Uncertainty Quantification of Elliptic Eigenvalue Problems

Abstract

This thesis considers the uncertainty quantification of elliptic eigenvalue problems (EVPs) with a special focus on degenerate eigenvalues. Elliptic EVPs are problems to find a pair of eigenvalues and eigenfunctions of an elliptic operator. We consider the stochastic moments of these eigenvalues and eigenfunctions to describe their uncertainty given a stochastic perturbation of the elliptic operator, for example, by material coefficients or shape deformations.<br/> In a multiparametric stochastic model, the identification of eigenvalues and eigenfunctions as functions of the parameter is not trivial. Assuming analytic dependence of the elliptic operator with respect to the parameter, we first investigate the bifurcation behavior of these eigenvalue trajectories in a neighborhood of some reference parameter value. In the general degenerate case, these trajectories can only be defined with respect to the eigenspace of the eigenpairs. Assuming analyticity of the EVPs, these trajectories with respect to the eigenspace are also analytic and can thus be described by their derivatives. We characterize Fréchet derivatives of arbitrary order with respect to the eigenspace using saddle point equations. The trajectories in the traditional sense and with respect to the eigenspace are related via a pathwise-defined parameterized polarization matrix, which we also characterize including its derivatives.<br/> Equipped with the (locally) well-defined and measurable trajectories of the eigenpairs with respect to the eigenspace, we investigate the uncertainty quantification of eigenvalues in a neighborhood of the reference point using a perturbation ansatz. We discuss the efficient implementation of this perturbation ansatz and benchmark it against other methods of calculating stochastic means and covariances of the eigenpairs like quasi-Monte Carlo. As an application of our setting, we consider stochastic shape deformation models of the Laplace and Maxwell EVP in more detail.<br/> Lastly, we consider the possibility of incorporating measurement data into our model using a Bayesian inverse model. This leads to an adaptation of the perturbation approximations of the mean and correlation by amending terms that reflect the influence of the Radon-Nikodým derivative of the posterior measure with respect to the prior measure. We also consider the possibility of using the perturbation approximation of the posterior mean in an iteration to improve the parameter reference point. This iteration is related to the corresponding regularized inverse problem.