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By reducing the number of global synchronization bottlenecks per iterationand hiding communication behind useful computational work, pipelined Krylovsubspace methods achieve significantly improved parallel scalability onpresent-day HPC hardware. However, this typically comes at the cost of areduced maximal attainable accuracy. This paper presents and compares severalstabilized versions of the communication-hiding pipelined Conjugate Gradientsmethod. The main novel contribution of this work is the reformulation of themulti-term recurrence pipelined CG algorithm by introducing shifts in therecursions for specific auxiliary variables. These shifts reduce theamplification of local rounding errors on the residual. The stability analysispresented in this work provides a rigorous method for selection of the optimalshift value in practice. It is shown that, given a proper choice for the shiftparameter, the resulting shifted pipelined CG algorithm restores the attainableaccuracy and displays nearly identical robustness to local rounding errorpropagation compared to classical CG. Numerical results on a variety of SPDbenchmark problems compare different stabilization techniques for the pipelinedCG algorithm, showing that the shifted pipelined CG algorithm is able to attaina high accuracy while displaying excellent parallel performance.

Numerically Stable Variants of the Communication-hiding Pipelined  Conjugate Gradients Algorithm for the Parallel Solution of Large Scale  Symmetric Linear Systems