galerkin methods for stochastic elliptic partial ... · galerkin methods for stochastic elliptic...

Galerkin Methods for Stochastic Elliptic PartialDifferential Equations of Flow through Porous Media

Hermann G. Matthiesgemeinsam mit Andreas Keese

Institut fur Wissenschaftliches RechnenTechnische Universitat Braunschweig

[email protected]://www.wire.tu-bs.de

2

Why Probabilistic or Stochastic Models?

Many descriptions (especially of future events) contain elements,which are uncertain and not precisely known.

• Future rainfall, or discharge from a river.

• More generally, the action from the surrounding environment.

• The system itself may contain only incompletely known parame-ters, processes or fields (not possible or too costly to measure)

• There may be small, unresolved scales in the model, they act as akind of background noise.

All these items introduce a certain kind of uncertainty in the model.

TU Braunschweig Institute of Scientific Computing

3

Ontology and Modelling

A bit of ontology:

• Uncertainty may be aleatoric, which means random and not redu-cible, or

• epistemic, which means due to incomplete knowledge.

Stochastic models give quantitative information about uncertainty.They can be used to model both types of uncertainty.

Possible areas of use: Reliability, heterogeneous materials, upscaling,incomplete knowledge of details, uncertain [inter-]action with envi-ronment, random loading, etc.


4

Quantification of Uncertainty

Uncertainty may be modelled in different ways:

Intervals / convex sets do not give a degree of uncertainty, quantification only throughsize of sets.

Fuzzy and possibilistic approaches model quantitative possibility with certain rules.Mathematically no measure.

Evidence theory models basic probability, but also (as a generalisation) plausability (akind of lower bound) and belief (a kind of upper bound) in a quantitative way.Mathematically no measures.

Stochastic / probabilistic methods model probability quantitatively, have mostdeveloped theory.


5

Physical Models

Models for a system S may be stationary with state u, exterior action f and randommodel description (realisation) ω, which may include random fields

S(u, ω) = f(ω).

Evolution in time may be discrete (e.g. Markov chain), may be driven by discreterandom process

un+1 = F(un, ω),

or continuous, (e.g. Markov process, stochastic differential equation), may be driven byrandom processes

du = (S(u, ω)− f(ω, t))dt + B(u, ω)dW (ω, t) + P(u, ω)dQ(ω, t)

In this Ito evolution equation, W (ω, t) is the Wiener process, and Q(ω, t) is the(compensated) Poisson process.


6

References (Incomplete)

Formulation of PDEs with random coefficients,i.e. Stochastic Partial Differntial Equations (SPDEs):Babuska, Tempone; Glimm; Holden, Øksendal; Xiu, Karniadakis; M., Keese; Schwab, Tudor

Spatial/temporal expansion of stochastic processes/ random fields:Adler; Fourier; Karhunen, Loeve; Kree; Wiener

White noise analysis/ polynomial chaos/ multiple Ito integrals:Cameron, Martin; Hida, Potthoff; Holden, Øksendal; Ito; Kondratiev; Malliavin; Wiener

Galerkin methods for SPDEs:Babuska, Tempone; Benth, Gjerde; Cao; Ghanem, Spanos; Xiu, Karniadakis; M., Keese

Adjoint Methods:Giles; Johnson; Kleiber; Marchuk; M., Meyer; Rannacher, Becker


7

Statistics

Desirable: Uncertainty Quantification or Optimisation under uncertainty:

The goal is to compute statistics which are—deterministic—functionals of the solution:

Ψu = 〈Ψ(u)〉 := E (Ψ(u)) :=∫

Ω

Ψ(u(ω), ω) dP (ω)

e.g.: u = E (u), or varu = E((u)2

), where u = u− u, or Pru ≤ u0 = E

(χu≤u0

)Principal Approach:

1. Discretise / approximate physical model (e.g. via finite elements, finite differences),and approximate stochastic model (processes, fields) in finitely many independentrandom variables (RVs), ⇒ stochastic discretisation.

2. Compute statistics:• Via direct integration (e.g. Monte Carlo, Smolyak (= sparse grids)). Each

integration point requires one PDE solution (with rough data).• Or approximate solution with some response-surface, then integration by

sampling a “cheap” expression at each integration point.


8

Model Problem

Aquifer

0

0.5

1

1.5

2

00.5

11.5

2

Geometry

2D Model

simple stationary model of groundwater flow (Darcy)

−∇ · (κ(x)∇u(x)) = f(x) & b.c., x ∈ G ⊂ Rd

−κ(x, u)∇u(x) = g(x), x ∈ Γ ⊂ ∂G,

u hydraulic head, κ conductivity, f and g sinks and sources.


9

Stochastic Model

• Uncertainty of system parameters—e.g. κ = κ(x, ω) = κ(x) + κ(x, ω),f = f(x, ω), g = g(x, ω) are stochastic fields

ω ∈ Ω = probability space of all realisations, with probability measure P .

• Assumption: 0 < κ0 ≤ κ(x, ω) < κ1. Possibilities: Bounded distributions

(such as beta), or transformation / translationof Gaussian field γ

κ(x, ω) = φ(x, γ(x, ω)) := F−1κ(x)(Φ(γ(x, ω)))

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

e.g. log-normal distribution (but this violates assumption)

κ(x, ω) = a(x) + exp(γ(x, ω))

,


10

Realisation of κ(x, ω)


11

Stochastic PDE and Variational Form

Insert stochastic parameters into PDE ⇒ stochastic PDE

−∇ · (κ(x, ω)∇u(x, ω)) = f(x, ω), x ∈ G & b.c. on ∂G

⇒ Solution u(x, ω) is a stochastic field—in tensor product form

W⊗ S 3 u(x, ω) =∑

µ

vµ(x)⊗ u(µ)(ω) =∑

µ

vµ(x)u(µ)(ω)

W is a normal spatial Sobolev space, and S a space of random variables,

e.g. S = L2(Ω, P ). Then W⊗ S ≡ L2(Ω, P ;W).

Variational formulation: Find u ∈ W⊗ S, such that ∀v ∈ W⊗ S :

a(v, u) :=∫

Ω

∫G

∇v(x, ω) · (κ(x, ω)∇u(x, ω)) dx dP (ω) =∫Ω

[∫G

v(x, ω)f(x, ω) dx +∫

∂G

v(x, ω)g(x, ω) dS(x)]

dP (ω) =: 〈〈f, v〉〉.


12

Mathematical Results

To find a solution u ∈ W⊗ S such that for ∀v : a(v, u) = 〈〈f, v〉〉

• is guaranteed by Lax-Milgram lemma, problem is well-posed in the sense ofHadamard (existence, uniqueness, continuous dependence on data f, g in L2 and onκ in L∞-norm).

• may be approximated by Galerkin methods, convergence established with Cea’slemma

• Galerkin methods are stable, if no variational crimes are committed

Good approximating subspaces of W⊗ S have to be found, as well as efficientnumerical procedures worked out.

Different ways to discretise: Simultaneously in W⊗ S, or first in W (spatial), or first inS (stochastic).


13

Computational Approach

Principal Approach:

1. Discretise spatial (and temporal) part (e.g. via finite elements, finite differences).

2. Represent computationally stochastic fields which are input to problem.

3. Compute solution/statistics:• Directly via high-dimensional integration (e.g. Monte Carlo, Smolyak, FORM).• Or approximate solution with response-surface, then integration

Variational formulation discretised in space, e.g. via finite element ansatzu(x, ω) =

∑n`=1 u`(ω)N`(x) = [N1(x), . . . , Nn(x)][u1(ω), . . . , un(ω)]T = N(x)Tu(ω):

K(ω)[u(ω)] = f(ω).

Remark on Computing with Averages: 〈Ku〉 = 〈f〉 6= 〈K〉〈u〉,as K and u are not independent, but if K and f are independent

〈u〉 = 〈K−1f〉 = 〈K−1〉〈f〉 ⇒ 〈K−1〉−1〈u〉 = 〈f〉


14

Example Solution

0

0.5

1

1.5

2

0

0.5

1

1.5

2

Geometry

flow out

Dirichlet b.c.

flow = 0 Sources

7

8

9

10

11

12

0

1

2

0

1

2

5

10

15

Realization of κ

5.5

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6.5

7

7.5

8

8.5

9

9.5

10

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1

2

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1

2

4

6

8

10

Realization of solution

4

5

6

7

8

9

10

0

1

2

0

1

2

0

5

10

Mean of solution

1

2

3

4

5

0

1

2

0

1

2

0

2

4

6

Variance of solution

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

0

0.2

0.4

0.6

0.8

y

x

Pru(x) > 8


15

First Summary

• Probabilistic formulation to quantify uncertainty, or find variation / distribution ofresponse,

• Mathematical formulation in variational form gives well-posed problem

• Numerical approach may first perform familiar space / time semi-discretisation,

• Computational representation of input stochastic fields is needed,

• Statistics / Expectations may be computed

– either through direct integration with various methods(best known is Monte Carlo)

– or indirectly by first computing representation of solution u(ω) in terms ofstochastic input, and then perform integration


16

Computational Requirements

• How to represent a stochastic process for computation, both simulation orotherwise?

• Best would be as some combination of countably many independent randomvariables (RVs).

• How to compute the required integrals or expectations numerically?

• Best would be to have probability measure as a product measure P = P1⊗ . . .⊗P`,then integrals can be computed as iterated one-dimensional integrals via Fubini’stheorem, ∫

Ω

Ψ dP (ω) =∫

Ω1

. . .

∫Ω`

Ψ dP1(ω1) . . . dP`(ω`)


17

Random Fields

Consider region in space G, a random field is a RV κx at each x ∈ G,

alternatively for each ω ∈ Ω a random function—a realisation— κω(x) on G

Mean κ(x) = E (κω(x))—now a function of x—and fluctuating part κ(x, ω).

Covariance may be considered at different positions Cκ(x1, x2) := E (κ(x1, ·)⊗ κ(x2, ·))

If κ(x) ≡ κ, and Cκ(x1, x2) = cκ(x1 − x2), process is (weakly) homogeneous.

Here representation through spectrum as a Fourier sum or integral is well known.

We have to deal with another function (RV) at each point x!

• Need to discretise spatial aspect (generalise Fourier representation).One possibility is the Karhunen-Loeve Expansion.

• Need to discretise each of the random variables in Fourier synthesis.One possibility is the Polynomial Chaos Expansion.


18

Karhunen-Loeve Expansion I

−1

−0.5

0

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1

−1

−0.5

0

0.5

1

0

0.2

0.4

0.6

0.8

1

KL−mode 1

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

−2

−1.5

−1

−0.5

0

0.5

1

KL−mode 15

Eigenvalue-Problem for Karhunen-Loeve Expansion KLE. Other names:Proper Orth. Decomp.(POD), Sing. Value Decomp.(SVD), Prncpl. Comp. Anal.(PCA):∫

G

Cκ(x, y)g(y) dy = κ2g(x)

gives spectrum κ2 and orthogonal KLE eigenfunctions g(x) ⇒ Representation of κ:

κ(x, ω) = κ(x) +∞∑

=1

κ g(x)ξ(ω) =:∞∑

=0

κ g(x)ξ(ω) =∞∑

=0

κ g(x)⊗ ξ(ω)

with centred, uncorrelated random variables ξ(ω) with unit variance.Truncate after m largest eigenvalues ⇒ optimal—in variance—expansion in m RVs.A sparse representation in tensor products.


19

Karhunen-Loeve Expansion II

Singular Value Decomposition: To every random field with vanishing meanw(x, ω) ∈ W⊗ L2(Ω) (W Hilbert) associate a linear map W : W → L2(Ω).

W : W 3 v 7→ W(v)(ω) = 〈v(·), w(·, ω)〉W =∫

G

v(x)w(x, ω) dx ∈ L2(Ω).

KLE is SVD of the map W, the covariance operator is Cw := W∗W, and ∀u, v ∈ W :

〈u, Cwv〉W = 〈W(u),W(v)〉L2(Ω) =

E (W(u)W(v)) = E(∫

G

u(x)w(x, ω) dx

∫G

v(y)w(y, ω) dy

)=∫

G

∫G

u(x)E (w(x, ω)w(y, ω)) v(y) dy dx =∫

G

u(x)∫

G

Cw(x, y)v(y) dy dx

The covariance operator Cw is represented by the covariance kernel Cw(x, y).

Truncating the KLE is therefore the same as what is done when truncating a SVD,finding a sparse representation.


20

Representation in Gaussian Random Variables

How to handle the RVs ξ(ω) in the KLE?

In L2(Ω, P ) take the Hilbert subspace Θ of centred Gaussian RVs.Let θk(ω) be an orthonormal (uncorrelated and unit variance) basis in theGaussian Hilbert space Θ. For Gaussian RVs: uncorrelated ⇒ independent.

On a m-dimensional subspace Θ(m) = spanθ1, . . . , θm we have(with the notation θm(ω) = (θ1(ω), . . . , θm(ω)) ∈ Rm):

Θ(m) can be represented by Rm with a Gaussian product measure Γm:

dΓm(θm) = (2π)−m/2 exp(−|θm|2/2) dθm

=m∏

=1

dΓ1(θ) =m∏

=1

(2π)−1/2 exp(−θ2/2) dθ

Represent ξ(ω) as functions of those Gaussian RVs, i.e. as functions of basis.


21

Polynomial Chaos Expansion

As Θ ⊂ Lp(Ω) 1 ≤ p < ∞, finite products of Gaussians are in L2(Ω):

Theorem[Polynomial Chaos Expansion]: Any RV r(ω) ∈ L2(Ω, P ) can be represented inorthogonal polynomials of Gaussian RVs θm(ω)∞m=1 =: θ(ω):

r(ω) =∑α∈J

%(α)Hα(θ(ω)), with Hα(θ(ω)) =∞∏

=1

hα(θ(ω)),

where J := N(N)0 = α |α = (α1, . . . , α, . . .), α ∈ N0, |α| :=

∑∞=1 α < ∞,

are multi-indices, where only finitely many of the α are non-zero,and h`(ϑ) are the usual Hermite polynomials. The series converges in L2(Ω).In other words, the Hα are an orthogonal basis in L2(Ω),with 〈Hα,Hβ〉L2(Ω) = E (HαHβ) = α! δαβ, where α! :=

∏∞=1(α!).

Hn = spanHα : |α| = n is homogeneous chaos of degree n, and L2(Ω) =⊕∞

n=0Hn.

Especially each ξ(ω) =∑

α ξ(α) Hα(θ(ω)) from KLE may be expanded in PCE.


22

Second Summary

• Representation in a countable number of uncorrelated ⇒ independent Gaussian RVs

– Allows direct simulation—independent Gaussian RVs in computer by randomgenerators (quasi-random), or direct hardware (e.g. sample noise from sounddevice).

– Each finite dimensional joint Gaussian measure Γm(θm) is a product measure,allows Fubini’s theorem (Γm(θ) =

∏m`=1 Γ1(θ`)) .

– Polynomial expansion theorem allows approximation of any random variablewhich depends on this representation, especially also the output / responseu(x, ω) = u(x, θ), resp. in the spatially discretised case u(ω) = u(θ).

• Still open:

– How to actually compute u(θ) ?– How to perform integration ?– In which order ?


23

Computational Approaches

The principal computational approaches are:

Direct Integration (Monte Carlo) Directly compute statistic by quadrature:Ψu = E (Ψ(u(ω), ω)) =

∫Θ

Ψ(u(θ),θ) dΓ (θ) by numerical integration.

Perturbation Assume that stochastics is a small perturbation around mean value, doTaylor expansion and truncate—usually after linear or quadratic term.

Response Surface Try to find a functional fit u(θ) ≈ v(θ), then compute with v.Integrand is now “cheap”.

Stochastic Galerkin Use an ansatz for the solution u(θ) =∑

β u(β)Hβ(θ) in the

stochastic dimension, then do Galerkin method to obtain u(β).

Observe that this is one possible functional form of response surface.


24

Stability Issues

For direct integration approaches direct expansion (both KLE and PCE) pose stabilityproblems: Both expansions only converge in L2, not in L∞ (uniformly) as required ⇒spatially discrete problems to compute u(θz) for a specific realisation θz may not bewell posed.

Convergence of KLE may be uniform if covariance Cκ(x1, x2) smooth enough.

E.g. not possible for Cκ(x1, x2) = exp(a|x1 − x2|+ b)⇒ here for truncated KLE there are always regions in space where κ is negative.

Truncation of PCE gives a polynomial, as soon as one α is odd, there are θ valueswhere κ is negative

— compare approximating exp(ξ) with a truncated Taylor poplynomial at odd power.

This can not be repaired. Like negative Jacobian in normal FEM.

For methods which directly require u(θz) for a specific realisation, transformation

method κ(x, ω) = φ(x, γ(x, ω)) possible with KLE of Gaussian γ(x, ω).


25

Stochastic Galerkin I

Recipe: Stochastic ansatz and projection in stochastic dimensions

u(θ) =∑

β

u(β)Hβ(θ) = [. . . , u(β), . . .][. . . , Hβ(θ), . . .]T = uH(θ)

Goal: Compute coefficients u(β)

1. Directly through projection as explained, has problems.

2. Through stochastic Galerkin Methods (weighted residuals),

∀α : E(Hα(θ)(f(θ)−K(θ)u(θ))

)= 0,

requires solution of one huge system, only integrals of residua

3. For stochastic Galerkin methods the convergence issues of KLE and PCE may besolved. A (forgivable) variational crime like numerical integration in normal FEM.


26

Stochastic Galerkin II

Of course we can not use all α ∈ J , but we limit ourselves to a finite subset

Jk,m = α ∈ J | |α| ≤ k, ı > m ⇒ αı = 0 ⊂ J .

Let Sk,m = spanHα : α ∈ Jk,m,

then dim Sk,m =(

m + p + 1p + 1

)

m k dim Sk,m

3 3 355 84

5 3 1265 462

10 3 10015 8008

20 3 106265 230230

10 ≈ 8.5 · 107

100 3 ≈ 4.6 · 106

5 ≈ 1.7 · 109

10 ≈ 4.7 · 1014


27

Third Summary

• Integration may require many evaluations of integrand, if combined with expensiveto evaluate integrand ⇒ tremendous effort (this hits direct Monte Carlo).

• Direct integration and response surface method is conceptually simple, but mayhave stability problems.

• Stochastic Galerkin methods require solution of huge system, and a bit morechanges to existing programs.

• Stochastic Galerkin methods require only much cheaper integrands of residua.

• Stochastic Galerkin methods are stable (here (variational) crime pays).


28

Results of Galerkin Method

err. ·104

in meanm = 6k = 2

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

0

1

2

3

4

y

x

err. ·104

in std devm = 6k = 2

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

0

0.1

0.2

0.3

0.4

y

x

u(α) forα = (0, 0, 0, 1, 0, 0).

−1−0.8−0.6−0.4−0.200.20.40.60.81

−1−0.8

−0.6−0.4

−0.20

0.20.4

0.60.8

1

0

0.02

0.04

0.06

0.08

y

x

Error·104 inu(α)

Galerkinscheme.

−1−0.8−0.6−0.4−0.200.20.40.60.81

−1−0.8

−0.6−0.4

−0.20

0.20.4

0.60.8

1

−0.50

0.51

1.52

2.53

y

x


29

Galerkin-Methods for the Linear Case

∀α satisfy:

∑β

[∫G

∇N(x) E (Hα(θ)κ(x,θ)Hβ(θ))∇N(x)T dx

]u(β) = E (f(θ)Hα(θ))︸︷︷︸

=:α! f (α)

More efficient representation through direct expansion of κ in KLE and PCEand analytic computation of expectations.

κ(x,θ) =∞∑

=0

κ ξ(θ)g(x) ≈r∑

=0

∑γ∈J2k,m

κ ξ(γ) Hγ(θ)g(x).


30

Resulting Equations

Insertion of expansion of κ (∀α satisfy):∑β

∑γ

∑

κ ξ(γ) E (HαHβHγ)︸︷︷︸

=:∆(γ)α,β

∫∇N(x) g(x)∇N(x)T dx︸︷︷︸

K

u(β) = α! f (α)

• K is a sparse stiffness matrix of a FEM discretisation for the material g(x).

• As →∞, κ → 0, and as |γ| → ∞, ξ(γ) → 0.

• The Hermite polynomials form an algebra, i.e. HβHγ =∑

ε c(ε)β,γHε; the c

(ε)β,γ are

known, they are the structure constants of the Hermite algebra.

• Therefore the matrix ∆(γ) = (∆(γ)α,β) may be evaluated analytically.

∆(γ)α,β = E

(Hα

∑ε

c(ε)β,γHε

)=∑

ε

c(ε)β,γ E (HαHε) =

∑ε

c(ε)β,γ α! δαε = α! c

(α)β,γ


31

Tensor Product Structure

The equations have the structure of a Kronecker or tensor product

∑γ

∑

κ ξ(γ)

∆(γ)α1,β1

K ∆(γ)α2,β1

K · · ·∆(γ)

α1,β2K ∆(γ)

α2,β2K · · ·

... ... . . .

u(β1)

...u(βN)

=

f (α1)

...f (αN)

.

or with tensors u = [u(β1), . . . ,u(βN)] and f = [f (α1), . . . ,f (αN)]

Ku =

∑

∑γ

κ ξ(γ) ∆(γ) ⊗K

u = f

• Exploit parallelism in the multiplication, parallel operator-sum

• Block-matrix efficiently stored and used in tensor-representation.

• Use sparse low-rank representation of tensors f (via KLE of RHS) and u via SVD.


32

Sparsity Structure of ∆

Non-zero blocks for increasing degree of Hγ


33

Approximation Theory

• Stability of discrete approximation under truncated KLE and PCE. Matrix staysuniformly positive definite.

• Convergence follows from Cea’s Lemma.

• Convergence rates under stochastic regularity in stochastic Hilbertspaces—stochastic regularitry theory?.

• Error estimation via dual weighted residuals possible.

Stochastic Hilbert spaces—start with formal PCE: R(θ) =∑

α∈J R(α)Hα(θ). Define

for |ρ| ≤ 1 and p ≥ 0 norm (with (2N)β :=∏

∈N(2)β):

‖R‖2ρ,p =∑α

‖R(α)‖2(α!)1+ρ(2N)pα.


34

Stochastic Hilbert Spaces—Convergence Rates

Define for 1 ≥ ρ ≥ 0, p ≥ 0:

(S)ρ,p = R(θ) =∑α∈J

R(α)Hα(θ) : ‖R‖ρ,p < ∞.

These are Hilbert spaces, the duals are denoted by (S)−ρ,−p, and L2(Ω) = (S)0,0. LetPk,m be orthogonal projection from (S)ρ,p into the finite dimensional subspace Sk,m.

Theorem: Let p > 0, r > 1 and let |ρ| ≤ 1. Then for any R ∈ (S)ρ,p:

‖R− Pk,m(R)‖2ρ,−p ≤ ‖R‖2ρ,−p+r c(m, k, r)2,

where c(m, k, r)2 = c1(r)m1−r + c2(r)2−kr.

dim Sk,m grows too quickly with k and m. Needed are sparser spaces and errorestimates for them.


35

Matrix Expansion Errors

Hγ is orthogonal on polynomials of degree < |γ|

⇒ ∆(γ)αβ = E (HαHβHγ) = 0 for |γ| > |α + β|,

hence sum over γ is finite.

K in a finite dimensional space ⇒ One can choose J such that expansion

J∑=0

∑γ

κj ξ(γ) ∆(γ) ⊗K u

is arbitrarily close to

=∑

β

∫G

∇N(x) E (κ(x,θ)Hα(θ)Hβ(θ))∇N(x)T dx u(β),

and hence global matrix uniformly positive definite. This stability argument is only validfor the Galerkin method.


36

Properties of Global Equations

Ku =∑

∑α

κ ξ(α) ∆(α) ⊗K u = f

• Each K is symmetric, and each ∆(α).⇒ Block-matrix K is symmetric.

• SPDE ist positive definite.Appropriate expansion of κ ⇒ K is uniformly positive definite ⇒ stability.

Solving the Equations:

• Never assemble block-matrix explicitly.

• Use K only as multiplication.

• Use Krylov Method (here CG) with pre-conditioner.


37

Block-Diagonaler Pre-Conditioner

Let K = K0 = stiffness-matrix for average material κ(x).

Use deterministic solver as pre-conditioner:

P =

K . . . 0... . . . ...0 . . . K

= I ⊗K

Good pre-conditioner, when variance of κ not too large.Otherwise use P = block-diag(K).This may again be done with existing deterministic solver.

Block-diagonal P is well suited for parallelisation.


38

Parallelising the Matrix-Vector Product

∀α : (K u)(α) =J∑

∑γ

∑β

κj ξ(γ) ∆(γ)

α,β ·K uβ

• K = deterministic solver.This may be a (lower-level) parallel program to do K uβ.

• Parallelise operator-sum in ⇒ several instances of deterministic solver in parallel.

• Distribute u and f ⇒ Parallelise sum in β.

• Sum in α may also be done in parallel, but usually not essential.


39

Further Sparsification

One term of matrix vector product may also be written as

κ ξ(γ) ∆(γ) ⊗Ku = κ ξ(γ)

Ku(∆(γ))T .

Use discretised version of KLE of right hand side f(ω) with KLE eigenvectors f `:

f(ω) = f +∑

`

ϕ` φ`(ω)f ` = f +∑

`

∑α

ϕ` φ(α)` Hα(ω)f `.

In particular α! f (α) =∑

` ϕ` φ(α)` f `. As ϕ` → 0, only a few ` are needed.

Let φ = (φ(α)` ), ϕ = diag(ϕ`), and f = [. . . , f `, . . .]; then the SVD of f(ω) is

f = [. . . , f (α), . . .] = fϕφT =∑

`

ϕ` f ` ⊗ φ`.

Similar sparsification needed for u via SVD with small m:

u = [. . . , u(β), . . .] = [. . . , um, . . .]diag(υm)(y(β)m )T = uυyT .

Then Ku(∆(γ))T = (K u)υ (∆(γ) y)T is much cheaper.


40

Computation of Moments

Let M(k)f = E

k times︷︸︸︷f(ω)⊗ . . .⊗ f(ω)

= E(f⊗k), or clearer (as M

(k)f is symmetric)

M(k)f = E

(fk), with M

(1)f = f and M

(2)f = Cf and symmetric tensor product .

KLE of Cf is Cf =∑

` ϕ` f ` f `. For deterministic operator K, just computeKv` = f `, and then Ku = f , and

M (2)u = Cu = E

(uk

)=∑

`

ϕ` v` v`.

As u(ω) =∑

`

∑α ϕ` φ

(α)` Hα(ω)v`, it results that

M (k)u =

∑`1≤...≤`k

k∏m=1

ϕ`m

∑α(1),...,α(k)

k∏n=1

φα(n)

`nE(Hα(1)

· · ·Hα(k)

)v`1 . . . v`k

.


41

Non-Linear Equations

Example: Use κ(x, u, ω) = a(x, ω) + b(x, ω)u2, and a, b random (similarily as before).

Space discretisation generates a non-linear equation A(∑

β u(β)Hβ(θ)) = f(θ).Projection onto PCE:

a[u] = [. . . , E

Hα(θ)A(∑

β

u(β)Hβ(θ))

, . . .] = f = fϕφT

Expressions in a need high-dimensional integration (in each iteration), e.g. Monte Carloor Smolyak (sparse grid) quadrature. After that, a should be sparsified or compressed.

The residual equation to be solved is

r(u) := f − a[u] = 0.


42

Solution of Non-Linear Equations

As model problem is gradient of some functional, use a quasi-Newton (here BFGS withline-search) method, to solve in k-th iteration

(∆u)k = −Hk−1r(u)

uk = uk−1 + (∆u)k−1

Hk = H0 +k∑

=1

(ap ⊗ p + bq ⊗ q)

Tensors p and q computed from residuum and last increment. Notice tensor productsof (hopefully sparse) tensors.

Needs pre-conditioner H0 for good convergence: May use linear solver as describedbefore, or just preconditioner (uses again deterministic solver), i.e.

H0 = Dr(u0) or H0 = I ⊗K0.


43

Fourth Summary

• Stochastic Galerkin methods work.

• They are computationally possible on todays hardware.

• They are numerically stable, and have variational convergence theory behind them.

• They can use existing software efficiently.

• Software framework is being built for easy integration of existing software.


44

Important Features of Stochastic Galerkin

• For efficency try and use sparse representation throughout: ansatz in tensorproducts, as well as storage of solution and residuum—and matrix in tensorproducts, sparse grids for integration.

• In contrast to MCS, they are stable and have only cheap integrands.

• Can be coupled to existing software, only marginally more complicated than withMCS.


45

Outlook

• Stochastic problems at very beginning (like FEM in the 1960’s), when to choosewhich stochastic discretisation?

• Nonlinear (and instationary) problems possible (but much more work).

• Development of framework for stochastic coupling and parallelisation.

• Computational algorithms have to be further developed.

• Hierarchical parallelisation well possible.


galerkin methods for stochastic elliptic partial ... · galerkin methods for stochastic elliptic...

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