分析和改进通信隐藏流水线BiCGStab方法的最大可实现精度
Analyzing and improving maximal attainable accuracy in the communication hiding pipelined BiCGStab method
AI总结：
本文研究了流水线BiCGStab方法在通信隐藏下的最大可实现精度，通过数值稳定性分析量化了局部舍入误差对多项递推中残差的最大可实现精度的影响，并通过数值实验验证了残差替换策略的有效性。
摘要：
流水线Krylov子空间方法通过减少全局同步瓶颈的数量并利用有用的计算工作隐藏全局通信来避免通信延迟。在精确算术中，流水线Krylov子空间算法等同于经典Krylov子空间方法并生成相同的迭代序列。然而，由于算法改写以提高并行性，流水线方法在实际有限精度设置中可能会导致严重减少的可实现精度。本文提出了一个数值稳定性分析，描述并量化了局部舍入误差传播对预处理流水线BiCGStab方法中多术语递推的残差的最大可实现精度的影响。推导了真残差与计算残差之间的差距的理论表达式以及其他算法中使用的辅助变量的差距，并分析了这些差距在各种递归计算向量变量之间的基本依赖关系。对应的传播矩阵和向量的范数提供了局部舍入误差在整个算法中可能放大的见解。通过数值实验将流水线BiCGStab方法的稳定性与流水线CG方法在对称基准问题中的稳定性进行比较。此外，提供了支持使用残差替换类型策略来提高流水线BiCGStab方法最大可实现精度有效性的数值证据。
英文摘要：
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.
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