Scheduling Massively Parallel Multigrid for Multilevel Monte Carlo Methods
Scheduling Massively Parallel Multigrid for Multilevel Monte Carlo Methods
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DOI：
10.1137/16m1083591
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发表时间：
2016-07
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通讯作者：
D. Drzisga;B. Gmeiner;U. Rüde;Robert Scheichl;B. Wohlmuth
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作者：
D. Drzisga;B. Gmeiner;U. Rüde;Robert Scheichl;B. Wohlmuth
基于采样的三维偏微分方程不确定性量化的计算复杂度是非常高的。多层蒙特卡罗（MLMC）等多层方法与快速多网格求解器相结合可以显著降低复杂性，但要在并行环境中充分利用它们，需要复杂的调度策略。我们优化了MLMC方法的三层并发执行：跨层并行化、跨样本并行化和跨空间网格并行化。在一系列数值测试中，说明了对多网格求解器的“可伸缩性窗口”的整体性能的影响（即，可以保持良好并行效率的处理器数量范围）。提出并讨论了不同的同构调度策略和异构调度策略。最后，进行了大规模的三维缩放实验，包括自适应实验。
The computational complexity of naive, sampling-based uncertainty quantification for 3D partial differential equations is extremely high. Multilevel approaches, such as multilevel Monte Carlo (MLMC), can reduce the complexity significantly when they are combined with a fast multigrid solver, but to exploit them fully in a parallel environment, sophisticated scheduling strategies are needed. We optimize the concurrent execution across the three layers of the MLMC method: parallelization across levels, across samples, and across the spatial grid. In a series of numerical tests, the influence on the overall performance of the “scalability window” of the multigrid solver (i.e., the range of processor numbers over which good parallel efficiency can be maintained) is illustrated. Different homogeneous and heterogeneous scheduling strategies are proposed and discussed. Finally, large 3D scaling experiments are carried out, including adaptivity.