Rational Krylov Methods for Nonlinear Eigenvalue Problems

Eigenvalue problems arise in all fields of science and engineering. The mathematical properties and numerical solution methods for standard, linear eigenvalue problems are well understood. However, recent advances in several application areas resulted in a new type of eigenvalue problem, i.e., the nonlinear eigenvalue problem which exhibits nonlinearity in the eigenvalue parameter. The goal of this thesis is to develop new rational Krylov methods for solving both small-scale and large-scale nonlinear eigenvalue problems. Firstly, by using polynomial and rational interpolation of the matrix-valued functions, we obtain methods which are globally convergent inside the region of interest. Secondly, linearization of the corresponding polynomial and rational eigenvalue problems results in linear pencils. Thirdly, the exploitation of the special structure of the linearization pencils and possibly a low rank structure results in efficient and reliable software which is publicly available. We propose the Compact Rational Krylov (CORK) method as a generic class of numerical methods for solving nonlinear eigenvalue problems. CORK is characterized by a uniform and simple representation of structured linearization pencils. The structure of these linearization pencils is fully exploited and the subspace is represented in a compact form. Consequently, we are able to solve problems of high dimension and high degree in an efficient and reliable way. The family of CORK methods has a lot of flexibility for solving the nonlinear eigenvalue problem. We discuss three particular types of CORK methods. The first one is the Newton Rational Krylov method which makes use of dynamic polynomial interpolation. The second one is the Fully Rational Krylov method which uses rational interpolation and has three viable variants: a static, dynamic, and hybrid variant. The third one is the Infinite Arnoldi method which uses an operator setting to solve the nonlinear eigenvalue problem. Finally, the proposed methods are used to solve applications from mechanical engineering, quantum physics, and civil engineering which were not solved earlier with the same efficiency and reliability.

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