On uncertainty quantification of eigenvalues and eigenspaces with higher multiplicity

TL;DR

This paper studies the uncertainty quantification of eigenvalues and eigenspaces with higher multiplicity in stochastic operator eigenvalue problems, proposing new derivatives, sampling strategies, and perturbation methods.

Contribution

It introduces a new linear characterization of eigenpairs with higher multiplicity and a Monte Carlo sampling strategy for their uncertainty quantification.

Findings

Eigenvalues with higher multiplicity require eigenspace analysis, not individual eigenpair analysis.

The paper proves local analyticity of eigenspaces under perturbations.

Numerical examples validate the proposed methods and theoretical results.

Abstract

We consider generalized operator eigenvalue problems in variational form with random perturbations in the bilinear forms. This setting is motivated by variational forms of partial differential equations with random input data. The considered eigenpairs can be of higher but finite multiplicity. We investigate stochastic quantities of interest of the eigenpairs and discuss why, for multiplicity greater than 1, only the stochastic properties of the eigenspaces are meaningful, but not the ones of individual eigenpairs. To that end, we characterize the Fr\'echet derivatives of the eigenpairs with respect to the perturbation and provide a new linear characterization for eigenpairs of higher multiplicity. As a side result, we prove local analyticity of the eigenspaces. Based on the Fr\'echet derivatives of the eigenpairs we discuss a meaningful Monte Carlo sampling strategy for multiple eigenvalues and develop an uncertainty quantification perturbation approach. Numerical examples are presented to illustrate the theoretical results.