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.
Highlights
In this work we consider a problem of computing a large number of eigenvalues in an open real interval J ⊂ R and the corresponding eigenvectors of the nonlinear eigenvalue problem (NEP) T (λ)x = 0, (1)where T (λ) ∈ Cn×n is a family of large and sparse Hermitian matrices for every λ ∈ J
In this work we propose a new restart technique which allows to project the NEP (1) only onto search spaces of a fixed, relatively small dimension throughout the iteration
The new restart technique can be integrated with iterative projection methods such as the Nonlinear Arnoldi or Jacobi-Davidson method making them capable of computation of a large number of eigenpairs, possibly in the interior of the spectrum
Summary
In this work we consider a problem of computing a large number of eigenvalues in an open real interval J ⊂ R and the corresponding eigenvectors of the nonlinear eigenvalue problem (NEP). We assume that the eigenvalues of (1) in J can be characterized as minmax values of a Rayleigh functional [35] Such problems routinely arise in simulation of acoustic properties of e.g.vehicles or their parts in order to minimize the noise exposure to the passengers as well as to the environment. The new restart technique can be integrated with iterative projection methods such as the Nonlinear Arnoldi or Jacobi-Davidson method making them capable of computation of a large number of eigenpairs, possibly in the interior of the spectrum. The resulting framework for restarting of nonlinear iterative projection methods for interior eigenvalue computation is summarized in Sect.
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