Pipelined Krylov subspace methods achieve significantly improved parallel scalability compared to standard Krylov methods on present-day HPC hardware. However, this typically comes at the cost of a reduced maximal attainable accuracy. This paper presents and compares several stabilized versions of the communication-hiding pipelined Conjugate Gradients method. The main novel contribution of this work is the reformulation of the multi-term recurrence pipelined CG algorithm by introducing shifts in the recursions for specific auxiliary variables. These shifts reduce the amplification of local rounding errors on the residual. Given a proper choice for the shift parameter, the amplification of local rounding errors is reduced and the resulting algorithm improves the attainable accuracy. Numerical results on a variety of SPD benchmark problems compare different stabilization techniques for the pipelined CG algorithm, showing that the stabilized algorithms are able to attain a high accuracy while displaying excellent parallel performance.
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