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A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.ORCID iD: 0000-0001-9443-8772
2016 (English)In: Numerical Linear Algebra with Applications, ISSN 1070-5325, E-ISSN 1099-1506Article in journal (Refereed) PublishedText
Abstract [en]

Summary: We consider the nonlinear eigenvalue problem M(λ)x = 0, where M(λ) is a large parameter-dependent matrix. In several applications, M(λ) has a structure where the higher-order terms of its Taylor expansion have a particular low-rank structure. We propose a new Arnoldi-based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldi's method applied to an operator whose reciprocal eigenvalues are solutions to the nonlinear eigenvalue problem. The iterates in the algorithm are functions represented in a particular structured vector-valued polynomial basis similar to the construction in the infinite Arnoldi method [Jarlebring, Michiels, and Meerbergen, Numer. Math., 122 (2012), pp. 169-195]. In this paper, the low-rank structure is exploited by applying an additional operator and by using a more compact representation of the functions. This reduces the computational cost associated with orthogonalization, as well as the required memory resources. The structure exploitation also provides a natural way in carrying out implicit restarting and locking without the need to impose structure in every restart. The efficiency and properties of the algorithm are illustrated with two large-scale problems.

Place, publisher, year, edition, pages
John Wiley & Sons, 2016.
Keyword [en]
Arnoldi method, Low-rank, Nonlinear eigenvalue problem
National Category
URN: urn:nbn:se:kth:diva-186762DOI: 10.1002/nla.2043ScopusID: 2-s2.0-84959517210OAI: diva2:927938

QC 20160513

Available from: 2016-05-13 Created: 2016-05-13 Last updated: 2016-05-17Bibliographically approved

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Jarlebring, Elias
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Numerical Analysis, NA
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