Seminar in Numerical Analysis: Jürgen Dölz (University of Bonn)

On uncertainty quantification of eigenvalue problems

We consider generalized symmetric operator eigenvalue problems with random symmetric perturbations of the operators. This implies that the eigenpairs of the eigenvalue problem are also random. We investigate stochastic quantities of interest of eigenpairs of higher but finite multiplicity and discuss why for multiplicity larger than one, only the stochastic quantities of interest of the eigenspaces are meaningful. To do so, 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. We present numerical examples to illustrate the theoretical results.

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