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 (Numerical Algorithms, 83, 1441-1487, 2020). 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.
Geometric Multigrid for Darcy and Brinkman models of flows in highly heterogeneous porous media: A numerical study
Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR), Universität Heidelberg（海德堡大学计算科学跨学科中心）；Department of Mathematics, Texas A&M University（德克萨斯农工大学数学系）；Department of Geology & Geophysics, Texas A&M University（德克萨斯农工大学地质与地球物理系）
A two-grid preconditioner with an adaptive coarse space for flow simulations in highly heterogeneous media
School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）；Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学系）
An Interior Penalty Method with $C^0$ Finite Elements for the Approximation of the Maxwell Equations in Heterogeneous Media: Convergence Analysis with Minimal Regularity
Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media
Texas A&M University（德克萨斯农工大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Chinese University of Hong Kong（香港中文大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
Higher-Order Multiscale Computational Method for Multi-Continuum Problems in Highly Heterogeneous Media
School of Mathematics and Statistics, Xidian University（西安电子科技大学数学与统计学院）；School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）
