A quasi-Monte Carlo multiscale method for the wave propagation in random media

TL;DR

This paper introduces a novel multiscale numerical method combining quasi-Monte Carlo sampling and boundary-corrected discretization to efficiently simulate wave propagation in media with random refractive indices, achieving high accuracy and convergence.

Contribution

It develops a boundary-corrected multiscale method integrated with qMC sampling for Helmholtz problems with random media, providing rigorous convergence analysis and superconvergence results.

Findings

Achieves superconvergence rates of $ ext{O}(H^4)$ in $L^2$-error

Almost first-order convergence in the random space due to qMC

Numerical experiments confirm theoretical accuracy and convergence rates

Abstract

In this paper, we propose and analyze an accurate numerical approach to simulate the Helmholtz problem in a bounded region with a random refractive index, where the random refractive index is denoted using an infinite series parameterized by stochastic variables. To calculate the statistics of the solution numerically, we first truncate the parameterized model and adopt the quasi-Monte Carlo (qMC) method to generate stochastic variables. We develop a boundary-corrected multiscale method to discretize the truncated problem, which allows us to accurately resolve the Robin boundary condition with randomness. The proposed method exhibits superconvergence rates in the physical space (theoretical analysis suggests $\mathcal{O}(H^4)$ for $L^2$-error and $\mathcal{O}(H^2)$ for a defined $V$-error). Owing to the employment of the qMC method, it also exhibits almost the first-order convergence rate in the random space. We provide the wavenumber explicit convergence analysis and conduct numerical experiments to validate key features of the proposed method.