A rank‐exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems |
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| Authors: | Roel Van Beeumen Elias Jarlebring Wim Michiels |
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| Affiliation: | 1. Department of Computer Science, KU 2. Leuven, University of Leuven, Heverlee, Belgium;3. Department of Mathematics, NA group, KTH Royal Institute of Technology, Stockholm, Sweden |
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| 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. The efficiency and properties of the algorithm are illustrated with two large‐scale problems. Copyright © 2016 John Wiley & Sons, Ltd. |
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| Keywords: | nonlinear eigenvalue problem Arnoldi method low‐rank |
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