While generally considered computationally expensive, Uncertainty Quantification using Monte Carlo sampling remains beneficial for applications with uncertainties of high dimension. As an extension of the naive Monte Carlo method, the Multi-Level Monte Carlo method reduces the overall computational effort, but is unable to reduce the time to solution in a sufficiently parallel computing environment. In this work, we propose a Uncertainty Quantification method combining Multi-Level Monte Carlo sampling and Parallel-in-Time integration for select samples, exploiting remaining parallel computing capacity to accelerate the computation. While effective at reducing the time-to-solution, Parallel-in-Time integration methods greatly increase the total computational effort. We investigate the tradeoff between time-to-solution and total computational effort of the combined method, starting from theoretical considerations and comparing our findings to two numerical examples. There, a speedup of 12 - 45% compared to Multi-Level Monte Carlo sampling is observed, with an increase of 15 - 18% in computational effort.

翻译：尽管通常被认为计算成本高昂，但基于蒙特卡洛采样的不确定性量化方法在处理高维不确定性应用时仍具优势。作为朴素蒙特卡洛方法的扩展，多级蒙特卡洛方法降低了总体计算量，但在充分并行的计算环境中无法有效缩短求解时间。本研究提出一种不确定性量化方法，该方法结合了多级蒙特卡洛采样与针对选定样本的并行时间积分技术，通过利用剩余并行计算能力来加速计算过程。虽然并行时间积分方法能有效缩短求解时间，但会显著增加总计算量。我们从理论分析出发，通过两个数值算例验证研究结果，深入探讨了该混合方法在求解时间与总计算量之间的权衡关系。实验表明，相较于单纯的多级蒙特卡洛采样，该方法可获得12%-45%的加速效果，同时计算量仅增加15%-18%。