Time-parallel time integration has received a lot of attention in the high performance computing community over the past two decades. Indeed, it has been shown that parallel-in-time techniques have the potential to remedy one of the main computational drawbacks of parallel-in-space solvers. In particular, it is well-known that for large-scale evolution problems space parallelization saturates long before all processing cores are effectively used on today's large scale parallel computers. Among the many approaches for time-parallel time integration, ParaDiag schemes have proved themselves to be a very effective approach. In this framework, the time stepping matrix or an approximation thereof is diagonalized by Fourier techniques, so that computations taking place at different time steps can be indeed carried out in parallel. We propose here a new ParaDiag algorithm combining the Sherman-Morrison-Woodbury formula and Krylov techniques. A panel of diverse numerical examples illustrates the potential of our new solver. In particular, we show that it performs very well compared to different ParaDiag algorithms recently proposed in the literature.

翻译：时间并行积分方法在过去二十年中受到了高性能计算领域的广泛关注。事实上，并行时间技术已被证明具有弥补空间并行求解器主要计算缺陷的潜力。具体而言，众所周知，对于大规模演化问题，空间并行化在当今大规模并行计算机上所有处理核心被有效利用之前很久就已达到饱和状态。在众多时间并行积分方法中，ParaDiag方案已被证明是一种非常有效的方法。在该框架下，时间步进矩阵或其近似通过傅里叶技术进行对角化，从而使得不同时间步上的计算能够真正并行执行。本文提出了一种结合Sherman-Morrison-Woodbury公式与Krylov技术的新型ParaDiag算法。一系列多样化的数值算例展示了我们新求解器的潜力。特别地，与近期文献中提出的几种ParaDiag算法相比，我们的方法表现出了优异的性能。