Stochastic collocation method for computing eigenspaces of parameter-dependent operators

TL;DR

This paper develops a stochastic collocation method to efficiently compute eigenspaces of parameter-dependent elliptic operators, leveraging their complex-analytic dependence for convergence of sparse polynomial approximations.

Contribution

It introduces a new approach combining complex-analytic extension of eigenspaces with sparse grid collocation for parameter-dependent eigenproblems.

Findings

Method converges rapidly in numerical examples

Eigenspaces extend analytically in parameters

Applicable to stochastic diffusion operators

Abstract

We consider computing eigenspaces of an elliptic self-adjoint operator depending on a countable number of parameters in an affine fashion. The eigenspaces of interest are assumed to be isolated in the sense that the corresponding eigenvalues are separated from the rest of the spectrum for all values of the parameters. We show that such eigenspaces can in fact be extended to complex-analytic functions of the parameters and quantify this analytic dependence in way that leads to convergence of sparse polynomial approximations. A stochastic collocation method on an anisoptropic sparse grid in the parameter domain is proposed for computing a basis for the eigenspace of interest. The convergence of this method is verified in a series of numerical examples based on the eigenvalue problem of a stochastic diffusion operator.