Parallel-in-time integration has been the focus of intensive research efforts over the past two decades due to the advent of massively parallel computer architectures and the scaling limits of purely spatial parallelization. Various iterative parallel-in-time (PinT) algorithms have been proposed, like Parareal, PFASST, MGRIT, and Space-Time Multi-Grid (STMG). These methods have been described using different notations, and the convergence estimates that are available are difficult to compare. We describe Parareal, PFASST, MGRIT and STMG for the Dahlquist model problem using a common notation and give precise convergence estimates using generating functions. This allows us, for the first time, to directly compare their convergence. We prove that all four methods eventually converge super-linearly, and also compare them numerically. The generating function framework provides further opportunities to explore and analyze existing and new methods.

翻译：过去二十年来,由于出现了大规模平行的计算机结构以及纯空间平行化的尺度限制,平行整合一直是集中研究工作的重点,提出了各种迭代平行实时算法,如Parareal、PFASST、MGRIT和空间时多格(STMG)等,这些方法使用不同的标记加以描述,现有的趋同估计数字难以比较。我们用共同的标记来描述Dahalquist模型问题,并用生成功能提供精确的趋同估计。这使我们第一次能够直接比较它们的趋同。我们证明,所有四种方法最终都会以超线方式汇合,并用数字加以比较。生成功能框架为探索和分析现有和新方法提供了进一步的机会。