Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

Abstract

Summary This paper discusses techniques for computing a few selected eigenvalue–eigenvector pairs of large and sparse symmetric matrices. A recently developed class of techniques to solve this type of problems is based on integrating the matrix resolvent operator along a complex contour that encloses the interval containing the eigenvalues of interest. This paper considers such contour integration techniques from a domain decomposition viewpoint and proposes two schemes. The first scheme can be seen as an extension of domain decomposition linear system solvers in the framework of contour integration methods for eigenvalue problems, such as FEAST. The second scheme focuses on integrating the resolvent operator primarily along the interface region defined by adjacent subdomains. A parallel implementation of the proposed schemes is described, and results on distributed computing environments are reported. These results show that domain decomposition approaches can lead to reduced run times and improved scalability.

Krylov eigenvalue strategy using the FEAST algorithm with inexact system solves

“…A recent parallel FEAST (PFEAST) implementation was proposed for solving larger system sizes of this kind, taking advantage of distributed-memory sparse linear system solvers and domain decomposition techniques. 5,14 In very large-scale applications, however, the structure of the matrix A causes the factorization step to be extremely slow and expensive to perform, and the storage of the factorization may even be impossible due to memory constraints. In some other cases, matrix A is too large and dense to be stored at all and is instead being represented implicitly by a rule for performing fast matrix-vector products (see the work of Giannozzi et al 15 for an example of an application where this approach is used).…”

“…A direct solver requires that one be able to form and store a factorization of the matrices ( z k I − A ). A recent parallel FEAST (PFEAST) implementation was proposed for solving larger system sizes of this kind, taking advantage of distributed‐memory sparse linear system solvers and domain decomposition techniques . In very large‐scale applications, however, the structure of the matrix A causes the factorization step to be extremely slow and expensive to perform, and the storage of the factorization may even be impossible due to memory constraints.…”

Recursive integral method with Cayley transformation

“…RIM uses an analog of bisection for a search region on the complex plane. This is different than the domain decomposition technique used in the work of Kalantzis et al, in which iterative solutions of the linear systems encountered in the FEAST algorithm are accelerated by utilizing domain decomposition preconditioners to solve the complex linear systems with multiple right‐hand sides. Note that a contour integral method was used for multiplicity counting of eigenvalues in the work of Di Napoli et al…”

Solving generalized eigenvalue problems for large scale fluid-structure computational models with mid-power computers

“…The algorithms for solving eigenvalue problems (including generalized eigenvalue problems for which one matrix is positive definite) have received a very great attention this last 40 years from a mathematical point of view (see for instance, [1,2,3,4,5,6,7,8,9]), for algorithms adapted to parallel computation (see for instance, [10,11,12,13,14,15,16,17,18]), and also for massively parallel computers (see for instance, [19,20,21,22,23]). The majority of the efficient algorithms have been implemented in a mathematical library for computers, parallel computers, and massively parallel computers (see for instance, [24,25,26]).…”