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Title: A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems

Abstract

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. In conclusion, the efficiency and properties of the algorithm are illustrated with two large-scale problems.

Authors:
 [1];  [2];  [1]
  1. Univ. of Leuven, Heverlee (Belgium)
  2. KTH Royal Institute of Technology, Stockholm (Sweden)
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1525163
Grant/Contract Number:  
AC02-05CH11231
Resource Type:
Accepted Manuscript
Journal Name:
Numerical Linear Algebra with Applications
Additional Journal Information:
Journal Volume: 23; Journal Issue: 4; Journal ID: ISSN 1070-5325
Publisher:
Wiley
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; nonlinear eigenvalue problem; Arnoldi method; low‐rank

Citation Formats

Van Beeumen, Roel, Jarlebring, Elias, and Michiels, Wim. A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems. United States: N. p., 2016. Web. doi:10.1002/nla.2043.
Van Beeumen, Roel, Jarlebring, Elias, & Michiels, Wim. A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems. United States. doi:10.1002/nla.2043.
Van Beeumen, Roel, Jarlebring, Elias, and Michiels, Wim. Thu . "A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems". United States. doi:10.1002/nla.2043. https://www.osti.gov/servlets/purl/1525163.
@article{osti_1525163,
title = {A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems},
author = {Van Beeumen, Roel and Jarlebring, Elias and Michiels, Wim},
abstractNote = {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. In conclusion, the efficiency and properties of the algorithm are illustrated with two large-scale problems.},
doi = {10.1002/nla.2043},
journal = {Numerical Linear Algebra with Applications},
number = 4,
volume = 23,
place = {United States},
year = {2016},
month = {3}
}

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