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，随着大规模并行计算机架构的出现以及纯空间并行化的扩展限制，时间并行积分方法成为研究热点。各类迭代并行时间算法（如Parareal、PFASST、MGRIT和时空多重网格法）被相继提出。这些方法采用不同符号体系描述，且现有的收敛性估计难以直接比较。本文针对Dahlquist模型问题，使用统一符号体系描述Parareal、PFASST、MGRIT与STMG算法，并通过生成函数给出精确的收敛性估计，从而首次实现四种方法收敛性的直接对比。我们证明所有方法最终均呈现超线性收敛，并进行了数值比较。该生成函数框架为探索和分析现有及新方法提供了更多可能性。