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Parallel implementations of Krylov subspace methods often help to acceleratethe procedure of finding an approximate solution of a linear system. However,such parallelization coupled with asynchronous and out-of-order execution oftenenlarge the non-associativity impact in floating-point operations. Theseproblems are even amplified when communication-hiding pipelined algorithms areused to improve the parallelization of Krylov subspace methods. Introducingreproducibility in the implementations avoids these problems by getting morerobust and correct solutions. This paper proposes a general framework forderiving reproducible and accurate variants of Krylov subspace methods. Theproposed algorithmic strategies are reinforced by programmability suggestionsto assure deterministic and accurate executions. The framework is illustratedon the preconditioned BiCGStab method and its pipelined modification, which infact is a distinctive method from the Krylov subspace family, for the solutionof non-symmetric linear systems with message-passing. Finally, we verify thenumerical behaviour of the two reproducible variants of BiCGStab on a set ofmatrices from the SuiteSparse Matrix Collection and a 3D Poisson's equation.

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