Inverse subspace iteration for spectral stochastic finite element methods

TL;DR

This paper introduces a stochastic inverse subspace iteration method for efficiently computing eigenvalues and eigenvectors of parameter-dependent matrices using spectral stochastic finite element techniques, with applications to vibration analysis.

Contribution

It develops a novel stochastic inverse subspace iteration algorithm based on spectral stochastic finite elements, enabling accurate computation of interior eigenvalues and multiple eigenvectors.

Findings

The method accurately computes eigenvalues and eigenvectors in stochastic settings.

It outperforms Monte Carlo and stochastic collocation in accuracy.

Numerical experiments demonstrate effectiveness on vibration analysis problems.

Abstract

We study random eigenvalue problems in the context of spectral stochastic finite elements. In particular, given a parameter-dependent, symmetric positive-definite matrix operator, we explore the performance of algorithms for computing its eigenvalues and eigenvectors represented using polynomial chaos expansions. We formulate a version of stochastic inverse subspace iteration, which is based on the stochastic Galerkin finite element method, and we compare its accuracy with that of Monte Carlo and stochastic collocation methods. The coefficients of the eigenvalue expansions are computed from a stochastic Rayleigh quotient. Our approach allows the computation of interior eigenvalues by deflation methods, and we can also compute the coefficients of multiple eigenvectors using a stochastic variant of the modified Gram-Schmidt process. The effectiveness of the methods is illustrated by numerical experiments on benchmark problems arising from vibration analysis.