Paper Info
Title
A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems.
Abstract
We consider the nonlinear eigenvalue problem M(lambda)x = 0, where M(lambda) is a large parameter-dependent matrix. In several applications, M(lambda) 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 (C) 2016 John Wiley & Sons, Ltd.
Year
DOI
Venue
2016
10.1002/nla.2043
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
Keywords
Field
DocType
nonlinear eigenvalue problem,Arnoldi method,low-rank
Polynomial basis,Mathematical optimization,Matrix (mathematics),Arnoldi iteration,Algorithm,Operator (computer programming),Divide-and-conquer eigenvalue algorithm,Orthogonalization,Eigenvalues and eigenvectors,Mathematics,Taylor series
Journal
Volume
Issue
ISSN
23.0
4.0
1070-5325
Citations 
PageRank 
References 
1
0.35
17
Authors
3
Name
Order
Citations
PageRank
Roel Van Beeumen1373.81
Jarlebring Elias28411.48
Wim Michiels351377.24