Chapter

Matching Moments and Model Reduction View

Jörg Liesen and Zdenek Strakos

in Krylov Subspace Methods

Published in print October 2012 | ISBN: 9780199655410
Published online January 2013 | e-ISBN: 9780191744174 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199655410.003.0003

Series: Numerical Mathematics and Scientific Computation

Matching Moments and Model Reduction View

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The projected system matrix in Krylov subspace methods consists of moments of the original system matrix with respect to the initial residual(s). This hints that Krylov subspace methods can be viewed as matching moments model reduction. Through the simplified Stieltjes moment problem, orthogonal polynomials, continued fractions, and Jacobi matrices, we thus obtain the Gauss–Christoffel quadrature representation of the conjugate gradient method (CG). It is described how generalisations to the non-Hermitian case can easily be achieved using the Vorobyev method of moments. Finally, the described results and their historical roots are linked with the model reduction of large-scale dynamical systems. The chapter demonstrates the strong connection between Krylov subspace methods used in state-of-the-art numerical calculations and classical topics of analysis and approximation theory. Since moments represent very general objects, this suggests that Krylov subspace methods might have much wider applications beyond their immediate context of solving algebraic problems.

Keywords: moment problem; model reduction; Hauss-Christoffel quadrature; continued fractions; orthogonal polynomials; Jacobi matrices; Vorobyev method of moments; estimates in quadratic forms; minimal partial realisation; P. l. Chebyshev; A. A. Markoff; T. J. Stieltjes

Chapter.  52476 words.  Illustrated.

Subjects: Applied Mathematics

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