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 solvin g 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. Mul ti-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 deterior ates. The SPAI type preconditioner is inherently parallel and is gaining po pularity. Finally, for the 5-point Laplacian matrix SOR method is known to have a nondeteriorating rate of convergence when the multi-coloring order i s adopted. Also, Block SOR is expected to incur less communication overhead s in a message-passing machine. Hence, Multi-Color Block SOR method is expe cted to have a good performance. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying u p to 1024 x 1024. MPI library was used for interprocess communications. The results show that ILU(0) in the multi-coloring ordering gives the best per formance.