Methods for eigenvalue problems with applications in model order reduction

Abstract

Physical structures and processes are modeled by dynamical systems in a wide range of application areas. The increasing demand for complex components and large structures, together with an increasing demand for detail and accuracy, makes the models larger and more complicated. To be able to simulate these large-scale systems, there is need for reduced-order models of much smaller size, that approximate the behavior of the original model and preserve the important characteristics. In this thesis, algorithms are presented that compute the important characteristics and construct reduced-order models. The eigenvalue problems related to these large-scale dynamical systems are usually too large to be solved completely. The algorithms described in this thesis are efficient and effective methods for the computation of specific dominant eigenvalues. Here, the interpretation of the adjective dominant depends on the application from which the eigenproblem arises. In stability analysis, one is interested in the rightmost eigenvalues, while in other applications the dominant eigenvalues are the natural frequencies. For general linear time invariant dynamical systems, the dominant eigenvalues are the poles of the transfer function that contribute significantly to the frequency response. Many of the methods are based on Newton's method and are similar to (two-sided) Rayleigh quotient iteration. It is shown that compared to (two-sided) Rayleigh quotient iteration, the new methods, Subspace Accelerated Dominant Pole Algorithm (SADPA) and Subspace Accelerated MIMO Dominant Pole Algorithm (SAMDP), have better convergence to the dominant poles of SISO and MIMO transfer functions, respectively. Because SADPA and SAMDP take advantage of efficient deflation for the computation of more than one dominant pole and compute the dominant poles automatically, they are preferred over (two-sided) Jacobi-Davidson methods and Arnoldi based methods. SADPA and SAMDP can be generalized to methods for the computation of dominant transfer function zeros and dominant poles of higher-order transfer functions. The computed eigenvalues and corresponding eigenvectors can be used to construct reduced-order models in the form of modal approximations, but also to improve reduced-order models computed by Krylov subspace based techniques. Large-scale generalized eigenvalue problems arise, for instance, in stability analysis of discretized Navier-Stokes equations and other systems. Due to singularity of one of the matrices in the pencil, there may be eigenvalues at infinity. These eigenvalues at infinity have no physical relevance, but their presence complicates the computation of the rightmost finite eigenvalues. Standard Arnoldi and Jacobi-Davidson approaches may fail because they may interpret approximations of eigenvalues at infinity as approximations to finite eigenvalues. New strategies that combine shift-and-invert and Cayley transformations with purification techniques, and exploit the structure of the eigenvalue problem, are presented to compute the rightmost finite eigenvalues successfully.