Implementation of a preconditioned eigensolver using Hypre
Andrew V. Knyazev 1 , and Merico E. Argentati 1
1 Department of Mathematics, University of Colorado at Denver, USA
SUMMARY
This paper describes a parallel preconditioned algorithm for solution of partial eigenvalue problems for large sparse symmetric matrices on massively parallel computers, taking advantage of advances in the scalable linear solvers, in particular in multigrid technology and in incomplete factorizations, developed under the Hypre project, at the Lawrence Livermore National Laboratory (LLNL), Center for Applied Scientific Computing. The algorithm implements a ”matrix free” locally optimal block preconditioned conjugate gradient method (LOBPCG), suggested earlier by the first author, to compute one or more of the smallest eigenvalues and the corresponding eigenvectors of a symmetric matrix. We discuss our Hypre specific implementation approach for a flexible parallel algorithm, and the capabilities of the developed software. We demonstrate parallel performance on a set of test problems, using Hypre algebraic multigrid and other preconditioners. The code is written in MPI based C-language and uses Hypre and LAPACK libraries. It has been included in Hypre starting with revision 1.8.0b, which is publicly available on the Internet at LLNL web site. Copyright c 2005 John Wiley & Sons, Ltd.
KEY WORDS: Eigenvalue; eigenvector; symmetric eigenvalue problem; preconditioning; sparse matrices; parallel computing; conjugate gradients; algebraic multigrid; additive Schwarz; incomplete factorization; Hypre. AMS(MOS) subject classifications. 65F15, 65N25, 65Y05.
1. Introduction
We implement a parallel preconditioned algorithm of the Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG) [3, 4] for the solution of eigenvalue problems Ax = λx for large sparse symmetric matrices A on massively parallel computers. We take advantage of advances in the scalable linear solvers, in particular in multigrid technology and in incomplete factorizations, developed under the High Performance Preconditioners (Hypre) project [5], at the Lawrence Livermore