A Parallel Implementation of the Block Preconditioned GCR Method

The parallel implementation of GCR is addressed, with particular focus on communication costs associated with orthogonalization processes. This consideration brings up questions concerning the use of Householder reflections with GCR. To precondition the GCR method a block Gauss-Jacobi method is used. Approximate solvers are used to obtain a solution of the diagonal blocks. Experiments on a cluster of HP workstations and on a Cray T3E are given. 1 Introduction This paper addresses the parallel implementation of a Krylov accelerated block Gauss-Jacobi method for the DeFT Navier-Stokes solver described in [15], and is the continuation of work summarized in [5]. Results from a parallel implementation of a Krylov-accelerated Schur complement domain decomposition method are presented in [3]. We report results for a Poisson problem on a square domain, which is representative of the system which must be solved for the pressure correction method used in DEFT. Aside from the preconditioning, the main parallel operations required in these methods are distributed matrix-vector multiplications and inner products. For many problems, the matrix-vector multiplications require only nearest neighbor communications, and are very efficient. Inner products, on the other hand, require global communications; therefore, the focus has been on reducing the number of inner products [8, 16], overlapping inner product communications with computation [6], or increasing the number of inner products that can be computed with a single communication [2, 13]. The block Gauss-Jacobi preconditioner is described in Section 1.1. Parallel implementations of orthogonalization procedures for the GCR (Generalized Conjugate