Vortragsankündigung
Im Rahmen des Kolloquiums spricht
Herr Prof. Dr. Jürgen Dölz (Universität Bonn)
über
Title: On uncertainty quantification of eigenvalue problems
Abstract:
Eigenvalue problems have manifold applications in engineering, physics and medicine, and their (numerical) solution is nowadays well understood in most cases. However, since real-world circumstances typically deviate in an unforeseeable and seemingly random fashion from the ideal (computational) environment, the real-world solutions of the eigenvalue problem also deviate in a random fashion from the simulation results. 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 discuss numerical examples to illustrate the theoretical results, including an example of a three dimensional particle accelerator based on real-world data.
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