Department of Mathematics - Seminar on Applied Mathematics - Numerical method for random Maxwell's equations

A numerical method is developed for efficiently computing the mean field and variance of solutions to three-dimensional Maxwell's equations with random interfaces, utilizing shape calculus and pivoted low-rank approximation. By applying perturbation theory and shape calculus, we describe the statistical moments of these solutions in relation to the perturbation magnitude through a first-order shape-Taylor expansion. To achieve high-resolution oscillation capture near the interface, an adaptive finite element method using Nedelec's third-order edge elements of the first kind is employed, solving the deterministic Maxwell's equations with the mean interface to approximate the expected solutions. For computing the second moment, a low-rank approximation based on the pivoted Cholesky decomposition is introduced to efficiently estimate the two-point correlation function and approximate the variance. Numerical experiments are provided to support our theoretical findings.