Vorträge

We consider eigenvalue problems which are nonlinear in the eigenvalue
parameter. Newton-based methods are well-established techniques for determining individual eigenpairs of such nonlinear eigenvalue problems. If a larger number of eigenpairs is sought, however, the tendency of these methods to re-converge to previously discovered eigenpairs is a hindrance. In this talk, a deflation strategy for nonlinear eigenvalue problems will be presented, which overcomes this limitation in a natural way. Furthermore, we will comment on how this deflation approach can be implemented in a Jacobi-Davidson framework with only minimal overhead.

The concept of invariant subspaces is fundamental to linear eigenvalue problems and provides an important theoretical foundation in the design of numerical eigenvalue solvers. It turns out that there is no straightforward extension of this concept to eigenvalue problems that are nonlinear in the eigenvalue parameter. One obstacle is that eigenvectors belonging to different eigenvalues may become linearly dependent in the nonlinear case. Invariant pairs offer an elegant way to avoid this obstacle and appear to be the most natural extension of invariant subspaces. In this talk, we give an overview of the properties of invariant pairs and explain how they can be used in the design of numerical algorithms for nonlinear eigenvalue problems, as they arise for example in band diagram calculations for photonic crystals and fluid-structure interaction problems.

Ein klassischer Satz von Friedrichs besagt, dass Schrödingeroperatoren kompakte Resolvente besitzen, wenn das zugrundeliegende Potential bei Unendlich gegen Unendlich geht. In diesem Vortrag werden wir einen einfachen Beweis einer Verallgemeinerung präsentieren, basierend auf einer gemeinsamen Arbeit mit D. Lenz (Jena) und D. Wingert.