Parallel algorithms for large sparse generalized eigenproblems

Zhanye Tong, Purdue University

Abstract

This dissertation discusses parallel algorithms for the generalized eigenvalue problem Ax = λBx where both A and B are real matrices with B being symmetric positive definite. We are interested in the case that both A and B are large and sparse, and only a few of the eigenpairs are desired. Although many methods for such problems have been developed, most of them require factorizing the involved matrices or solving a large number of linear systems to high order accuracy. Moreover, they are usually designed exclusively for the traditional vector machines and don't perform well in distributed computing environments—multiprocessor machines with distributed memory or networks of workstations (NOW). In this thesis, we develop algorithms that use the basic matrix vector operations only, that don't require solving any linear systems exactly, and that are efficient in distributed computing environments. Among all the practical eigenvalue methods known today, we have found the trace minimization method and the Jacobi-Davidson method more attractive than others in the sense that they admit both data parallelism and task parallelism. We first study the trace minimization method, give a convergence analysis for practical situations, introduce a preconditioning technique, and develop more efficient shifting strategies. We then study the Jacobi-Davidson method, explore its connection to the trace minimization method, analyze its weaknesses, and incorporate some of the techniques developed in the trace minimization method into it to obtain a new algorithm. The implementation details of the new algorithm in distributed computing environments are then discussed. Numerical experiments have shown that the modified algorithm is more efficient than the original block Jacobi-Davidson algorithm for a variety of problems.

Degree

Ph.D.

Advisors

Sameh, Purdue University.

Subject Area

Computer science

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