ N and for some real number r ≥ 1
u0 ∈ Lrk(Ω;L1(Rd)) . (11)
Then, for every 0 < T <∞ and every
0 < t1, t2, . . . , tk ≤ T <∞ (12)
the spatial k-point correlation function
u(x1, t1;ω)⊗ · · · ⊗ u(xk, tk;ω) (13)
is well-defined as an element of Lr(Ω;L1(Rkd)). In particular, the k-th moment
(Mku)(t1, . . . , tk) := E[u(·, t1;ω)⊗ · · · ⊗ u(·, tk;ω)] (14)
is well-defined for any choice of tj as in (12) as an element of L1(Rkd), and it satisfies
∥∥∥(Mku
)(t1, ..., tk)
∥∥∥(L1(Rd))(k)
≤∥∥∥
k⊗
j=1
u(·, tj; ·)∥∥∥L1(Ω;(L1(Rd))(k))
≤ ‖u0‖kLk(Ω;L1(Rd)). (15)
goal
Goal of Computation
For the operator equation
Au = f
with f ∈ Lk(Ω, V ),
given M(k)f , find M(k)
u .
given law of f , find law of u .
Approaches:
• Monte-Carlo FEM (“Collocation in ω”):
1. multilevel MC
2. sparse tensor approximation of higher moments
• Sparse Wavelet FEM for deterministic approximation of M(k)
MonteCarlo
Monte Carlo - I
Given data ensemble
f(ωj), j = 1, ...,M ⊂ V ′
generate (in parallel) solution ensemble
u(ωj), j = 1, ...,M ⊂ V
Theorem 3Assume (4) and (5) and that f ∈ L2k(Ω, V ′).Estimate M(k)u by the k-th moment of ensemble u(ωj) : j = 1, ...,M, i.e. by
EM [M(k)u] := u⊗ · · · ⊗ uM
=1
M
M∑
j=1
u(ωj)⊗ ...⊗ u(ωj) ∈ V (k).
Then ex. C(k) > 0 such that for every M ≥ 1 and every 0 < ε < 1 holds
P
‖M(k)u − EM [M(k)u]‖V⊗...⊗V ≤ C
‖M2k(f)‖1/2V ′(2k)√
εM
≥ 1− ε (16)
FEM
Monte Carlo - II
Lemma (Law of iterated logarithm in Hilbert spaces):
V separable Hilbert and X ∈ L2(Ω, V ). Then
lim supM→∞
∥∥∥XM − E(X)∥∥∥V
(2M−1 log logM)1/2≤ ‖X − E(X)‖L2(Ω,V ) with probability 1.
Proof: Classical law of iterated logarithm: for real valued Y (ω) holds
lim supM→∞
∣∣∣Y M − E(Y )∣∣∣2
2M−1 log logM= VarY with probability 1. (17)
Let Z := X − E(X). V separable ⇒ w.l.o.g V = ℓ2 = spanej∞j=1 and Y := (ej, Z) = Zj ∈ R. Apply (17)
with
VarY = (ej ⊗ ej,M2Z) = (M2Z)j,j.
Add estimates for j = 1, 2, . . . and obtain
lim supM→∞
∑∞j=1 |Zj|
2
2M−1 log logM≤
∞∑
j=1
(M2Z)j,j with probability 1.
FEM
Monte Carlo - III
P-a.s. convergence of MCM (Semidiscrete Case !).
Convergence of MCM without Second Moments?
Theorem 4Consider Moments of order k ∈ N and assume
f ∈ Lαk(Ω, V ′) for some α ∈ (1, 2].
Then ex. C such that for every M ≥ 1 and every 0 < ε < 1
P
(‖Mku− EM [Mku]‖V (k) ≤ C
‖f‖kLαk(Ω,V ′)
ε1/αM 1−1/α
)≥ 1− ε (18)
So far: MCM assuming that Au = f solved exactly (“Semidiscrete MCM”).
Next: Galerkin FEM in V .
FEM
Galerkin FEM
Dense sequence of subspaces:
S0 ⊂ S1 ⊂ S2 ⊂ · · · ⊂ Sℓ ⊂ Sℓ+1 ⊂ . . . V
Galerkin FEM: given f ∈ Lk(Ω, V ′), find
uL(ω) ∈ Lk(Ω, SL) such that 〈vL, AuL(ω)〉 = 〈vL, f(ω)〉 ∀vL ∈ SL
Galerkin Projection: GL : V → SL defined by
∀vL ∈ SL : 〈AGLu, vL〉 = 〈f, vL〉
is stable: ex. L0 > 0 s.t.
∀L ≥ L0 : ‖GLu‖V ≤ C‖u‖Vand converges quasioptimally:
∀L ≥ L0 ∀vL ∈ SL : ‖u(ω)− uL(ω)‖V ≤ C‖u(ω)− v‖V P− a.e.ω ∈ Ω.
ConvRates
Convergence Rates
Smoothness Spaces:
Xss≥0, X0 = V, Xs ⊆ V, Yss≥0, Y0 = V ′, Ys ⊆ V ′
Regularity:
A−1 : Ys ∋ f → u ∈ Xs, s ≥ 0.
Convergence Rate:
‖u(ω)− uL(ω)‖V ≤ CΦ(s,NL) ‖f‖Ys where Φ(s,Nℓ) := supv∈Xs
infvℓ∈Sℓ
‖v − vℓ‖V‖v‖Xs
.
MC Galerkin: given f(ωj) : j = 1, ...,M, compute uL(ωj) : j = 1, ...,M and sample average
EM,L
Mku:=
1
M
M∑
j=1
uL(ωj)⊗ ...⊗ uL(ωj)︸ ︷︷ ︸k−times
∈ S(k)L .
Work:
O(MNkL) where NL = dimSL DOFs for “mean field” problem.
WaveletFEM
Wavelet FEM (Cohen, Dahmen, Kunoth, Schneider, ...)
Wavelet Scale:
W0 := S0, Sℓ = Sℓ−1 ⊕Wℓ, ℓ = 1, 2, . . . ,
Sparse Tensor Product Space (Smol’yak, Teml’yakov, Zenger, Griebel,...):
V(k)L =
∑
~ℓ∈Nk0|~ℓ|≤L
Wℓ1 ⊗Wℓ2 ⊗ · · · ⊗Wℓk .
Sparse Projection (quasi-interpolation):
P(k)L : V (k) → V
(k)L given by (P
(k)L v)(x) :=
∑
0≤ℓ1+···+ℓk≤L1≤jν≤nℓν ,ν=1,...,k
vℓ1...ℓjj1...jk
ψℓ1j1 (x1) . . . ψℓkk (xk)
or
P(k)L =
∑
0≤ℓ1+···+ℓk≤LQℓ1 ⊗ · · · ⊗Qℓk where Qℓ := Pℓ − Pℓ−1, ℓ = 0, 1, ... and P−1 := 0.
Wavelets
Biorthogonal Spline Wavelets in 1− d, degree p = 1.
V3
(Nodal basis)
W0
ψ10
W1
ψ11 ψ
21
W2
ψ12 ψ
22 ψ
32 ψ
42
W3
ψ13 ψ
23 ψ
33 ψ
43 ψ
53 ψ
63 ψ
73 ψ
83
WaveletFEM
Sparse Tensor Product Space(Zenger 1990, Griebel & Bungartz Acta Numerica 2004)
x
y
W0⊗ W
0W
1⊗ W
0
W0⊗ W
1
MCWaveletFEM
Monte Carlo IV – Sparse Monte Carlo FEM
Sparse Tensor Product MC estimate of Mku :
EM,L[Mku] :=1
M
M∑
j=1
P(k)L [uL(ωj)⊗ ...⊗ uL(ωj)] ∈ S
(k)L .
Work:
O(MNL(log2NL)k−1) operations and NL(log2NL)
k−1 memory
Theorem 5Assume 1 < α ≤ 2 and
f ∈ Lk(Ω, Ys) ∩ Lαk(Ω, V ′) for some 0 ≤ s < s0.
Then
Mku ∈ Xs ⊗ ...⊗Xs =: X(k)s
and there is C(k) > 0 such that for all M ≥ 1, L ≥ L0 and all 0 < ε < 1 holds
P(‖Mku− EM,L[Mku]‖V (k) < λ
)≥ 1− ε
with λ = C(k)[Φ(s,NL)(logNL)
(k−1)/2 ‖f‖kLk(Ω,Ys) + ε−1/αM−(1−1/α) ‖f‖kLαk(Ω,V ′)
].
SparseGalerkinFEM
Sparse Tensor FEM
Idea: A linear and deterministic allows to
Compute Mku directly, without MC
Proposition 6Assume A satisfies (4), (5) and that f ∈ Lk(Ω, V ′) for k > 1.
Then
(A⊗ ...⊗ A)Z = Mkf , (19)
has a unique solution Z ∈ V (k) and
Z = Mku.
For f ∈ Lk(Ω, Ys), s > 0, holds
‖Mku‖Xs⊗...⊗Xs ≤ Ck,s ‖Mkf‖Ys⊗...⊗Ys, 0 ≤ s < s0, k ≥ 1
Regularity of Mku in spaces of mixed highest derivative!
SparseGalerkinFEM
Sparse Tensor FEM
Theorem 7
Then for all L ≥ kL0 sparse Galerkin approximation ZL of Mku is uniquely defined and
‖Mku− ZL ‖V⊗...⊗V ≤ C(k)hsL| log hL|(k−1)/2‖f‖Lk(Ω,Ys) ∼ N−sL logbNL‖f‖Lk(Ω,Ys) 0 ≤ s < s0.
ZL can be computed in O(NL(logNL)k+1) work and memory.
Note: Full Tensor Galerkin FEM gives
‖Mku− ZL ‖V⊗...⊗V ≤ C(k)hsL‖f‖Lk(Ω,Ys) ∼ N−s/kL ‖f‖Lk(Ω,Ys) 0 ≤ s < s0
ZL can be computed in O(NkL) work and memory.
SparseGalerkinFEM
Multilevel MC (for Hyperbolic Conservation Laws)
FVM: based on Tℓ∞ℓ=0 nested triangulations of D ⊂ Rd such that mesh width
∆xℓ = ∆x(Tℓ) = supdiam(K) : K ∈ Tℓ = O(2−ℓ∆x0), ℓ ∈ N0 (20)
Assume shape regular cells, CFL condition and
Sℓ := S(Tℓ), Pℓ := PTℓ, Tℓ ∈ M, ℓ = 0, 1, ... . (21)
generate a sequence of stable FV approximations, vℓ(·, t)∞ℓ=0 on Tℓ for a number of time steps of sizes ∆tℓadapted to grid Tℓ ∈ M. Set in what follows v−1(·, t) := 0.
Then, given level L ∈ N of spatial resolution, by the linearity of the expectation
E[vL(·, t)] = E
[ L∑
ℓ=0
(vℓ(·, t)− vℓ−1(·, t))]. (22)
SparseGalerkinFEM
Multilevel MC (for Hyperbolic Conservation Laws)
Estimate each term in (22) statistically by a MCM with a level-dependent number of samples, Mℓ; this gives
the MLMC estimator
EL[u(·, t)] =L∑
ℓ=0
EMℓ[vℓ(·, t)− vℓ−1(·, t)] (23)
where EM [vℓ(·, t)] is defined by
M1(u(·, t)) ≈ EM [vℓ(·, t)] :=1
M
M∑
i=1
viℓ(·, t) , (24)
and, for k > 1, the kth moment (or k point correlation function) Mk(u(·, t)) = E[(u(·, t))(k)] is estimated by the
sparse tensor MLMC estimate of Mk[u(·, t)] defined by
EL,(k)[u(·, t)] :=L∑
ℓ=0
EMℓ[Pℓ
(k)(vℓ(·, t))(k) − Pℓ−1
(k)(vℓ−1(·, t))(k)] . (25)
SparseGalerkinFEM
Multilevel MC (for Hyperbolic Conservation Laws)
Error Bounds: Assume
u0 ∈ L2k(Ω;W s,1(Rd)) for some 0 < s < 1 .
The MLMC-FVM estimate EL,(k)[u(·, t)] in (25) satisfies, for every sequence MℓLℓ=0 of MC samples,
‖Mku(·, t)− EL,(k)[u(·, t;ω)]‖L2(Ω;L1(Rkd))
. (1 ∨ t)∆xsL| log∆xL|k−1‖TV(u0(·, ω))‖kLk(Ω;dP) + ‖u0(· ;ω)‖kL∞(Ω;W s,1(Rd))
+
L∑
ℓ=0
∆xsℓ| log∆xℓ|k−1
M1/2ℓ
‖u0(· ;ω)‖kL2k(Ω;W s,1(Rd))
+ t‖TV(u0(· ;ω))‖kL2k(Ω;dP).
The total work to compute the MLMC estimates EL,(k)[u(· ; t)] on compact domains D ⊂ Rd is therefore (with
O(·) depending on the size of D)
WorkMLMC
L = O
(L∑
ℓ=0
Mℓ∆x−(d+1)ℓ | log∆x|k−1
).
SparseGalerkinFEM
Multilevel MC (for Hyperbolic Conservation Laws)
Choice of samples?
M−1
2ℓ ∆xsl
!= C∆xsL, ℓ = 0, . . . , L .
Then
‖Mku(·, t)− EL,(k)[u(·, t;ω)]‖L2(Ω;L1(Dk)) ≤ C(WorkMLMC
L )−s′/(d+1)
for any 0 < s′ < s with the constant depending on D and growing as 0 < s′ → s ≤ 1.
Same accuracy vs. Work as a single deterministic FVM run.
SparseGalerkinFEM
FoSM on Domains with Stochastic Boundaries
Dirichlet Problem:
−∆u = f in D, u = g on ∂D.
How does u depend on D?
Boundary Perturbation of amplitude ε > 0 in direction U:
U(x) : ∂D → R3, ||U||2 = 1, U(x) · n(x) > 0
∂Dε := x + εU(x) : x ∈ ∂D , Dε := interior∂Dε
Specifically:
U(x) = κ(x)n(x)
with κ(x) ∈ C4(∂D,R)
Dirichlet Problem on perturbed domain:
−∆uε = f in Dε, uε = g on ∂Dε.
Idea: for ε > 0 sufficiently small
uε = u + εdu[U] +ε2
2d2u[U,U] +O(ε3)
FoSM on Domains with Stochastic Boundaries
Thm (Hadamard 1909, F. Murat & J. Simon (1976), J. Sokolowski & P. Zolesio,J. Simon (1980)):
u depends Frechet-differentiably on D.
The first derivative of u w.r. to D, the local shape derivative du[U], is solution of the Dirichlet problem
∆du = 0 in D, du = 〈∇(g − u),U〉 = 〈U,n〉∂(g − u)
∂non ∂D
where u is the solution of the Dirichlet Problem on D.
Shape Hessian: bilinear form on pairs of boundary perturbation fields (U,U′), denoted by
d2u = d2u[U,U′].
It is obtained from the Dirichlet problem (Hettlich & Rundell SINUM 2000, Eppler 2003):
∆d2u = 0 in D,
d2u = 〈H[g − u])U′,U〉 − 〈∇du[U],U′〉 − 〈∇du[U′],U〉 on ∂D.
FoSM on Domains with Stochastic Boundaries
Random domain variation:
U(x, ω) = κ(x, ω)n(x),
where κ is P-measurable and
κ(x, ω) : Ω → X = Ck(∂D,R), k = 4.
Finite second moments of κ(x, ω) in X with respect to P :
Eκ(x) :=
ˆ
Ω
κ(x, ω)dP(ω) = E(κ(x, ω)
)= 0, x ∈ ∂D,
and
Covarκ(x,y) :=
ˆ
Ω
κ(x, ω)κ(y, ω)dP(ω) = E(κ(x, ω)κ(y, ω)
), x,y ∈ ∂D,
of κ(x, ω) exist and are known:
Eκ = 0 =⇒ Covarκ = M2[κ].
FoSM on Domains with Stochastic Boundaries
Lemma 8
For sufficiently small ε > 0, uε(ω)
uε(z, ω) = u(z) + εdu(z, ω) +ε2
2d2u(z, ω) +O(ε3) for P − a.e. ω ∈ Ω,
where u ∈ H1(D) solves the deterministic Dirichlet problem
−∆u = f in D, u = g on ∂D,
where
du(z, ω) := du[κ(·, ω)n](z),and
d2u(z, ω) := d2u[κ(z, ω)n(z), κ(z′, ω)n(z′)]|z′=z.
The remainder term is O(ε3) for P -a.e. ω ∈ Ω.
FoSM on Domains with Stochastic Boundaries
How to compute second moments of u(x, ω)?
It holds
E(du(z, ω)
)= 0.
and, for ε > 0 sufficiently small,
Eu(z) = u(z) +O(ε2), z ∈ Dε.
and Varu(z) satisfies
Varu(z) = ε2Var(du(z, ω)
)+O(ε3) = ε2E (du(z, ω)2) +O(ε3).
How to compute E(du(z, ω)2
)deterministically?
Since E(du(x, ω)) = 0,
Var(du(z, ω)
)= Covar
(du(z, ω), du(z′, ω)
)∣∣z=z′.
Approximate Var(du(z, ω)
)by trace of two-point correlation of shape gradient du in the
“random” direction κ(x, ω)n(x).
FoSM on Domains with Stochastic Boundaries
Theorem 9
Covardu(z, z′) := Covar
(du(z, ω), du(z′, ω)
)
is the unique solution in H1,1(D ×D) of the tensor product boundary value problem on D ×D ⊂ R2n
(∆z ⊗∆z′) Covardu(z, z′) = 0, z, z′ ∈ D,
Covardu(x,y) = Covarκ(x,y)
[∂(g − u)
∂n(x)⊗ ∂(g − u)
∂n(y)
], x,y ∈ ∂D.
Moreover, Covardu ∈ Hs+1/2,s+1/2(D ×D) provided that ∂(g − u)/∂n ∈ Hs(∂D) for some s ≥ 1/2.
Figure 1: The domain D and the potential evaluation points.
FoSM on Domains with Stochastic Boundaries
Numerical Results
f = 1, g = −x2/2, u = −x2/2, Covarκ = xyz exp(−x2 − y2 − z2)
J NJ ‖ρ− ρJ‖L2(∂D) ‖u− uJ‖∞ cpu-time1 24 2.9e-1 5.6e-1 12 96 3.5e-1 (0.8) 5.1e-2 (11) 13 384 1.7e-1 (2.1) 2.0e-2 (2.5) 24 1536 8.4e-2 (2.0) 3.4e-3 (5.9) 95 6144 4.2e-2 (2.0) 4.4e-4 (7.9) 476 24576 2.1e-2 (2.0) 9.1e-5 (4.8) 4137 98304 1.0e-2 (2.0) 1.6e-5 (5.6) 20028 393216 1.3e-2 (2.0) 3.7e-6 (4.3) 13097
Table 1: Numerical results with respect to the mean field equation.
102
103
104
105
10−6
10−5
10−4
10−3
10−2
10−1
100
Number of Unknowns
Mea
n Fi
eld
Equ
atio
n
L2−Error of Normal DerivativeSlope 0.50PotentialSlope 1.22
Figure 2: Asymptotic behaviour of the errors for the mean field equation.
FoSM on Domains with Stochastic Boundaries
J NJ ‖QJ − QJ‖L2(∂D×∂D) ‖ΣJ − ΣJ‖L2(∂D×∂D) ‖CJ − CJ‖∞ cpu-time1 252 1.2e-1 1.3e-1 1.6 12 1440 3.4e-1 (0.4) 7.3e-1 (0.2) 2.0e-1 (7.8) 13 7488 2.5e-1 (1.4) 7.2e-1 (1.0) 1.7e-1 (1.2) 34 36864 9.6e-2 (2.6) 5.2e-1 (1.4) 2.5e-2 (6.6) 145 175104 2.8e-2 (3.5) 4.2e-1 (1.2) 8.5e-3 (3.0) 1246 811008 8.8e-3 (3.1) 3.7e-1 (1.1) 1.0e-3 (8.6) 12107 3.7 mio 4.2e-3 (2.1) 3.1e-1 (1.2) 1.6e-4 (6.4) 3 hrs8 16.5 mio 2.1e-3 (2.0) 3.3e-1 (0.9) 9.4e-5 (1.6) 24 hrs
Table 2: Errors in the covariance approximation by the sparse tensor product approach.
102
103
104
105
10−4
10−3
10−2
10−1
100
Number of Unknowns
Spa
rse
Grid
Err
or
L2−Error of Right Hand SideSlope 0.66
L2−Error of DensitySlope 0.11
L∞−Error of PotentialSlope 1.03
Figure 3: Asymptotic behaviour of the errors of the sparse tensor product approach.
Conclusions
• Monte-Carlo, MLMC for Galerkin FEM and FVM: framework, convergence analysis.
Error bounds in probability (L1), mean square (L2) and P-a.s.
For low order discretizations in physical space, MLMC optimal (also vs. gpc methods).
• Sparse Tensor Galerkin FEM for k-point correlations:
regularity in anisotropic spaces; sparse tensor product spaces,
log-linear complexity of k-point correlation computations.
• Given data statistics, get solution statistics by deterministic computation
• trade stochasticity and MC for high-dimensionality + deterministic FEM
• Use sparse tensor products of wavelet spaces to avoid O(NkL) complexity
• Fast Matrix Vector Multiplication (Sch. & Todor: Numer. Math. 2003)
• a-priori and a-posteriori error estimates, adaptivity: for elliptic PDEs
→ framework of Cohen, Dahmen, DeVore in tensor product Besov spaces
(P.A. Nitsche: Constr. Approx. 2006, Stevenson and Sc.: MathComp 2008)
Mk(u) ∈ Bαq (Lq(D))⊗q . . .⊗q B
αq (Lq(D))
for arbitrarily large α with
q = [α/2 + 1/2]−1 < 1 indep. of k.
• nonlinear (Frechet-differentiable) problems: First Order Second Moment (FoSM)
– linearize around “nominal” solution,
– get 2nd order statistics of random solution from gradient and Hessian at “nominal” solution,
– Sparse Tensor Discretization of FoSM problems
(H. Harbrecht, R. Schneider and Sch. Numer. Math. (2008))
• Wavelets in physical domain (e.g. D, Γ) essential?
No, any hierarchic basis [BPX, spectral, hp] will work;
for PDEs frames are sufficient... :
H. Harbrecht, R. Schneider and Ch. Sch. (Numer. Math. (2008)),
p-FEM, Spectral FEM: A. Chernov and Ch. Sch.: AppNum. (2009) .