Comparisons of Parallel Preconditioners for the Computation of Interior Eigenvalues by a CG-type Method on a Parallel Computer

Abstract

Recently iterative algorithms based on the optimization of the Rayleigh quotient have been developed, and a CG scheme for the optimization of the Rayleigh quotient has been proven to be a very attractive and promising technique for large sparse eigenproblems for interior eigenvalues. Ax = /spl lambda/Bx (1) The given matrices A, and B are assumed to be large and sparse, and symmetric and B is further assumed to be positive definite. Also, the method is very amenable to parallel computations. A proper choice of the preconditioner significantly improves the convergence of the CG scheme. We compare the parallel preconditioners for the computation of the interior eigenvalues of a symmetric matrix by CG-type method. The considered preconditioners are ILU(0) in the natural order, ILU(0) in the multi-coloring order, and multi-color block SSOR (symmetric successive overrelaxation). Our results were implemented on the CRAY-T3E with 128 nodes, assuming B = I. The MPI (Message Passing Interface) library was adopted for the interprocessor communications. The test matrices are up to 512/spl times/512 in dimensions and were created from the discretizations of the elliptic PDE. All things considered the MC-BSSOR seems to be most robust preconditioner.