Pipelined Preconditioned Conjugate Gradient Methods for Distributed Memory Systems

Abstract

Preconditioned Conjugate Gradient (PCG) method has been one of the widely used methods for solving linear systems of equations for sparse problems. Pipelined PCG (PIPECG) attempts to eliminate the dependencies in the computations in the PCG algorithm and overlap non-dependent computations by reorganizing the traditional PCG code and using non-blocking allreduces. We have developed a novel pipelined PCG algorithm called PIPECG-OATI (One Allreduce per Two Iterations) that provides large overlap of global communication and computations at higher number of cores in distributed memory CPU systems. Our method achieves this overlapping by using iteration combination and by introducing new non-recurrence computations. We compare our method with other pipelined CG methods on a variety of problems and demonstrate that our method always gives the least runtimes. Our method gives up to 3x speedup over PCG method and 1.73x speedup over PIPECG method at large number of cores.