SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 1, Page 106-134, January 2021.
We propose and analyze a quasi-Monte Carlo (QMC) algorithm for efficient simulation of wave propagation modeled by the Helmholtz equation in a bounded region in which the refractive index is random and spatially heterogenous. Our focus is on the case in which the region can contain multiple wavelengths. We bypass the usual sign-indefiniteness of the Helmholtz problem by switching to an alternative sign-definite formulation recently developed by Ganesh and Morgenstern Numer. Algorithms, 83 (2020), pp. 1441--1487. The price to pay is that the regularity analysis required for QMC methods becomes much more technical. Nevertheless we obtain a complete analysis with error comprising stochastic dimension truncation error, finite element error, and cubature error, with results comparable to those obtained for the diffusion problem.

中文翻译：

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非均质随机介质中波传播的准蒙特卡罗有限元分析

SIAM / ASA不确定性量化杂志，第9卷，第1期，第106-134页，2021年1月。
我们提出并分析准蒙特卡罗（QMC）算法，以有效地模拟由Helmholtz方程建模的有限区域中折射率随机且空间异质的波传播。我们的重点是该区域可以包含多个波长的情况。我们通过改用Ganesh和Morgenstern Numer最近开发的替代符号定义公式来绕过亥姆霍兹问题的通常符号不确定性。算法，83（2020），pp.1441--1487。需要付出的代价是QMC方法所需的规律性分析变得更加技术化。然而，我们获得了一个完整的分析，其误差包括随机尺寸截断误差，有限元误差和空间误差，其结果可与扩散问题获得的结果相媲美。