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Stochastic Finite Elements andFast Iterative Solvers
David Silvester & Catherine PowellSchool of Mathematics
University of Manchester
http://www.maths.manchester.ac.uk/djs/
Silvester 2007 – p. 1/54
Joint work with
• Jonathan Boyle (EPSRC EP/C000528/1)• Oliver Ernst & Elisabeth Ullmann (TU Bergakademie
Freiberg, ARC Project 1279)
Silvester 2007 – p. 2/54
Outline
A Toy Problem (deterministic)
• P-IFISS Toolbox• HSL_MI20 Algebraic Multigrid
A Practical Problem (uncertain/stochastic)
• Monte Carlo Method• Stochastic Finite Element Method
• Dundee 1981–1987
Silvester 2007 – p. 3/54
A Toy Problem
Given boundary “pressure head” data g, and an isotropicpermeability matrix A = µI2, such that
0 < µ∗ ≤ µ(~x) ≤ µ∗ <∞ ∀~x ∈ D ⊂ R2 :
we want to compute p such that
−∇ · A∇p = 0 in D,
p = g on ΓD,
A∇p · ~n = 0 on ΓN .
Silvester 2007 – p. 4/54
A Toy Problem
Given boundary “pressure head” data g, and an isotropicpermeability matrix A = µI2, such that
0 < µ∗ ≤ µ(~x) ≤ µ∗ <∞ ∀~x ∈ D ⊂ R2 :
we want to compute p such that
−∇ · A∇p = 0 in D,
p = g on ΓD,
A∇p · ~n = 0 on ΓN .
Or alternatively, to compute the pair (~u, p) such that
A−1~u+ ∇p = 0 in D,
∇ · ~u = 0 in D,
p = g on ΓD; ~u · ~n = 0 on ΓN .Silvester 2007 – p. 4/54
Flow Problem I
Silvester 2007 – p. 5/54
Flow Problem III
Silvester 2007 – p. 6/54
Flow Problem IV
Streamlines of interpolated RT0 flux solution
Silvester 2007 – p. 7/54
PIFISSPotential (Incompressible) Flow & Iterative
Solution Software Guide ∗
David J. SilvesterSchool of Mathematics, University of Manchester
d.silvester@manchester.ac.uk.
Catherine E. PowellSchool of Mathematics, University of Manchester
c.powell@manchester.ac.uk.
Version 1.0, released 7 February 2007
Contents
1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 A Model Steady-State Diffusion Problem . . . . . . . . . . . . 34 Potential Flow Problems (Groundwater flow) . . . . . . . . . . 75 Directory structure . . . . . . . . . . . . . . . . . . . . . . . . 126 Help facility and function glossary . . . . . . . . . . . . . . . 137 Appendix: List of Test Problems . . . . . . . . . . . . . . . . . 15
∗This software was developed with support from the UK Engineering and Physical Sci-ences Research Council and the British Council under the British–German Academic Re-search Collaboration (ARC) scheme.
Silvester 2007 – p. 8/54
Mixed formulation
Define
Flux space X := ~v ∈ H(div;D) : ~v · ~n = 0 on ΓN )
Pressure space M := L2(D)
We want to find (~u, p) ∈ X ×M such that
(A−1~u, ~v) − (p,∇ · ~v) =< g, ~v · ~n > ∀~v ∈ X,
−(∇ · ~u, q) = 0 ∀q ∈M ;
with (·, ·) denoting the L2(D) inner product and < ·, · >denoting the L2(ΓD) inner product.
Silvester 2007 – p. 9/54
RT(0) mixed approximation
Silvester 2007 – p. 10/54
Discrete Formulation
Introducing the basis sets
Xh = span~φini=1, Flux basis functions;
Mh = spanψjmj=1; Pressure basis functions.
gives the finite element problem(
M BT
B 0
)(
u
p
)
=
(
g
0
)
,
in terms of the associated discrete matrices
Mij = (µ−1~φi, ~φj), local mass
Bij = −(∇ · ~φj , ψi), divergence
Silvester 2007 – p. 11/54
Fast Solver
We solve the (symmetric–) system Lx = f
(
M BT
B 0
)(
u
p
)
=
(
g
0
)
with M ∈ Rn×n, and B ∈ R
m×n, using MINRES with theblock diagonal preconditioning
P−1 =
(
M−1∗
0
0 Q−1∗
)
.
Silvester 2007 – p. 12/54
Given that the blocks M∗ (mass matrix diagonal) and Q∗
(via amg) satisfy
γ2 ≤ uTMu
uTM∗u≤ Γ2 ∀u ∈ R
n,
θ2 ≤ pTBM−1∗BTp
pTQ∗p≤ Θ2 ∀p ∈ R
m,
then the eigenvalues of the preconditioned problem,(
M BT
B 0
)(
u
p
)
= λ
(
M∗ 0
0 Q∗
)(
u
p
)
,
lie in the union of intervals that are bounded away from zeroand ±∞, independently of h and µ.
Silvester 2007 – p. 13/54
Algebraic Multigrid Ingredients (HSL_MI20)
Fine Grid : Ωh = 1, 2, . . . , NSplitting Routine : Ωh → C ∪ F
Coarse Grid : ΩH = C ⊂ Ωh
Smoother : Symmetric Gauss-Seidel
Interpolation Operator : IhH
Restriction Operator : IHh = (Ih
H)T
Coarse-grid Operator : AH = IHh A
hIhH
Convergence theory available for M-matrices.
BM−1∗BT is symmetric, positive definite and diagonally
dominant with positive diagonal entries and negativeoff-diagonal entries.
Silvester 2007 – p. 14/54
Sample eigenvalue bounds: AMG V-cycle
θ2 ≤ pTBM−1∗BTp
pTQ∗p≤ Θ2 ∀p ∈ R
m
1. µ = 1
2. µ =(
1 + 103(
x2 + y2))−1
3. µ = 10−3 in Ω∗, µ = 1 in Ω\Ω∗
1. 2. 3.
θ2 0.954 0.950 0.954Θ2 1 1 1
Silvester 2007 – p. 15/54
Sample MINRES iteration counts
h 116
132
164
1128
1. Ideal 26 26 26 26AMG 26 (0.18) 26 (0.48) 26 (1.90) 26 (9.06)
2. Ideal 26 26 26 26AMG 26 (0.20) 26 (0.65) 26 (2.57) 26 (11.57)
3. Ideal 25 25 25 25AMG 25 (0.20) 26 (0.60) 27 (2.72) 27 (10.64)
Silvester 2007 – p. 16/54
For details, check out
Catherine Powell & David SilvesterOptimal Preconditioning for Raviart-Thomas MixedFormulation of Second-Order Elliptic ProblemsSIAM J. Matrix Anal. Appl., 25, 2004.
David Silvester & Catherine PowellPIFISS Potential (Incompressible) Flow & IterativeSolution Software guide,MIMS Eprint 2007.14, February 2007.
Silvester 2007 – p. 17/54
A Toy Problem (deterministic)
• P-IFISS Toolbox• HSL_MI20 Algebraic Multigrid
A Practical Problem (uncertain/stochastic)
• Monte Carlo Method• Stochastic Finite Element Method
• Dundee 1981–1987
Silvester 2007 – p. 18/54
Waste Isolation Pilot Plant (WIPP) Site
• Operational nuclear waste repository in ChihuahuanDesert in New Mexico.
• Extensive measurements, research and testing prior tostart of operation; hence a lot of data available.
• Domain: rectangular horizontal 2D section throughmost transmissive rock layer (Culebra Dolomite)
• Stochastic data obtained from measurements ofpermeability and pressure head at 41 boreholes[LaVenue et al. (1990)] using geostatistical techniques[Cliffe et al. (2000)].
• Boundary data for pressure head on the domainboundary is obtained through a calibration process[LaVenue et al. (1990)].
Silvester 2007 – p. 19/54
0 0.5 1 1.5 2
x 104
0
0.5
1
1.5
2
2.5
3
x 104
mE
mN
WIPP Site: Bore Locations
H1H2H3
H4
H5H6
H7
H9
H10
H11
H12
H14
H15H16
H17
H18
DOE1
DOE2
P14
P15
P17
P18
WIPP12WIPP13
WIPP18WIPP19WIPP21WIPP22
WIPP25
WIPP26
WIPP27
WIPP28
WIPP30
ERDA9
CB1
ENGLE
USGS1
D268
AEC7
0 0.5 1 1.5 2
x 104
0
0.5
1
1.5
2
2.5
3
x 104 WIPP Site: Sandia Grid
mE
mN
Silvester 2007 – p. 20/54
Modelling Uncertainty
We model permeability A = µI2 as a random field withstochastic transmissivity µ(~x) ∼ T (~x, ω). More precisely,
• (Ω,B, P ) is a complete probability space;
• T : D × Ω → R is a family of random variables indexedby ~x ∈ D: or a “random field” (RF);
• ∀~x ∈ D: T (~x, ·) is a random variable (RV);
• ∀ω ∈ Ω: T (·, ω) is a realization of the transmissivity field.
Silvester 2007 – p. 21/54
Basic Definitions
〈ξ〉 :=
∫
Ω
ξ(ω) dP (ω) expected valueof RV ξ : Ω → R
T (~x) := 〈T (~x, ·)〉 meanof RF T at ~x ∈ D
covT (~x,~y) :=〈(T (~x, ·) − T (~x))(T (~y, ·) − T (~y))〉 covarianceof RF T at ~x, ~y ∈ D
varT (~x) := covT (~x, ~x) varianceof RF T at ~x ∈ D
σT (~x) :=√
varT (~x) standard deviationof RF T at ~x ∈ D.
Silvester 2007 – p. 22/54
Randomization
We restrict random variables to those with finite varianceL2
P (Ω): the space of square integrable random variableswith respect to P :
ξ : Ω → R, 〈ξ2〉 :=
∫
Ω
ξ(ω)2 dP (ω) <∞.
We look for a weak solution (~u, p) in the tensor productspace (X ×M) ⊗ L2
P (Ω).
Silvester 2007 – p. 23/54
Randomization
We restrict random variables to those with finite varianceL2
P (Ω): the space of square integrable random variableswith respect to P :
ξ : Ω → R, 〈ξ2〉 :=
∫
Ω
ξ(ω)2 dP (ω) <∞.
We look for a weak solution (~u, p) in the tensor productspace (X ×M) ⊗ L2
P (Ω). The Stochastic Mixed Formulationis:
Find (~u, p) ∈ (X ×M) ⊗ L2P (Ω) such that
〈(T−1~u, ~v)〉 − 〈(p,∇ · ~v)〉 = 〈< g, ~v · ~n >〉 ∀~v ∈ X ⊗ L2P (Ω),
−〈(q,∇ · ~u)〉 = 0 ∀q ∈M ⊗ L2P (Ω).
Silvester 2007 – p. 23/54
The associated Stochastic Primal Formulation is to findp ∈ Xg ⊗ L2
P (Ω) such that
〈(T ∇p,∇q)〉 = 0 ∀q ∈ X ⊗ L2P (Ω).
where Xg := ~v ∈ H1(D) : p = g on ΓD and X := X0.
Silvester 2007 – p. 24/54
The associated Stochastic Primal Formulation is to findp ∈ Xg ⊗ L2
P (Ω) such that
〈(T ∇p,∇q)〉 = 0 ∀q ∈ X ⊗ L2P (Ω).
where Xg := ~v ∈ H1(D) : p = g on ΓD and X := X0.
If this problem is to be well posed we require that
0 < µ∗ ≤ T (~x, ω) ≤ µ∗ <∞ a.e. in D × Ω.
which rules out Gaussian random fields ...
Silvester 2007 – p. 24/54
... unless the mean value (T (~x), assumed constant) is largecompared to the variance and we cheat!
To apply Monte Carlo we generate NR realizations ofGaussian RF T−1 with mean 1, variance 0.01, and solve adeterministic flow problem for each realization:
NR max varp max varuxmax varuy
100 2.0668e-4 0.0113 3.5023e-41000 2.3266e-4 0.0084 2.5289e-45000 2.3108e-4 0.0087 2.7190e-4
50000 2.3467e-4 0.0084 2.7288e-4∞ 2.29· · ·e-4
Such a naive approach will not work for “larger” variancehowever ...
Silvester 2007 – p. 25/54
In practice a lognormal distribution is assumed T = exp(Z)where Z is a homogeneous, isotropic, random field withzero mean which is associated with the covariance function
C(~x, ~y) = σ2 exp (−‖~x− ~y‖λ
).
For the WIPP site we have “known” values of σ and λ.Silvester 2007 – p. 26/54
Stochastic Finite Element Discretization
Primal problem: find p ∈ Xg ⊗ L2P (Ω) such that
〈(T ∇p,∇q)〉 = 0 ∀q ∈ X ⊗ L2P (Ω).
Step 1: Define deterministic discretization.
We choose a suitable finite dimensional subspace
Xh = spanφ1(~x), φ2(~x), . . . , φnx(~x) ⊂ X.
This can be done independently of the stochasticdiscretization.
Silvester 2007 – p. 27/54
Step 2: Define finite dimensional stochasticity.
We choose M independent random variablesξm : Ω → R, m = 1, 2, . . . ,M, which sufficiently capture thestochastic variability of the problem.Consequence: the RF occurring in stochastic PDE satisfies
T (~x, ω) ∼ T (~x, ξ1(ω), . . . , ξM (ω)) =: T (~x, ξ(ω)),
with independent RVξmMm=1. Set further
Γm := ξm(Ω) range space of RV ξmMm=1,
ρm : Γm → [0,∞) density function of ξm,
Γ := Γ1 × · · · × ΓM joint range of vector ξ = (ξ1, . . . , ξM ),
ρ(ξ) := ρ1(ξ1) · . . . · ρM (ξM ) joint probability density of ξ1, . . . , ξM .Silvester 2007 – p. 28/54
Replacing L2P (Ω) with L2
ρ(Γ), the variational formulationbecomes
Find p ∈ Xg ⊗ L2ρ(Γ) such that
〈a(p, q)〉 = 0 ∀q ∈ X ⊗ L2ρ(Γ)
where
〈a(p, q)〉 =
∫
Γ
ρ(ξ)
∫
D
T (~x, ξ)∇p(~x, ξ) · ∇q(~x, ξ) d~x dξ,
We have thus transformed the stochastic problem into adeterministic problem with an M -dimensional parameter ξ.
Silvester 2007 – p. 29/54
Step 3: Define discretization of W := L2ρ(Γ)
Given a finite dimensional subspace (common choices aregiven below)
W h = spanψ1(ξ), ψ2(ξ), . . . , ψnξ(ξ) ⊂W = L2
ρ(Γ),
the tensor product space X ⊗W is approximated by
Xh⊗W h =
v ∈ L2(D×Γ) : v ∈ spanφ(~x)ψ(ξ) : φ ∈ Xh, ψ ∈W h
.
The trial function ph ∈ Xh ⊗W h is thus of the form
ph(~x, ξ) =∑
i,j
pi,j φi(~x)ψj(ξ)
with a set of nx · nξ coefficients pi,j, nx = dimXh,nξ = dimW h.
Silvester 2007 – p. 30/54
Construction of W h
We discretize each variable of W = L2ρ1
(Γ1) ⊗ · · · ⊗ L2ρM
(ΓM )
independently and form tensor product space
W h = span
ψα(ξ) =M∏
j=1
ψαj(ξj) : ψαj
∈W hj ⊂ Lρj
(Γj)
, α ∈ NM0 .
piecewise polynomials (requires bounded intervals Γj)[Deb, Babuška & Oden (2001)] [Babuška et al. (2004)]
global polynomials known as Polynomial Chaos (PC)when ξm independent Gaussian, and GeneralizedPolynomial Chaos for other probability distributions,[Ghanem & Spanos (1991)] [Xiu & Karniadakis (2002)][Babuška et al. (2004)]
reduced tensor products fixed total degree, sparse grids[Schwab & Todor (2003)] [Matthies & Keese (2004)] [Xiu(2007)] Silvester 2007 – p. 31/54
Galerkin Approximation
Inserting the trial and test functions
ph(~x, ξ) =∑
i,j
pi,j φi(~x) ψj(ξ), qh(~x, ξ) = φk(~x) ψℓ(ξ)
into the variational formulation gives
〈a(ph, qh)〉 = 0 ∀qh,
with
〈a(ph,qh)〉=∑
i,j
(∫
Γ
ρ(ξ)ψj(ξ)ψℓ(ξ)
∫
D
T (~x, ξ)∇φi(~x)·∇φk(~x) d~x dξ
)
pi,j
=:∑
i,j
(∫
Γ
ρ(ξ)ψj(ξ)ψℓ(ξ)[K(ξ)]i,k dξ
)
pi,j .
Silvester 2007 – p. 32/54
Semi-Discrete System
The resulting systemAp = f
has dimension nx · nξ and has the block form
A =
A1,1 . . . A1,nξ
......
Anξ,1 . . . Anξ,nξ
, p =
p1
...pnξ
, f =
f1...
fnξ
,
with deterministic problem matrices defining the blocks
Aℓ,j = 〈ψjψℓK(ξ)〉 ∈ Rnx×nx , ℓ, j = 1, . . . , nξ.
Silvester 2007 – p. 33/54
Semi-Discrete System
The resulting systemAp = f
has dimension nx · nξ and has the block form
A =
A1,1 . . . A1,nξ
......
Anξ,1 . . . Anξ,nξ
, p =
p1
...pnξ
, f =
f1...
fnξ
,
with deterministic problem matrices defining the blocks
Aℓ,j = 〈ψjψℓK(ξ)〉 ∈ Rnx×nx , ℓ, j = 1, . . . , nξ.
The final ingredient is the representation of K(ξ) : there aretwo generic approaches ...
Silvester 2007 – p. 33/54
(a) Karhunen-Loève (KL) expansion of RF (linear inξ)
If our postulated random field T has a continuouscovariance function, then we can define a set ofuncorrelated random variables ξ1, ξ2, . . . via
T (~x, ξ) = T (~x) + σ∞∑
m=1
√
λm Tm(~x)ξm,
where Tm are the eigenfunctions of the covariance operatorassociated with the random field T .
Truncating the series after M terms (setting T0 := T ) gives
[K(ξ)]i,k = [K0]i,k + σM∑
m=1
√
λm[Km]i,k ξm, [Km]i,k = (Tm∇φk,∇φi),
[G0]ℓ,k = 〈ψkψℓ〉, [Gm]ℓ,k = 〈ξmψkψℓ〉,
A = G0 ⊗ K0 +σ∑
m
√
λmGm ⊗ Km.Silvester 2007 – p. 34/54
Diagonalization of the Galerkin system
When the stochastic FE space consists of polynomials ofseparate degree d in M random variables, i.e.,nξ = (d+ 1)M , then the stochastic basis ψj can be chosenin such a way that Gm are diagonal, and hence A is blockdiagonal, [Babuška et al. (2004)].
This is not possible when using the (smaller) space ofcomplete polynomials of degree d, i.e., nξ =
(
M+dd
)
. [Ernst &Ullmann (in preparation)].
Silvester 2007 – p. 35/54
Example: M = 4 Gaussian RVs, G0, G1, G2, G3, G4.
0 2 4 6 8 10 12 14 160
2
4
6
8
10
12
14
16
nnz = 55
total degree ≤ 2
0 5 10 15 20 25 30 350
5
10
15
20
25
30
35
nnz = 155
total degree ≤ 3
Silvester 2007 – p. 36/54
Example: flow in a square
Ω = [0, 1] × [0, 1], with boundary conditions: p = 1 on0 × [0, 1], p = 0 on 1 × [0, 1] , ~u · ~n = 0 on (0, 1) × 0, 1 .
Mean problem solution
~u(x, y) =
[
1
0
]
, p(x, y) = 1 − x.
Deterministic discretization:
• RT (0) mixed approximation
• uniform square grid of size h
Silvester 2007 – p. 37/54
Example: flow in a square
Stochastic discretization:
• KL expansion of Gaussian RF T−1:
T (~x, ω) = T (~x)+σM∑
m=1
√
λmTm(~x)ξm(ω), T (~x) = 1, σ = 0.1
• Covariance function: double exponential,
c(~x, ~y) = σ2 exp(−|x1 − y1|c1
− |x2 − y2|c2
), c1 = c2 = 1.
• Stochastic FE space: Hermite polynomial chaos of(individual) degree ≤ d in M random variables.
• Biorthogonal basis, Galerkin system decouples into2 × 2 block diagonal system.
Silvester 2007 – p. 38/54
Computed Solutions
0
0.5
1
0
0.5
10
1
2
3
4
x 10−4
x
Variance of uy
y 0
0.5
1
0
0.5
16.5
7
7.5
x 10−3
x
Variance of ux
y
Silvester 2007 – p. 39/54
Convergence of Computed Variance
1 2 3 4 52.22
2.23
2.24
2.25
2.26
2.27
2.28
2.29
2.3
2.31x 10
−4
d
M=5M=6M=7M=8M=9
max varp
1 2 3 4 50
1
2
x 10−4
d
M=5M=6M=7M=8M=9
max varuy
Silvester 2007 – p. 40/54
Fast Solver
We solve the (symmetric–) block systems Lkx = fk
(
M0 + σ∑M
m=1Gkm
√λi Mm BT
B 0
)(
u
p
)
=
(
gk
0
)
with weighted mass matrices [Mm]i,j = (Tmφj , φi) andcoefficients Gk
m = 〈ξmψkψk〉 using ...
Silvester 2007 – p. 41/54
Fast Solver
We solve the (symmetric–) block systems Lkx = fk
(
M0 + σ∑M
m=1Gkm
√λi Mm BT
B 0
)(
u
p
)
=
(
gk
0
)
with weighted mass matrices [Mm]i,j = (Tmφj , φi) andcoefficients Gk
m = 〈ξmψkψk〉 using ... MINRES with the blockdiagonal preconditioning
P−1 =
(
M∗
−1 0
0 Q∗
−1
)
where• M∗ ∼M0 (mass matrix diagonal);
• Q∗ ∼ BM∗
−1BT (via amg).
Silvester 2007 – p. 41/54
Solver performance : dependence onh
Example: M = 4, d = 2, (nξ = 81)
h−1 av. its/system av. solve time setup time32 41 0.48 0.4764 41 1.92 1.60
128 42 9.03 11.3256 40 36.35 125.6
Silvester 2007 – p. 42/54
Solver performance: dependence ond,M
M = 2 M = 3 M = 4
d nξ its/system nξ its/system nξ its/system
1 4 39 8 40 16 40
2 9 39 27 41 81 41
3 16 40 64 42 256 42
4 25 40 125 42 625 43
M = 5 M = 6 M = 7
1 32 40 64 40 128 41
2 243 41 729 42 2187 42
3 1024 43 4096 43 16384 43
4 3125 43 15625 43 78125 43
Silvester 2007 – p. 43/54
Solver performance : dependence onσ
Example: M = 4, d = 2, (nξ = 81)
σ av. its/system0.1 41
0.2 46
0.3 53
0.4 indef. (1,1) block
Silvester 2007 – p. 44/54
Performance Summary
Our AMG preconditioner is robust with respect to:
• spatial mesh size h;• stochastic discretization parameters M and d.
Iteration counts are sensitive to the ratio σ/µ:the solver breaks down if σ/µ ∼ 1 (when the discreteproblem is not well-posed).
Silvester 2007 – p. 45/54
For further details, check out
Michael Eiermann, Oliver Ernst & Elisabeth UllmannComputational Aspects of the Stochastic Finite ElementMethodComputing and Visualization in Science, 10, 3–15,2007.
Catherine Powell & Howard ElmanBlock-Diagonal Preconditioning for Spectral StochasticFinite Element SystemsMIMS Eprint 2007.xx, June 2007.
Next question:
Silvester 2007 – p. 46/54
For further details, check out
Michael Eiermann, Oliver Ernst & Elisabeth UllmannComputational Aspects of the Stochastic Finite ElementMethodComputing and Visualization in Science, 10, 3–15,2007.
Catherine Powell & Howard ElmanBlock-Diagonal Preconditioning for Spectral StochasticFinite Element SystemsMIMS Eprint 2007.xx, June 2007.
Next question: How do we deal with lognormal randomfields?
Silvester 2007 – p. 46/54
(b) Expansion in polynomial chaos (nonlinear inξ)
T (~x, ξ) = T0(~x) +
nξ∑
m=1
tm(~x) ψm(ξ), ξ = ξ(ω) ∈ RM ,
This generates a “less sparse” Galerkin matrix
A = G0 ⊗ K0 +
nξ∑
m=1
Hm ⊗ Km,
[Hm]ℓ,j = 〈ψm ψℓ ψj〉, [Km]i,k = (tm∇φk,∇φi).
Silvester 2007 – p. 47/54
(b) Expansion in polynomial chaos (nonlinear inξ)
T (~x, ξ) = T0(~x) +
nξ∑
m=1
tm(~x) ψm(ξ), ξ = ξ(ω) ∈ RM ,
This generates a “less sparse” Galerkin matrix
A = G0 ⊗ K0 +
nξ∑
m=1
Hm ⊗ Km,
[Hm]ℓ,j = 〈ψm ψℓ ψj〉, [Km]i,k = (tm∇φk,∇φi).
This technique can be used with a lognormal RF
T (~x, ω) = exp(G(~x, ω)), with G a Gaussian RF.
A Polynomial Chaos expansion of T can then be explicitlyconstructed from the KL expansion of G ...
Silvester 2007 – p. 47/54
For further details, look out for
Oliver Ernst, Catherine Powell, David Silvester &Elisabeth UllmannIterative Solution Methods for Mixed Formulations ofPotential Flow Problems with Random CoefficientsMIMS Eprint 2007.xx, August 2007.
Elisabeth UllmannPh.D Thesis, SFemLib software,December 2007.
Silvester 2007 – p. 48/54
Summary
There are two alternative SFEM formulations:
Expansion of RF linear in ξ using tensor products ofpolynomials of separate degree d in M randomvariables:• all Gm diagonal for doubly orthogonal basis,
• decoupled matrix: nξ = (d+ 1)M systems of size nx,
• beats Monte Carlo if nξ < nR (number ofrealizations) for same accuracy.
Expansion of RF nonlinear in ξ using polynomials oftotal degree d:• need to solve fully coupled system of size nξ · nx,
• significantly smaller problem overall: nξ =(
M+dd
)
.
Silvester 2007 – p. 49/54
A Toy Problem (deterministic)
• P-IFISS Toolbox• HSL_MI20 Algebraic Multigrid
A Practical Problem (uncertain/stochastic)
• Monte Carlo Method• Stochastic Finite Element Method
• Dundee 1981–1987
Silvester 2007 – p. 50/54
Dundee 1981
A R Conn G Dahlquist L M Delves J Douglas JrI S Duff R Fletcher C W Gear G H GolubJ G Hayes P J v.d. Houwen T E Hull P LancasterR J Y Macleod M J D Powell L B Wahlbin
Silvester 2007 – p. 51/54
Dundee 1983
R H Bartels A Bjorck C de Boor H BrunnerT Dupont D M Gay P W Hemker M J D PowellP Raviart L F Shampine A Spence H J StetterPh L Toint G A Watson M F Wheeler J R Whiteman
Silvester 2007 – p. 52/54
Dundee 1985
P Alefeld F Brezzi G J Cooper D GoldfarbS-P Han A Iserles C Van Loan P MarcusA R Mitchell K W Morton M L Overton A H G Rinnooy KanJ M Sanz-Sena L N Trefethen J G Verwer W L Wendland
Silvester 2007 – p. 53/54
Dundee 1987
I Babuska H Brunner J R Cash R E EwingP Gaffney G H Golub D F Griffiths T J R HughesR Jeltsch U Kulisch K Madsen C A MicchelliM J D Powell D M Sloan G Wanner B Werner
Silvester 2007 – p. 54/54
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