交织PFASST与并行多重网格
Interweaving PFASST and Parallel Multigrid
摘要：
时空并行全近似方案（PFASST）是由Emmett和Minion在2012年提出的一种用于常微分方程和离散偏微分方程时间并行化的迭代策略。顾名思义，PFASST在精神上类似于在多个时间步长上并行执行的时空FAS多重网格方法。然而，由于PFASST的原始重点一直是在时间并行性方面的方法的性能，任何空间系统的解决方案所产生的隐式或半隐式的时间方法PFASST内使用已简单地假设解决到一些所需的精度完全在每个子步骤和每次迭代的一些未指定的程序。因此，很自然地要研究空间维度上的迭代求解器如何与PFASST迭代交织在一起，以及这种策略是否会导致更有效的整体方法。本文提出了一个初步调查的相对性能不同的策略耦合PFASST迭代与多重网格方法隐式处理的扩散项偏微分方程。特别是，我们比较全精度多重网格解决在每个子步骤与一个小的固定数量的多重网格V-循环。这降低了每次PFASST迭代的成本，但可能以收敛所需的PFASST迭代次数的相应增加为代价。通过数值算例探讨了所得到的方法的并行效率。
英文摘要：
The parallel full approximation scheme in space and time (PFASST) introduced by Emmett and Minion in 2012 is an iterative strategy for the temporal parallelization of ODEs and discretized PDEs. As the name suggests, PFASST is similar in spirit to a space-time FAS multigrid method performed over multiple time-steps in parallel. However, since the original focus of PFASST has been on the performance of the method in terms of time parallelism, the solution of any spatial system arising from the use of implicit or semi-implicit temporal methods within PFASST have simply been assumed to be solved to some desired accuracy completely at each sub-step and each iteration by some unspecified procedure. It hence is natural to investigate how iterative solvers in the spatial dimensions can be interwoven with the PFASST iterations and whether this strategy leads to a more efficient overall approach. This paper presents an initial investigation on the relative performance of different strategies for coupling PFASST iterations with multigrid methods for the implicit treatment of diffusion terms in PDEs. In particular, we compare full accuracy multigrid solves at each sub-step with a small fixed number of multigrid V-cycles. This reduces the cost of each PFASST iteration at the possible expense of a corresponding increase in the number of PFASST iterations needed for convergence. Parallel efficiency of the resulting methods is explored through numerical examples.
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