Stochastic Finite Elements and Fast Iterative Solversnaconf/07/dun07djs.pdf · Stochastic Finite...

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

http://www.ma.umist.ac.uk/djs

http://www.ma.man.ac.uk

http://www.man.ac.uk

http://www.ma.umist.ac.uk/djs

Joint work with

• Jonathan Boyle (EPSRC EP/C000528/1)• Oliver Ernst & Elisabeth Ullmann (TU Bergakademie

Freiberg, ARC Project 1279)


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


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 .


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


Flow Problem III


Flow Problem IV

Streamlines of interpolated RT0 flux solution


PIFISSPotential (Incompressible) Flow & Iterative

Solution Software Guide ∗

David J. SilvesterSchool of Mathematics, University of Manchester

[email protected].

Catherine E. PowellSchool of Mathematics, University of Manchester

[email protected].

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.


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.


RT(0) mixed approximation


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


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∗

)

.


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.


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.


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


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)


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.






• Dundee 1981–1987


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)].


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


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.


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.


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).


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).


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.


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 ...


... 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 ...


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.


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.


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.


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 .


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.


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 ...


(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)].


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


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


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.


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


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


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 ...


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).


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


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


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


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).


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:


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?


(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).


(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 ...


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.


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

)

.






• Dundee 1981–1987


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


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


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


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


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