Analyzing and improving maximal attainable accuracy in the communication hiding pipelined BiCGStab method

Pipelined Krylov subspace methods avoid communication latency by reducing the number of global synchronization bottlenecks and by hiding global communication behind useful computational work. In exact arithmetic pipelined Krylov subspace algorithms are equivalent to classic Krylov subspace methods and generate identical series of iterates. However, as a consequence of the reformulation of the algorithm to improve parallelism, pipelined methods may suffer from severely reduced attainable accuracy in a practical finite precision setting. This work presents a numerical stability analysis that describes and quantifies the impact of local rounding error propagation on the maximal attainable accuracy of the multi-term recurrences in the preconditioned pipelined BiCGStab method. Theoretical expressions for the gaps between the true and computed residual as well as other auxiliary variables used in the algorithm are derived, and the elementary dependencies between the gaps on the various recursively computed vector variables are analyzed. The norms of the corresponding propagation matrices and vectors provide insights in the possible amplification of local rounding errors throughout the algorithm. Stability of the pipelined BiCGStab method is compared numerically to that of pipelined CG on a symmetric benchmark problem. Furthermore, numerical evidence supporting the effectiveness of employing a residual replacement type strategy to improve the maximal attainable accuracy for the pipelined BiCGStab method is provided.