A Robust Numerical Method for the Random Interface Grating Problem via Shape Calculus, Weak Galerkin Method, and Low-Rank Approximation

Abstract

We present an efficient numerical algorithm to solve random interface grating problems based on a combination of shape derivatives, the weak Galerkin method, and a low-rank approximation technique. By using the asymptotic perturbation approach via shape derivative, we estimate the expectation and the variance of the random solution in terms of the magnitude of the perturbation. To effectively capture the severe oscillations of the random solution with high resolution near the interface, we use weak Galerkin method to solve the Helmholtz equation related to the grating interface problem at each realization. To effectively compute the variance operator, we use an efficient low-rank approximation method based on a pivoted Cholesky decomposition to compute the two-point correlation function. Two numerical experiments are conducted to demonstrate the efficiency of our algorithm.