2018/09/06 by Siegfried Cools, Cools, Siegfried
Computer Science · Engineering · Physics and Astronomy · #65F10 #65G50 #65N12 #65N22 #65Y05 #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Scattering and Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1809.01948
openalex publication_date 2018/09/06 · openalex created_date 2022/08/03 · openalex updated_date 2026/07/28
Pipelined Krylov subspace methods avoid communication latency by reducing the\nnumber of global synchronization bottlenecks and by hiding global communication\nbehind useful computational work. In exact arithmetic pipelined Krylov subspace\nalgorithms are equivalent to classic Krylov subspace methods and generate\nidentical series of iterates. However, as a consequence of the reformulation of\nthe algorithm to improve parallelism, pipelined methods may suffer from\nseverely reduced attainable accuracy in a practical finite precision setting.\nThis work presents a numerical stability analysis that describes and quantifies\nthe impact of local rounding error propagation on the maximal attainable\naccuracy of the multi-term recurrences in the preconditioned pipelined BiCGStab\nmethod. Theoretical expressions for the gaps between the true and computed\nresidual as well as other auxiliary variables used in the algorithm are\nderived, and the elementary dependencies between the gaps on the various\nrecursively computed vector variables are analyzed. The norms of the\ncorresponding propagation matrices and vectors provide insights in the possible\namplification of local rounding errors throughout the algorithm. Stability of\nthe pipelined BiCGStab method is compared numerically to that of pipelined CG\non a symmetric benchmark problem. Furthermore, numerical evidence supporting\nthe effectiveness of employing a residual replacement type strategy to improve\nthe maximal attainable accuracy for the pipelined BiCGStab method is provided.\n