Nonlinear Eigenvalue Problems With Low-Rank Damping: Theories And Algorithms

In this thesis, we focus on the nonlinear eigenvalue problems with low-rank damp-ing, which can be found in various scientific and engineering problems.First, we consider a subclass of this kind of problems which can also be regarded as nonlinear low-rank modification of the symmetric eigenvalue problems. They arise from vibration analysis of mechanical structures, fiber optic design and fluid-solid in-teraction problem. We study the existence and distribution of eigenvalues for these problems. Then we study three numerical methods, which are the safeguarded Pi-card iteration, safeguarded nonlinear Rayleigh quotient iteration and successive linear approximation method (SLAM). For SLAM, the global convergence is proved, un-der some mild assumptions. Numerical examples show that SLAM is the most robust method.Then, we consider how to solve a quadratic eigenvalue problem (QEP) with low-rank damping. The damping term of a QEP arising from the structural and acoustic simulation is generally low-rank. We propose a method to explicitly exploit the low-rank damping property for solving the QEP. It is called the trimmed linearization via Pade approximation (TLP) method. The TLP method produces a linear eigenvalue problem of the dimension only slightly larger than the original QEP. Error analysis is made for TLP, based on which a new scaling strategy is also proposed. Numerical ex-periments show that the TLP method is effective and more efficient than the commonly used linearization-based method for solving the QEP.Further, we extend the TLP method to cover general nonlinear eigenvalue problems with low-rank damping. In numerical experiments from, for example, linear accelerator design, we show the efficiency of TLP compared with the nonlinear Arnoldi method. Keywords:nonlinear eigenvalue problem, quadratic eigenvalue problem, low-rank damping, low-rank modification...