In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems by time domain decomposition. The available algorithms enabling such decompositions present severe efficiency limitations and are an obstacle for the solution of large scale and high dimensional problems. Our main contribution is the improvement of the parallel efficiency of the parareal in time method. The parareal method is based on combining predictions made by a numerically inexpensive solver (with coarse physics and/or coarse resolution) with corrections coming from an expensive solver (with high-fidelity physics and high resolution). At convergence, the algorithm provides a solution that has the fine solver's high-fidelity physics and high resolution. In the classical version, the fine solver has a fixed high accuracy which is the major obstacle to achieve a competitive parallel efficiency. In this paper, we develop an adaptive variant that overcomes this obstacle by dynamically increasing the accuracy of the fine solver across the parareal iterations. We theoretically show that the parallel efficiency becomes very competitive in the ideal case where the cost of the coarse solver is small, thus proving that the only remaining factors impeding full scalability become the cost of the coarse solver and communication time. The developed theory has also the merit of setting a general framework to understand the success of several extensions of parareal based on iteratively improving the quality of the fine solver and re-using information from previous parareal steps. We illustrate the actual performance of the method in stiff ODEs, which are a challenging family of problems since the only mechanism for adaptivity is time and efficiency is affected by the cost of the coarse solver.

中文翻译：

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一种自适应 Parareal 算法

在本文中，我们考虑了通过时域分解来加速瞬态问题的数值模拟的问题。实现这种分解的可用算法存在严重的效率限制，并且是解决大规模和高维问题的障碍。我们的主要贡献是提高了平行时间方法的并行效率。Parareal 方法基于将数值上廉价的求解器（具有粗糙物理和/或粗糙分辨率）做出的预测与来自昂贵求解器（具有高保真物理和高分辨率）的校正相结合。在收敛时，该算法提供具有精细求解器的高保真物理和高分辨率的解决方案。在经典版本中，精细求解器具有固定的高精度，这是实现具有竞争力的并行效率的主要障碍。在本文中，我们开发了一种自适应变体，它通过跨平行迭代动态提高精细求解器的精度来克服这一障碍。我们从理论上表明，在粗求解器成本很小的理想情况下，并行效率变得非常有竞争力，从而证明阻碍完全可扩展性的唯一剩余因素成为粗求解器和通信时间的成本。所开发的理论还有一个优点，即基于迭代地提高精细求解器的质量和重用来自先前的parareal 步骤的信息，设置一个通用框架来理解parareal 的几个扩展的成功。