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

Marta M. Betcke, Heinrich Voss
2016 Numerische Mathematik  
In this work we present a new restart technique for iterative projection methods for nonlinear eigenvalue problems admitting minmax characterization of their eigenvalues. The variational property allows for the eigenvalues of such problems to be computed safely one after another and avoiding repetitions, as it is the case for linear Hermitian eigenvalue problems. This can be achieved applying iterative projection methods, which make use of the minmax induced enumeration of the eigenvalues in
more » ... inner iteration. However, to numerically exploit the minmax principle, it is necessary to include all the previously computed eigenvectors in the search subspace. This subspace growth is an obvious limitation if a large number of eigenvalues or eigenvalues in the interior of the spectrum are to be computed. In this paper we propose to overcome this problem employing a local numbering of eigenvalues. This local numbering only requires a presence of one eigenvector in the search subspace, effectively eliminating the search subspace growth and therewith the super-linear increase of the computational costs. Using this local numbering we derive a new restart technique for iterative projection methods and integrate it into nonlinear Arnoldi and Jacobi Davidson methods. The efficiency of our new method is demonstrated on a real-life conservative gyroscopic eigenvalue problem modeling free vibrations of a rolling tire.
doi:10.1007/s00211-016-0804-3 pmid:28615742 pmcid:PMC5445551 fatcat:y5dsgpxq3jgsndk6fm5pvwzdwm