具有深管道的通信隐式共轭梯度法
The Communication-Hiding Conjugate Gradient Method with Deep Pipelines
摘要：
Krylov子空间方法是求解大规模线性代数问题最有效的方法之一。然而，经典的Krylov子空间算法由于同步瓶颈而不能在大规模并行硬件上很好地扩展。通信隐藏流水线Krylov子空间方法通过将耗时的全局通信阶段与诸如spmvs的计算重叠来提供增加的并行可扩展性，从而减少全局同步的影响并避免处理器空闲。在这个类中最早发布的方法之一是流水线共轭梯度方法（p-CG）。然而，在大量处理器上，通信阶段可能比单个spmv的计算花费更长的时间。这项工作将流水线CG方法扩展到更深的流水线，表示为p（1）-CG，当全局通信阶段是主要的耗时因素时，这允许进一步缩放。通过将每次CG迭代中的全局全对全归约阶段与接下来的l个spmvs（深度流水线）重叠，该方法将通信延迟隐藏在额外的计算工作后面。p（1）-CG算法是从与现有p（1）-GMRES方法类似的原理并通过利用算子对称性导出的。还将p（l）-CG方法与其他Krylov子空间方法进行了比较，包括密切相关的经典CG和D-Lanczos方法以及Ghysels等人的流水线CG方法。通过分析p（1）-CG方法所能达到的最大精度，表明流水线技术在性能和数值稳定性之间进行了权衡。预处理版本的算法也提出了存储要求和性能估计进行了讨论。实验结果表明，可能的性能增益和可达到的精度更深的流水线CG解决大规模对称线性系统。
英文摘要：
Krylov subspace methods are among the most efficient solvers for large scale linear algebra problems. Nevertheless, classic Krylov subspace algorithms do not scale well on massively parallel hardware due to synchronization bottlenecks. Communication-hiding pipelined Krylov subspace methods offer increased parallel scalability by overlapping the time-consuming global communication phase with computations such as spmvs, hence reducing the impact of the global synchronization and avoiding processor idling. One of the first published methods in this class is the pipelined Conjugate Gradient method (p-CG). However, on large numbers of processors the communication phase may take much longer than the computation of a single spmv. This work extends the pipelined CG method to deeper pipelines, denoted as p(l)-CG, which allows further scaling when the global communication phase is the dominant time-consuming factor. By overlapping the global all-to-all reduction phase in each CG iteration with the next l spmvs (deep pipelining), the method hides communication latency behind additional computational work. The p(l)-CG algorithm is derived from similar principles as the existing p(l)-GMRES method and by exploiting operator symmetry. The p(l)-CG method is also compared to other Krylov subspace methods, including the closely related classic CG and D-Lanczos methods and the pipelined CG method by Ghysels et al.. By analyzing the maximal accuracy attainable by the p(l)-CG method it is shown that the pipelining technique induces a trade-off between performance and numerical stability. A preconditioned version of the algorithm is also proposed and storage requirements and performance estimates are discussed. Experimental results demonstrate the possible performance gains and the attainable accuracy of deeper pipelined CG for solving large scale symmetric linear systems.
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