General Framework for Deriving Reproducible Krylov Subspace Algorithms: BiCGStab Case

Abstract

Parallel implementations of Krylov subspace algorithms often help to accelerate the procedure to find the solution of a linear system. However, from the other side, such parallelization coupled with asynchronous and out-of-order execution often enlarge the non-associativity of floating-point operations. This results in non-reproducibility on the same or different settings. This paper proposes a general framework for deriving reproducible and accurate variants of a Krylov subspace algorithm. The proposed algorithmic strategies are reinforced by programmability suggestions to assure deterministic and accurate executions. The framework is illustrated on the preconditioned BiCGStab method for the solution of non-symmetric linear systems with message-passing. Finally, we verify the two reproducible variants of PBiCGStab on a set matrices from the SuiteSparse Matrix Collection and a 3D Poisson’s equation.