Helmholtz scattering by random domains: first-order sparse boundary element approximation

TL;DR

This paper develops a first-order boundary element method for efficiently approximating the statistical moments of acoustic scattering problems involving uncertain geometries, using sparse tensor discretizations and numerical validation.

Contribution

It introduces a novel first-order shape Taylor expansion approach combined with sparse tensor Galerkin discretization for stochastic boundary integral equations in acoustic scattering.

Findings

Error convergence rates confirmed numerically

Poly-logarithmic growth in degrees of freedom

Effective preconditioning strategies discussed

Abstract

We consider the numerical solution of time-harmonic acoustic scattering by obstacles with uncertain geometries for Dirichlet, Neumann, impedance and transmission boundary conditions. In particular, we aim to quantify diffracted fields originated by small stochastic perturbations of a given relatively smooth nominal shape. Using first-order shape Taylor expansions, we derive tensor deterministic first kind boundary integral equations for the statistical moments of the scattering problems considered. These are then approximated by sparse tensor Galerkin discretizations via the combination technique (Griebel et al.). We supply extensive numerical experiments confirming the predicted error convergence rates with poly-logarithmic growth in the number of degrees of freedom and accuracy in approximation of the moments. Moreover, we discuss implementation details such as preconditioning to finally point out further research avenues.