Inverse subspace iteration for spectral stochastic finite element methods
Provides a computational method for random eigenvalue problems in spectral stochastic finite elements, but the contribution is incremental as it adapts existing subspace iteration to a stochastic Galerkin framework.
The paper develops a stochastic inverse subspace iteration method for computing eigenvalues and eigenvectors of parameter-dependent matrices using polynomial chaos expansions, demonstrating accuracy comparable to Monte Carlo and stochastic collocation methods on vibration analysis benchmarks.
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.