Galerkin Methods for Stochastic Elliptic Partial ... · Galerkin Methods for Stochastic Elliptic...
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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
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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.
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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.
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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.
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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.
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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
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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.
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Model Problem
Aquifer
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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.
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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)))
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e.g. log-normal distribution (but this violates assumption)
κ(x, ω) = a(x) + exp(γ(x, ω))
,
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Realisation of κ(x, ω)
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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〉〉.
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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).
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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〉
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Example Solution
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Geometry
flow out
Dirichlet b.c.
flow = 0 Sources
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Realization of κ
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Realization of solution
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Mean of solution
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Variance of solution
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Pru(x) > 8
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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
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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`)
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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.
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Karhunen-Loeve Expansion I
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KL−mode 1
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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.
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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.
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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.
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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.
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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 ?
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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.
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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).
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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.
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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
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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).
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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
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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).
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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
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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.
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32
Sparsity Structure of ∆
Non-zero blocks for increasing degree of Hγ
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
.
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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.
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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.
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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.
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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.
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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.
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