提高跨方法并行谱延迟校正的效率
Improving Efficiency of Parallel Across the Method Spectral Deferred Corrections
AI总结：
针对跨方法并行谱延迟校正（SDC）依赖系数影响性能的问题，提出解析求最优参数的假设，所得方法稳定域和收敛阶接近成熟串行SDC，在80%效率假设下与多种方法相比具有竞争力。
摘要：
跨方法并行时间积分在求解初值问题时可提供小规模并行性。使用对角扫掠器的谱延迟校正（SDC）与Van der Houwen和Sommeijer提出的迭代龙格-库塔方法密切相关，其可用线程数等于底层配置方法中的求积节点数。然而，收敛速度、效率和稳定性严重依赖于所用系数。此前方法通过数值优化寻找良好参数，而我们提出一种假设，允许解析地找到最优参数。我们证明，所得并行SDC方法的稳定域和收敛阶与成熟的串行SDC变体非常相似。使用假设并行SDC实现效率为80%的计算成本模型，我们表明，我们的变体与串行SDC、先前发表的并行SDC系数、Picard迭代、显式RKM-4以及隐式四阶对角隐式龙格-库塔方法相比具有竞争力。
英文摘要：
Parallel-across-the method time integration can provide small scale parallelism when solving initial value problems. Spectral deferred corrections (SDC) with a diagonal sweeper, which is closely related to iterated Runge-Kutta methods proposed by Van der Houwen and Sommeijer, can use a number of threads equal to the number of quadrature nodes in the underlying collocation method. However, convergence speed, efficiency and stability depends critically on the used coefficients. Previous approaches have used numerical optimization to find good parameters. Instead, we propose an ansatz that allows to find optimal parameters analytically. We show that the resulting parallel SDC methods provide stability domains and convergence order very similar to those of well established serial SDC variants. Using a model for computational cost that assumes 80% efficiency of an implementation of parallel SDC we show that our variants are competitive with serial SDC, previously published parallel SDC coefficients as well as Picard iteration, explicit RKM-4 and an implicit fourth-order diagonally implicit Runge-Kutta method.
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