Efficient implementation of the improved quasi-minimal residual method on massively distributed memory computers

Abstract

For the solutions of linear systems of equations with unsymmetric coefficient matrices, we has proposed an improved version of the quasi-minimal residual (IQMR) method by using the Lanczos process as a major component combining elements of numerical stability and parallel algorithm design. For Lanczos process, stability is obtained by a couple two-term procedure that generates Lanczos vectors scaled to unit length. The algorithm is derived such that all inner products and matrix-vector multiplications of a single iteration step are independent and communication time required for inner product can be overlapped efficiently with computation time. Therefore, the cost of global communication on parallel distributed memory computers can be significantly reduced. In this paper, we describe an efficient implementation of this method which is particularly well suited to problems with irregular sparsity pattern. The corresponding communication cost is independent of the sparsity pattern with several performance improvement techniques such as overlapping computation and communication, balancing the computational load. The performance is demonstrated by numerical experimental results carried out on massively parallel distributed memory computer Parsytec GC/Power Plus.