General framework for re-assuring numerical reliability in parallel Krylov solvers: A case of BiCGStab methods

TL;DR

This paper introduces a general framework to enhance the numerical reliability of parallel Krylov subspace methods, specifically BiCGStab, by ensuring reproducibility and deterministic execution in high-performance computing environments.

Contribution

It proposes a novel framework for deriving reproducible and accurate variants of Krylov methods, with implementation strategies for deterministic parallel computations.

Findings

Reproducible BiCGStab variants show improved numerical stability.

Framework ensures deterministic results in parallel Krylov methods.

Validated on diverse matrices and 3D Poisson problem.

Abstract

Parallel implementations of Krylov subspace methods often help to accelerate the procedure of finding an approximate solution of a linear system. However, such parallelization coupled with asynchronous and out-of-order execution often enlarge the non-associativity impact in floating-point operations. These problems are even amplified when communication-hiding pipelined algorithms are used to improve the parallelization of Krylov subspace methods. Introducing reproducibility in the implementations avoids these problems by getting more robust and correct solutions. This paper proposes a general framework for deriving reproducible and accurate variants of Krylov subspace methods. The proposed algorithmic strategies are reinforced by programmability suggestions to assure deterministic and accurate executions. The framework is illustrated on the preconditioned BiCGStab method and its pipelined modification, which in fact is a distinctive method from the Krylov subspace family, for the solution of non-symmetric linear systems with message-passing. Finally, we verify the numerical behaviour of the two reproducible variants of BiCGStab on a set of matrices from the SuiteSparse Matrix Collection and a 3D Poisson's equation.