Inverse Obstacle Scattering in Two Dimensions with Multiple Frequency Data and Multiple Angles of Incidence

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.

Novel Methods for Electromagnetic Simulation and Design

“…• We completed a fast solver for inverse obstacle scattering and demonstrated what we believe are the most geometrically detailed, high frequency reconstructions to date (using simulated data [3]). The method is based on fast solvers coupled with recursive linearization.…”

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

“…The components of the right-hand-side in (10) corresponding to planewave u pw (1) are given by (16) F…”

“…Recall that matrix A is given by (14), and the right-hand side F corresponding to one angle of incidence in (16). This step also provides the value of J .…”

“…[19,58,61]. It is also the principle in Recursive linearization Algorithm [25], see discussions in [43,60,16]. Multifrequencies data are also used in [34,11,17] for location of homogeneities, [7,10,15] for medium reconstruction, [16] for shape reconstruction.…”

“…Quasi-Newton is less computational intensive than methods in the Newton-like family used in e.g. [47,44,46,42,16] and the references therein, while still offers good convergence rate of Newton's method cf. [47].…”