COMPARISONS OF THE PARALLEL PRECONDITIONERS FOR LARGE NONSYMMETRIC SPARSE LINEAR SYSTEMS ON A PARALLEL COMPUTER

Abstract

In this paper we compare various parallel preconditioners for solving large sparse nonsymmetric linear systems. They are Block Jacobi, Point-SSOR, ILU(0) in the wavefront order, ILU(0) in the multi-color order, SPAI(SParse Approximate Inverse), and Multi-Color Block SOR. The Block jacobi and Point-SSOR are well-known, and ILU(0) is a one of the most popular preconditioner, but it is inherently {\em serial}. ILU(0) in the wavefront order maximizes the parallelism, and ILU(0) in the multi-color order achieves the parallelism of order($N$), where $N$ is the order of the matrix. The SPAI tries to capture the approximate inverse in sparse form, which, then, is expected to be a scalable preconditioner. Finally, we implemented the Multi-Color Block SOR preconditioner combined with direct sparse matrix solver. For the Laplacian matrix the SOR method is known to have a nondeteriorating rate of convergence when used with Multi-Color ordering. Since most of the time is spent on the diagonal inversion, which is done on each processor, we expect it to be a good scalable preconditioner. Finally, due to the blocking effect, it will be effective for ill-conditioned problems. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying up to 1024 x 1024 , and for an ill-conditioned matrix from the shell problem from the Harwell-Boeing collection. CRAY-T3E with 128 nodes was used. MPI library was used for interprocess communications. The results show that Multi-Color Block SOR and ILU(0) with Multi-Color ordering give the best performances for the finite difference matrices and for the shell problem only the Multi-Color Block SOR and Block Jacobi converges. Based on this we recommend that the Multi-Color Block SOR is the most robust preconditioner out of the preconditioners considered.