The communication-hiding pipelined BiCGStab method for the parallel solution of large unsymmetric linear systems

TL;DR

This paper introduces a pipelined BiCGStab method that overlaps communication with computation to improve scalability in parallel computing environments for solving large unsymmetric linear systems.

Contribution

It develops a general framework for creating pipelined Krylov subspace algorithms and applies it to derive a pipelined BiCGStab method with a residual replacement strategy.

Findings

Significant speedups over standard BiCGStab on distributed systems

Enhanced scalability due to communication hiding

Maintains robustness with residual replacement strategy

Abstract

A High Performance Computing alternative to traditional Krylov subspace methods, pipelined Krylov subspace solvers offer better scalability in the strong scaling limit compared to standard Krylov subspace methods for large and sparse linear systems. The typical synchronization bottleneck is mitigated by overlapping time-consuming global communication phases with local computations in the algorithm. This paper describes a general framework for deriving the pipelined variant of any Krylov subspace algorithm. The proposed framework was implicitly used to derive the pipelined Conjugate Gradient (p-CG) method in "Hiding global synchronization latency in the preconditioned Conjugate Gradient algorithm" by P. Ghysels and W. Vanroose, Parallel Computing, 40(7):224--238, 2014. The pipelining framework is subsequently illustrated by formulating a pipelined version of the BiCGStab method for the solution of large unsymmetric linear systems on parallel hardware. A residual replacement strategy is proposed to account for the possible loss of attainable accuracy and robustness by the pipelined BiCGStab method. It is shown that the pipelined algorithm improves scalability on distributed memory machines, leading to significant speedups compared to standard preconditioned BiCGStab.