Iterative projection methods for symmetric nonlinear eigenvalue problems with applications

In this thesis we investigate those nonlinear eigenvalue problems, for which the spectrum of interest allows for a variational characterization. Large sparse problems of this type can therefore be efficiently handled by iterative projection methods, which directly work on the problem without destroying its structure or raising the dimension as it is the case for linearization approaches. The thesis splits in two parts. The first part deals with rational eigenvalue problems arising in the energy band calculation of semiconductor nanostructures described in the framework of k • c theory. We theoretically and numerically investigate various aspects of the modeling of quantum nanostructures. We develop possibly conditional variational principles for the stationary states of four types of Hamiltonians: effective one-band Hamiltonian with and without axial symmetry, including spin-orbit interaction or in the presence of an external magnetic field. Endowed with the minmax theory we devise efficient iterative projection methods especially tailored to the particular problem type. The second part of the thesis is concerned with the computation of eigenvalues in the interior of the spectrum. Such problems are often encountered in acoustics. An example is the gyroscopic eigenvalue problem arising in simulations of the noise radiation of rotating tires. However, even for linear problems the computation of high frequencies is a difficult task. It usually requires „shift-invert“ techniques to prevent convergence break down due to spectral pollution, which can be unpractical for very large problems. An analogous treatment of nonlinear eigenvalue problems would require prior linearization, doubling the size of the original gyroscopic problem and thereby making the approach even less feasible. To overcome these problems we devise a local restart technique for iterative projection methods allowing to compute the higher eigenvalues without maintaining the eigensubspace corresponding to all the eigenvalues computed so far in the search subspace. In this way we prevent an unlimited search subspace growth that would cause severe performance losses. We develop strategies for dealing with spectral pollution, which is intrinsic for higher eigenfrequencies computation. On this basis we devise a method for the computation of interior eigenvalues of gyroscopic problems. Finally, we generalize the ideas to rational eigenvalue problems arising for instance in simulations of fluid structure interaction.