On rounding error resilience, maximal attainable accuracy and parallel performance of the pipelined Conjugate Gradients method for large-scale linear systems in PETSc

Siegfried Cools 1 Wim Vanroose 1 Emrullah Fatih Yetkin 2 Emmanuel Agullo 2 Luc Giraud 2
1 Universiteit Antwerpen [Antwerpen]
2 HiePACS - High-End Parallel Algorithms for Challenging Numerical Simulations
LaBRI - Laboratoire Bordelais de Recherche en Informatique, Inria Bordeaux - Sud-Ouest
Abstract : Pipelined Krylov solvers typically display better strong scaling compared to standard Krylov methods for large linear systems. The synchronization bottleneck is mitigated by overlapping time-consuming global communications with computations. To achieve this hiding of communication, pipelined methods feature additional recurrence relations on auxiliary variables. This paper analyzes why rounding error effects have a significantly larger impact on the accuracy of pipelined algorithms. An algebraic model for the accumulation of rounding errors in the (pipelined) CG algorithm is derived. Furthermore, an automated residual replacement strategy is proposed to reduce the effect of rounding errors on the final solution. MPI parallel performance tests implemented in PETSc on an Intel Xeon X5660 cluster show that the pipelined CG method with automated residual replacement is more resilient to rounding errors while maintaining the efficient parallel performance obtained by pipelining.
Keywords : Pipelined Krylov methods Conjugate Gradients Parallelism Latency hiding Global communication Rounding error analysis
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https://hal.inria.fr/hal-01734422
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Siegfried Cools, Wim Vanroose, Emrullah Fatih Yetkin, Emmanuel Agullo, Luc Giraud. On rounding error resilience, maximal attainable accuracy and parallel performance of the pipelined Conjugate Gradients method for large-scale linear systems in PETSc. EASC 2016 - Exascale Applications and Software Conference, Apr 2016, Stockholm, Sweden. pp.1-10, ⟨10.1145/2938615.2938621⟩. ⟨hal-01734422⟩

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