Date of Award
Open Access Dissertation
Doctor of Philosophy in Mathematical Sciences (PhD)
Administrative Home Department
Department of Mathematical Sciences
Committee Member 1
Committee Member 2
Committee Member 3
This dissertation focuses on novel computational method for eigenvalue problems. In Chapter 1, preliminaries of functional analysis related to eigenvalue problems are presented. Some classical methods for matrix eigenvalue problems are discussed. Several PDE eigenvalue problems are covered. The chapter is concluded with a summary of the contributions. In Chapter 2, a novel recursive contour integral method (RIM) for matrix eigenvalue problem is proposed. This method can effectively find all eigenvalues in a region on the complex plane with no a priori spectrum information. Regions that contain eigenvalues are subdivided and tested recursively until the size of region reaches specified precision. The method is robust, which is demonstrated using various examples. In Chapter 3, we propose an improved version of RIM for non-Hermitian eigenvalue problems, called SIM-M. By incorporating Cayley transformation and Arnoldi’s method, the main computation cost of solving linear systems is reduced significantly. The numerical experiments demonstrate that RIM-M gains significant speed-up over RIM. In Chapter 4, we propose a multilevel spectral indicator method (SIM-M) to address the memory requirement for large sparse matrices. We modify the indicator of RIM-M such that it requires much less memory. Matrices from University of Florida Sparse Matrix Collection are tested, suggesting that a parallel version of SIM-M has the potential to be efficient. In Chapter 5, we develop a novel method to solve the elliptic PDE eigenvalue problem. We construct a multi-wavelet basis with Riesz stability in H1 0 ( ). By incorporating multi-grid discretization scheme and sparse grids, the method retains the optimal convergence rate for the smallest eigenvalue with much less computational cost.
Huang, Ruihao, "Novel Computational Methods for Eigenvalue Problems", Open Access Dissertation, Michigan Technological University, 2019.