Zugl.: Hamburg, Techn. Univ., Diss., 2008

The present paper is devoted to the investigation of the guided wave problem. This problem is formulated as the eigenvalue problem with a compact self-adjoint operator pencil. Applying the minimax principle for the compact operators in the Hilbert space we obtain a necessary and sufficient condition for the existence of a preassigned number of linearly independent guided modes. As a consequence of this result we also derive simple sufficient conditions, which can be easily applied in practice. We give a statement of the problem in a bounded domain and propose an efficient method for solving the problem.

This paper is concerned with the investigation of the nonlinear eigenvalue problem describing the natural oscillations of a plate with a load that elastically attached to it. We study properties of eigenvalues and eigenfunctions of this eigenvalue problem and prove the existence theorem for eigensolutions. Theoretical results are illustrated by numerical experiments.

This is a documentation of the software package COCOS. The purpose of COCOS is the computation of corner singularities of elliptic equations in polyhedral corners and crack tips. COCOS provides a self-contained library for the generation of structured 2D finite element meshes, including various routines for mesh manipulation, as well as several algorithms for the solution of quadratic eigenvalue problems with Hamiltonian structure. These and further features will be described in this documentation.

In this thesis we have studied several aspects of the minimal realization problem for descriptor systems. These aspects include purely theoretical questions such as that about the order of a minimal realization of a general improper rational matrix and problems of a numerical nature, like rounding error analysis of the computing a minimal realization from a nonminimal one. We have also treated the minimal partial realization problem for general descriptor systems with application to model reduction and to generalised eigenvalue problems.

In this paper we develop new preconditioned iterative methods for solving monotone nonlinear eigenvalue problems. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem.

We extend our results on the degree of ill-posedness for linear integration opera- tors A with weights mapping in the Hilbert space L^2(0,1), which were published in the journal 'Inverse Problems' in 2005 ([5]). Now we can prove that the degree one also holds for a family of exponential weight functions. In this context, we empha- size that for integration operators with outer weights the use of the operator AA^* is more appropriate for the analysis of eigenvalue problems and the corresponding asymptotics of singular values than the former use of A^*A.

We consider large and sparse eigenproblems where the spectrum exhibits special symmetries. Here we focus on Hamiltonian symmetry, that is, the spectrum is symmetric with respect to the real and imaginary axes. After briefly discussing quadratic eigenproblems with Hamiltonian spectra we review structured Krylov subspace methods to aprroximate parts of the spectrum of Hamiltonian operators. We will discuss the optimization of the free parameters in the resulting symplectic Lanczos process in order to minimize the conditioning of the (non-orthonormal) Lanczos basis. The effects of our findings are demonstrated for several numerical examples.

Large scale linear time invariant (LTI) systems arise in many physical and technical fields. An approximation, e.g. with model order reduction techniques, of this large systems is crucial for a cost efficient simulation. In this thesis we focus on a model order reduction method based on modal approximation, where the LTI system is projected onto the left and right eigenspaces corresponding to the dominant poles of the system. These dominant poles are related to the most dominant parts of the residue expansion of the transfer function and usually form a small subset of the eigenvalues of the system matrices. The computation of this dominant poles can be a formidable task, since they can lie anywhere inside the spectrum and the corresponding left eigenvectors have to be approximated as well. We investigate the subspace accelerated dominant pole algorithm and the two-sided and alternating Jacobi-Davidson method for this modal truncation approach. These methods can be seen as subspace accelerated versions of certain Rayleigh quotient iterations. Several strategies that admit an efficient computation of several dominant poles of single-input single-output LTI systems are examined. Since dominant poles can lie in the interior of the spectrum, we discuss also harmonic subspace extraction approaches which might improve the convergence of the methods. Extentions of the modal approximation approach and the applied eigenvalue solvers to multi-input multi-output are also examined. The discussed eigenvalue algorithms and the model order reduction approach will be tested for several practically relevant LTI systems.

Chemnitz, Techn. Univ., Studienarb., 2007.