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On uncertainty quantification of eigenvalues and eigenspaces with higher multiplicity

Jürgen Dölz, David Ebert

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 Fréchet 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 Fréchet 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.

On uncertainty quantification of eigenvalues and eigenspaces with higher multiplicity

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 Fréchet 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 Fréchet 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.
Paper Structure (33 sections, 14 theorems, 91 equations, 5 figures)

This paper contains 33 sections, 14 theorems, 91 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Examples
  4. Related work
  5. Contributions
  6. Outline
  7. Deterministic derivatives of eigenvalue problems
  8. Problem setting
  9. Notation for eigenvalues and -spaces with higher multiplicity
  10. A regularity, analyticity, and orthogonality result
  11. Derivatives of nondegenerate eigenvalues
  12. Derivatives of eigenfunctions to nondegenerate eigenvalues
  13. The problem with derivatives of degenerate eigenpairs
  14. Derivatives of eigenspaces to degenerated eigenvalues
  15. Polarization and derivatives of degenerated eigenvalues and eigenfunctions
  16. ...and 18 more sections

Key Result

Lemma 2.3

eq:eig_bfeq:eig_bf_normalized hold if and only if there is an orthogonal matrix $\mathbf{Q}\in\mathbb{R}^{m\times m}$ depending on $(\mu,\varepsilon)$ such that all $([\tilde{\mathbf{u}}]_i,\tilde{\lambda}_i)$, $i=1,\ldots,m$, $\mathop{\mathrm{diag}}\nolimits(\tilde{\lambda}_1,\ldots,\tilde{\lambda}

Figures (5)

  • Figure 1: Unperturbed (top) and perturbed (bottom) first three eigenpairs of the diffusion operator for $\alpha=\beta=1$. The corresponding eigenvalue trajectories are illustrated in \ref{['fig:splitting_eigvalue']}.
  • Figure 2: Illustration of the eigenvalue trajectories of $[\lambda]_{2,3}$ under a sample perturbation.
  • Figure 3: Convergence of residue of first-order approximations in the determinstic setting.
  • Figure 4: Convergence of Monte Carlo method for eigenvalues and -spaces.
  • Figure 5: Convergence of series approximations of the mean.

Theorems & Definitions (31)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • Corollary 2.6
  • Lemma 2.7
  • proof
  • ...and 21 more