Inverse Obstacle scattering in two dimensions with multiple frequency data and multiple angles of incidence

TL;DR

This paper develops a method for reconstructing the shape of sound-soft obstacles in 2D using multi-frequency and multi-angle scattering data, employing physical regularization and advanced numerical techniques.

Contribution

It introduces a physically motivated regularization approach and combines Newton's method with recursive linearization for improved inverse scattering reconstruction.

Findings

Effective shape reconstruction from multi-frequency data

Use of high-order integral equation discretizations and fast solvers

Demonstrated robustness of the method with multiple incident angles

Abstract

We consider the problem of reconstructing the shape of an impenetrable sound-soft obstacle from scattering measurements. The input data is assumed to be the far-field pattern generated when a plane wave impinges on an unknown obstacle from one or more directions and at one or more frequencies. It is well known that this inverse scattering problem is both ill posed and nonlinear. It is common practice to overcome the ill posedness through the use of a penalty method or Tikhonov regularization. Here, we present a more physical regularization, based simply on restricting the unknown boundary to be band-limited in a suitable sense. To overcome the nonlinearity of the problem, we use a variant of Newton's method. When multiple frequency data is available, we supplement Newton's method with the recursive linearization approach due to Chen. During the course of solving the inverse problem, we need to compute the solution to a large number of forward scattering problems. For this, we use high-order accurate integral equation discretizations, coupled with fast direct solvers when the problem is sufficiently large.