We present a novel approach aimed at high-performance uncertainty quantification for time-dependent problems governed by partial differential equations. In particular, we consider input uncertainties described by a Karhunen-Loève expansion and compute statistics of high-dimensional quantities-of-interest, such as the cardiac activation potential. Our methodology relies on a close integration of multilevel Monte Carlo methods, parallel iterative solvers, and a space-time discretization. This combination allows for space-time adaptivity, time-changing domains, and to take advantage of past samples to initialize the space-time solution. The resulting sequence of problems is distributed using a multilevel parallelization strategy, allocating batches of samples having different sizes to a different number of processors. We assess the performance of the proposed framework by showing in detail its application to the solution of nonlinear equations arising from cardiac electrophysiology. Specifically, we study the effect of spatially-correlated perturbations of the heart fibers' conductivities on the mean and variance of the resulting activation map. As shown by the experiments, the theoretical rates of convergence of multilevel Monte Carlo are achieved. Moreover, the total computational work for a prescribed accuracy is reduced by an order of magnitude with respect to standard Monte Carlo methods.

我们提出了一种新颖的方法，旨在针对偏微分方程控制的时间相关问题的高性能不确定性量化。特别是，我们考虑用Karhunen-Loève展开描述的输入不确定性，并计算高维感兴趣量的统计数据，例如心脏激活电位。我们的方法依赖于多级蒙特卡洛方法，并行迭代求解器和时空离散化的紧密集成。这种组合允许时空适应性，时域变化，并利用过去的样本来初始化时空解决方案。使用多级并行化策略分配问题的结果序列，将具有不同大小的一批样本分配给不同数量的处理器。我们通过详细显示其在解决由心脏电生理学引起的非线性方程组中的应用，来评估所提出框架的性能。具体而言，我们研究了心脏纤维电导率的空间相关扰动对激活图的均值和方差的影响。如实验所示，实现了多级蒙特卡洛的理论收敛速度。此外，相对于标准的蒙特卡洛方法，用于规定精度的总计算量减少了一个数量级。生成的激活图的均值和方差的电导率。如实验所示，实现了多级蒙特卡洛的理论收敛速度。此外，相对于标准的蒙特卡洛方法，用于规定精度的总计算量减少了一个数量级。生成的激活图的均值和方差的电导率。如实验所示，实现了多级蒙特卡洛的理论收敛速度。此外，相对于标准的蒙特卡洛方法，用于规定精度的总计算量减少了一个数量级。