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Restarting iterative projection methods for Hermitian nonlinear eigenvalue problems with minmax property

Betcke, MM; Voss, H; (2016) Restarting iterative projection methods for Hermitian nonlinear eigenvalue problems with minmax property. Numerische Mathematik , 135 (2) pp. 397-430. 10.1007/s00211-016-0804-3. Green open access

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Abstract

In this work we present a new restart technique for iterative projection methods for nonlinear eigenvalue problems admitting minmax characterization of their eigenvalues. Our technique makes use of the minmax induced local enumeration of the eigenvalues in the inner iteration. In contrast to global numbering which requires including all the previously computed eigenvectors in the search subspace, the proposed local numbering only requires a presence of one eigenvector in the search subspace. This effectively eliminates the search subspace growth and therewith the super-linear increase of the computational costs if a large number of eigenvalues or eigenvalues in the interior of the spectrum are to be computed. The new restart technique is integrated into nonlinear iterative projection methods like the Nonlinear Arnoldi and Jacobi-Davidson methods. The efficiency of our new restart framework is demonstrated on a range of nonlinear eigenvalue problems: quadratic, rational and exponential including an industrial real-life conservative gyroscopic eigenvalue problem modeling free vibrations of a rolling tire. We also present an extension of the method to problems without minmax property but with eigenvalues which have a dominant either real or imaginary part and test it on two quadratic eigenvalue problems.

Type: Article
Title: Restarting iterative projection methods for Hermitian nonlinear eigenvalue problems with minmax property
Open access status: An open access version is available from UCL Discovery
DOI: 10.1007/s00211-016-0804-3
Publisher version: http://dx.doi.org/10.1007/s00211-016-0804-3
Language: English
Additional information: Copyright © The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Keywords: Nonlinear eigenvalue problem, Iterative projection method, Nonlinear Arnoldi method, Jacobi-Davidson method, Minmax characterization, Restart, Purge and lock
UCL classification: UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
URI: https://discovery.ucl.ac.uk/id/eprint/1388874
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