# Asymptotic convergence of spectral inverse iterations for stochastic eigenvalue problems

## TL;DR

This paper analyzes the asymptotic convergence of spectral inverse iteration algorithms for computing eigenpairs of elliptic operators with random coefficients, combining stochastic and spatial approximations.

## Contribution

It provides an error analysis for the convergence of spectral inverse iteration and subspace iteration in stochastic eigenvalue problems, supported by numerical experiments.

## Key findings

- Spectral inverse iteration converges asymptotically to the smallest eigenvalue and eigenvector.
- Numerical experiments confirm the theoretical convergence results.
- The algorithms effectively handle eigenvalue crossings in stochastic parameter spaces.

## Abstract

We consider and analyze applying a spectral inverse iteration algorithm and its subspace iteration variant for computing eigenpairs of an elliptic operator with random coefficients. With these iterative algorithms the solution is sought from a finite dimensional space formed as the tensor product of the approximation space for the underlying stochastic function space, and the approximation space for the underlying spatial function space. Sparse polynomial approximation is employed to obtain the first one, while classical finite elements are employed to obtain the latter. An error analysis is presented for the asymptotic convergence of the spectral inverse iteration to the smallest eigenvalue and the associated eigenvector of the problem. A series of detailed numerical experiments supports the conclusions of this analysis. Numerical experiments are also presented for the spectral subspace iteration, and convergence of the algorithm is observed in an example case, where the eigenvalues cross within the parameter space. The outputs of both algorithms are verified by comparing to solutions obtained by a sparse stochastic collocation method.