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IntroductionApplications

RemarksExamples

References

Arnoldi Algorithm

NguyÔn, Thanh S¬n

Universitaet BremenZentrum fuer Technomathematik

25th March 2009

NguyÔn, Thanh S¬n Arnoldi Algorithm

RemarksExamples

References

Outline

1 Introduction

2 Applications

3 Remarks

4 Examples

5 References

RemarksExamples

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Introduction

Arnoldi algorithm/Arnoldi process is used to produce anorthonormal basis for a Krylov subspace. Given a squarematrix A, a non-zero vector x and an integer number m,find a matrix V s.t. VTV = I and

colspan(V) = span(x,Ax,A2x, ...,Am−1x).

Invented by W. E. Arnoldi in 1951 for eigenvalue problem.

Wide-used in approximate solvers.

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Introduction

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Introduction

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Introduction

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Coding

function [V,H] = Arnoldi(A,b,m, tol)

H = zeros(m + 1,m);

beta = norm(b);

V = b/beta;

for j = 1 : m

w = AV(:, j);

for i = 1 : j

H(i, j) = w′V(:, i);

for i = 1 : j

w = w− H(i, j) ∗ V(:, i);

H(j + 1, j) = norm(w);

if H(j + 1, j) < tol

H = H(1 : j,1 : j);

V = [V w/H(j + 1, j)];

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Coding

function [V,H] = Arnoldi(A,b,m, tol)

H = zeros(m + 1,m);

beta = norm(b);

V = b/beta;

for j = 1 : m

w = AV(:, j);

for i = 1 : j

H(i, j) = w′V(:, i);

for i = 1 : j

w = w− H(i, j) ∗ V(:, i);

H(j + 1, j) = norm(w);

if H(j + 1, j) < tol

H = H(1 : j,1 : j);

V = [V w/H(j + 1, j)];

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Three matrices A,V,H satisfy the relations

AVm = Vm+1H(m+1)xm; (1)

VTmAVm = Hmxm. (2)

Hmxm is a Hessenberg matrix, VTmVm = Im.

This structure is useful in many situations in linear algebra.

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Linear equations

GMRES ( Generalized Minimum RESidual method ): Givena square (usually large, sparse) system Ax = b of order N,intial guess x0, r0 = b−Ax0 called intial residual. One findsthe approximate solution in affine subspace x0 +Km(A, r0).

Let x = x0 + Vmy, y ∈ Rm, the minimization problem‖b− Ax‖2 subject to x ∈ RN is converted to minimizing‖‖r0‖e1 − H(m+1)xmy‖ subject to y ∈ Rm.

If A is nonsingular, GMRES breaks down at mth iffxm = x0 + Vmy is the exact solution.

Some other methods or variations: FOM, GCR,GMRES-DR, c.f [4, 6, 7, 8]

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Linear equations

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Linear equations

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Linear equations

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Eigenvalues problem

The original eigenvalue problem Ax = λx is replaced bythe "reduced" eigenvalue problem Hmxmzm = θzm wherethe Arnoldi basis Vm is started with a unit-normed initialeigenvector v1. Then, the approximate eigenpairs of A areselected form the Ritz pairs {(θi,Vmzm,i)}.This approach yields a better result in compare with Power

method which only uses Am−1v1 for approximation. c.f.[5]

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Eigenvalues problem

The original eigenvalue problem Ax = λx is replaced bythe "reduced" eigenvalue problem Hmxmzm = θzm wherethe Arnoldi basis Vm is started with a unit-normed initialeigenvector v1. Then, the approximate eigenpairs of A areselected form the Ritz pairs {(θi,Vmzm,i)}.This approach yields a better result in compare with Power

method which only uses Am−1v1 for approximation. c.f.[5]

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Model reduction

Consider a LTI dynamical system

Ex′(t) = Ax(t) + bu(t), (3)

y(t) = cx(t) + du(t); (4)

The transfer function is H(s) = c(sE− A)−1b + d. Momentsmatching method approximates H(s) near some point, say s0,by matching a few leading coefficients of Taylor expansion ats0.

H(s) = −c∞∑i=0

((A− s0E)−1E)i(A− s0E)−1b(s− s0)i + d

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Model reduction

This can be done by projecting the system (3)-(4) onto Krylovsubspace V = Km((A− s0E)−1E, (A− s0E)−1b) or bothV = Km((A− s0E)−1E, (A− s0E)−1b) andW = Ki+1(((A− s0E)−1E)T,CT). The reduced systems arethen,

One-sided

VTEVx′ = VTAVx + VTbu,

y = cVx + du;

Two-sided

WTEVx′ = WTAVx + WTbu,

y = cVx + du;

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Model reduction

One-sided

y = cVx + du;

Two-sided

y = cVx + du;

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Model reduction

One-sided

y = cVx + du;

Two-sided

y = cVx + du;

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Deflation may occur, if the process has not beenconvergent, restarting is required.

Restarting is also needed when the order of Krylovsubspace is big,( but under-convergent ) since thisrequires much computer memories.

Multiple starting vectors leads to so called block Arnoldi

algorithm which requires more efforts in implementation.

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Examples

Two applications of Arnoldi algorithm in solving linearequations and model reduction are provided.

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References

[Antoulas05] A. C. Antoulas, Approximation of Large-scale

Dynamical Systems, Advances in Design and ControlDC-06, SIAM, Philadenphia, 2005.

[Chahlaoui02] Y. Chahlaoui, P. V. Doreen, "A collection ofbenchmark examples for model reduction of linear timeinvariant dynamical systems", Technical report, SILICOTWorking Note 2002-2, 2002.

[Freund99] R. W. Freund, "Krylov-subspace methods forreduced-order modeling in circuit simulation", Numerical

Analysis Manuscript, No. 99-3-17, Bell Laboratories.

[Morgan02] R. B. Morgan, "GMRES with deflatedrestarting", SIAM J. Sci. Comput., Vol. 24, No. 1, pp. 20-37,2002.

RemarksExamples

References

[Saad92] Y. Saad, Numerical Methods for Eigenvalues

Problems, Manchester University Press, 1992.

[Saad96] Y. Saad, Iterative Methods for Sparse Linear

Systems, PWS Publishing Company, 1996.

[Sturler96], E. de Sturler, "Nested Krylov methods basedon GRC", J. Computational and Applied Mathematics, 67,pp. 15-41, 1996.

[Sturler99], E. de Sturler, "Truncation strategies for optimalKrylov subspace methods", SIAM J. Numer. Anal., Vol. 36,No. 3, pp. 864-889, 1999.

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