Orderings for Parallel Conjugate Gradient Preconditioners

Abstract

The efficient solution of the linear system of equations $A\mathbf{x} = \mathbf{b}$ on parallel computers sometimes requires a reordering of the unknowns. We investigate the effects of various orderings on the rate of convergence and the computation rate of the preconditioned conjugate method with SSOR as the preconditioner. As a model problem, we use a generalized Poisson equation $\nabla (K \nabla u ) = f$, discretized with finite differencing in two and three dimensions. The natural ordering has the highest rate of convergence yet results in poor to modest parallelism, whereas the classical red/black (RB) ordering fully exploits vector/parallel properties but has a deleterious effect on the rate of convergence. Intermediate between the RB and natural orderings are the ``many-color'' orderings studied by Harrar and Ortega, but we show that these have rather poor communication properties. Alternatives are domain decomposition orderings, which we consider with and without separator sets. We investigate the effectiveness of these orderings on parallel machines, the Intel iPSC/860 hypercube in particular. We also examine the effect of the remainder matrix on the rate of convergence.