Rational Krylov Methods for Nonlinear Eigenvalue Problems

Eigenvalue problems arise in all fields of scie nce and engineering. The mathematical properties a nd numerical solution methods for standard, linear eigenvalue problems are well understood. How ever, recent advances in several application areas resulted in a new type of eigenvalue pr oblem, i.e., the nonlinear eigenvalue problem whic h exhibits nonlinearity in the eigenvalue paramete r. The goal of this thesis is to develop new rational Krylov methods for solving both sma ll-scale and large-scale nonlinear eigenvalue prob lems. Firstly, by using polynomial and ration al interpolation of the matrix-valued functions, w e obtain methods which are globally convergent ins ide the region of interest. Secondly, lineari zation of the corresponding polynomial and ra tional eigenvalue problems results in linear pencils. Thirdly, the exploitation of the spe cial structure of the linearization pencils a nd possibly a low rank structure results in effici ent and reliable software which is publi cly available. We propose the Compact Ration al Krylov (CORK) method as a generic class of numerical methods for solving nonl inear eigenvalue problems. CORK is characteri zed by a uniform and simple representation of stru ctured linearization pencils. The structure of the se linearization pencils is fully exploited a nd the subspace is represented in a compact f orm. Consequently, we are able to solve problems of high dimension and high degr ee in an efficient and reliable way. Th e family of CORK methods has a lot of flexibi lity for solving the nonlinear eigenvalue pro blem. We discuss three particular types of CORK me thods. The first one is the Newton Rational K rylov method which makes use of dynamic polynomial interpolation. The second one is the Fu lly Rational Krylov method which uses rationa l interpolation and has three viable variants : a static, dynamic, and hybrid variant. The third one is the Infinite Arnoldi m ethod which uses an operator setting to solve the nonlinear eigenvalue prob lem. Finally, the proposed methods are u sed to solve applications from mechanical eng ineering, quantum physics, and civil eng ineering which were not solved earlier with t he same efficiency and reliability.nrpages: xix+175status: publishe