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General framework for deriving reproducible Krylov subspace algorithms: A BiCGStab case study

: Iakymchuk, Roman; Graillat, Stef; Aliaga, José

Fulltext ()

2021, Paper hal-03382119, 13 pp.
European Commission EC
H2020; 842528; Robust
Robust and Energy-Efficient Numerical Solvers Towards Reliable and Sustainable Scientific Computations
French National Agency for Research ANR
ANR-20-CE46-0009; InterFLOP
Report, Electronic Publication
Fraunhofer ITWM ()
reproducibility; accuracy; floating-point expansion; long accumulator; fused multiply-add; preconditioned BiCGStab; High-Performance Computing

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 in parallel environments with message-passing. Finally, we verify the two reproducible variants of PBiCGStab on a set matrices from the SuiteSparse Matrix Collection.