We design and implement a novel algorithm for computing a multilevel Monte Carlo (MLMC) estimator of the joint cumulative distribution function (CDF) of a vector-valued quantity of interest in problems with random input parameters and initial conditions. Our approach combines MLMC with stratified sampling of the input sample space by replacing standard Monte Carlo at each level with stratified Monte Carlo initialized with proportionally allocated samples. We show that the resulting stratified MLMC (sMLMC) algorithm is more efficient than its standard MLMC counterpart due to the additional variance reduction provided by the stratification of the random parameter's domain, especially at the coarsest levels. Additional computational cost savings are obtained by smoothing the indicator function with a Gaussian kernel, which proves to be an efficient and robust alternative to recently developed polynomial-based techniques.

我们设计并实现了一种新颖的算法，用于计算带有随机输入参数和初始条件的问题中向量值感兴趣量的联合累积分布函数（CDF）的多级蒙特卡洛（MLMC）估计量。我们的方法将MLMC与输入样本空间的分层采样相结合，方法是将每个级别的标准Monte Carlo替换为按比例分配的样本初始化的分层Monte Carlo。我们显示出，由于随机参数域的分层（尤其是在最粗糙的级别）提供了额外的方差减少，因此生成的分层MLMC（sMLMC）算法比其标准MLMC更为有效。通过使用高斯核对指标函数进行平滑处理，可以进一步节省计算成本，