Young Researchers' Minisymposium

YR Ma2 Nonlinear and structured eigenvalue problems
Elias Jarlebring (KU Leuven), Christian Schröder (TU Berlin)

Eigenvalue problems (EVPs) represents a useful tool in the analysis, characterization and solution of many current problems in physical sciences and engineering. They appear in a wide variety of fields and there exist very mature general purpose numerical methods for the most commonly appearing types of eigenvalue problems. The general purpose methods are, however, not always sufficient in practice, mostly due to the fact that there is an interest in models and problems with growing complexity.

Many of these more complex models can be formulated as nonlinear eigenvalue problems where the matrix depends on the eigenvalue parameter in a nonlinear way. The nonlinear eigenvalue problem is a very general type of problem and there are very few general purpose methods. In order to treat and analyze different types of nonlinearities adequately, the specific type of nonlinearity often has to be exploited. Examples are nonlinear EVPs stemming from delay-differential equations. New Arnoldi- and Jacobi-Davidson methods for these problems will be presented. Another source of nonlinear EVPs is the discretization of nonlinear PDEs. Preconditioning techniques can help during their solution.

Eigenvalue problems stemming from practical applications often exhibit some kind of inherent structure. In such situations it is essential to understand, characterize and preserve these structures in order to obtain physically meaningful results. Moreover, the exploitation of present structures may allow more efficient and robust methods and therefor answers the demand of ever increasing problem dimensions. Structures covered in this symposium include symmetry, low rank structure and palindromicity as well as two-parameter structure.

An important subclass of nonlinear EVPs are polynomial EVPs that arise e.g. from higher order differential equations. These problems can be reformulated as linear EVPs, but the well known companion form may not be the best linearization, especially for structured polynomials. Alternative choices for palindromic and two-parameter polynomials are covered by talks.

A further topic will be nonlinear structured eigenvalue perturbation theory.

With a minisymposium on these topics we will bring together young researchers working on nonlinear eigenvalue problems of different types and structures.

Kemal Yildiztekin (TU Hamburg-Harburg): Nonlinear low rank perturbation of symmetric nonlinear eigenvalue problems

Vanni Noferini (U Pisa): A new structured linearization for the palindromic eigenvalue problem

Elias Jarlebring (KU Leuven): Arnoldi on a function: an Arnoldi method for nonlinear eigenvalue problems

Cedric Effenberger (ETH ZĂĽrich): Preconditioned solution methods for nonlinear PDE eigenvalue problems

Andrej Muhic (U Ljubljana): Numerical solution of polynomial two-parameter eigenvalue problems

Christian Schröder (TU Berlin): A nonlinear eigenvalue problem in two real parameters arising in delay differential equations