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) .