Numerical robustness of single-layer method with Fourier basis for multiple obstacle acoustic scattering in homogeneous media

Abstract

We investigate efficient methods to simulate the multiple scattering of obstacles in homogeneous media. With a large number of small obstacles on a large domain, optimized pieces of software based on spatial discretization such as Finite Element Method (FEM) or Finite Difference lose their robustness. As an alternative, we work with an integral equation method, which uses singlelayer potentials and truncation of Fourier series to describe the approximate scattered field. In the theoretical part of the paper, we describe in detail the linear systems generated by the method for impenetrable obstacles, accompanied by a well-posedness study. For the numerical performance study, we limit ourselves to the case of circular obstacles. We first compare and validate our codes with the highly optimized FEM-based software Montjoie. Secondly, we investigate the efficiency of different solver types (direct and iterative of type GMRES) in solving the dense linear system generated by the method. We observe the robustness of direct solvers over iterative ones for closely-spaced obstacles, and that of GMRES with Lower-Upper Symmetric Gauss-Seidel and Symmetric Gauss-Seidel preconditioners for far-apart obstacles.

Localization of small obstacles from back-scattered data at limited incident angles with full-waveform inversion

“…We describe below how simulated data, corresponding to one incident angle, are computed via the single-layer potential solver (conveniently) called FSSL, for more details see [14,13].…”

“…See [14] for the definition of the exterior trace γ Discretization of the forward problem by FSSL Denote by Γ I the boundary of obstacle I, H (1) 0 the Hankel function of the first kind, and by w J,l the l-th order Fourier nodes on the boundary of obstacle J, i.e. (6) w J,l (θ J (x) , r J (x)) := e i l θ J (x) .…”

“…For circular obstacles, the multiple-scattering linear system can be described explicitly in the form of multipole expansion, using the Hankel functions of the first kind H (1) k and the Bessel function J k and their derivatives. The diagonal blocks A I are diagonal matrices, with diagonal components given by (14) (…”

“…For circular obstacles, to guarantee the invertibility of (10), the testing wavenumber and the radius of the obstacles have to satisfy: κ r are not in the set of zeros of Bessel functions. A 'cheap' requirement however sufficient for our tests is that 0 < κ r ≤ 2 (also holds for arbitrarily-shaped obstacles), for more details see [14,Remark 2]. With r = 0.5 (size of the obstacles taken in the experiments), this means 0 < κ ≤ 4.…”

“…It was shown in [14,13] that compared with highly-optimized software using Finite Element methods (FEM), a 'naive' implementation of FSSL is still drastically faster 5 , e.g. see [14,Experiment 3] in which it is 2000 times faster.…”