In the framework of uncertainty quantification, we consider a quantity of interest which depends non-smoothly on the high-dimensional parameter representing the uncertainty. We show that, in this situation, the multilevel Monte Carlo algorithm is a valid option to compute moments of the quantity of interest (here we focus on the expectation), as it allows to bypass the precise location of discontinuities in the parameter space. We illustrate how such lack of smoothness occurs for the point evaluation of the solution to a (Helmholtz) transmission problem with uncertain interface, if the point can be crossed by the interface for some realizations. For this case, we provide a space regularity analysis for the solution, in order to state converge results in the L1-norm for the finite element discretization. The latter are then used to determine the optimal distribution of samples among the Monte Carlo levels. Particular emphasis is given on the robustness of our estimates with respect to the dimension of the parameter space.

中文翻译：

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具有几何不确定性的传输问题的高维参数空间上的多级蒙特卡罗

在不确定性量化的框架中，我们考虑了一个非平滑依赖于表示不确定性的高维参数的感兴趣量。我们表明，在这种情况下，多级蒙特卡洛算法是计算感兴趣量的矩的有效选项（这里我们专注于期望），因为它允许绕过参数空间中不连续性的精确位置。我们说明了对于具有不确定界面的（亥姆霍兹）传输问题的解决方案的点评估，如果该点可以被界面穿过以实现某些实现，那么这种缺乏平滑性是如何发生的。对于这种情况，我们为解决方案提供空间正则性分析，以便在有限元离散化的 L1 范数中说明收敛结果。然后使用后者来确定样本在蒙特卡罗级别之间的最佳分布。特别强调了我们估计在参数空间维度方面的稳健性。