Comparisons of the Parallel Preconditioners on the Cray-T3E for Large Nonsymmetric Linear Systems

In this paper we consider five types of parallel preconditioners for solving large sparse nonsymmetric linear systems on the CRAY-T3E. They are ILU(0) in the wavefront ordering, ILU(0) in the multi-coloring ordering, SSOR in the wavefront ordering, the SPAI(SParse Approximate Inverse) preconditioner, and finally Multi-color Block SOR preconditioner. The ILU(0) is known to be robust and the wavefront ordering naturally exploits the parallelism but has a limited speedup due to the nonuniform lengths of the wavefronts. Multi-coloring is an efficient way of introducing the parallelism of order(N), where N is the order of the matrix but the convergence rate often deteriorates. The SPAI type preconditioner is inherently parallel and is gaining popularity. Finally, for the 5-point Laplacian matrix SOR method is known to have a nondeteriorating rate of convergence when the multi-coloring order is adopted. Also, Block SOR is expected to incur less communication overheads in a message-passing machine. Hence, Multi-Color Block SOR method is expected to have a good performance. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying up to 1024×1024. MPI library was used for interprocess communications. The results show that ILU(0) in the multi-coloring ordering gives the best performance.