Restarting iterative projection methods for Hermitian nonlinear eigenvalue problems with minmax property
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- 作者：Marta M. Betcke ; Heinrich Voss
- 关键词：Nonlinear eigenvalue problem ; Iterative projection method ; Nonlinear Arnoldi method ; Jacobi ; Davidson method ; Minmax characterization ; Restart ; Purge and lock
- 刊名：Numerische Mathematik
- 出版年：2017
- 出版时间：February 2017
- 年：2017
- 卷：135
- 期：2
- 页码：397-430
- 全文大小：1424KB
- 刊物类别：Mathematics and Statistics
- 刊物主题：Numerical Analysis; Mathematics, general; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation; Appl.Mathematics/Comput
- 出版者：Springer Berlin Heidelberg
- ISSN：0945-3245
- 卷排序：135
文摘
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.