A quasi Monte Carlo-based model reduction method for solving Helmholtz equation in random media

Abstract

Wave propagation in random media has broad applications in materials science and engineering. In this paper, we develop a quasi Monte Carlo (qMC)-based model reduction method for solving random Helmholtz equations. In the physical space, we construct multiscale reduced basis functions by using an optimization method together with the proper orthogonal decomposition method. Then, in the random space we employ the qMC method for discretization. Under mild conditions, we prove that the spatial grid size is only proportional to the wave number, and almost a first-order convergence rate is achieved in the random space with respect to the number of samples. Since the exact solution oscillates in both physical and random spaces, our approach provides an efficient strategy to find its numerical approximation. One significant advantage of our approach over existing methods is its applicability to generic random media which cannot be treated as random perturbations of homogeneous media. These are confirmed by a series of numerical examples.