Multifrequency inverse obstacle scattering with unknown impedance boundary conditions using recursive linearization

Abstract

In this paper, we consider the reconstruction of the shape and the impedance function of an obstacle from measurements of the scattered field at a collection of receivers outside the object. The data is assumed to be generated by plane waves impinging on the unknown obstacle from multiple directions and at multiple frequencies. This inverse problem can be reformulated as an optimization problem: that of finding band-limited shape and impedance functions which minimize the L 2 distance between the computed value of the scattered field at the receivers and the given measurement data. The optimization problem is highly non-linear, non-convex, and ill-posed. Moreover, the objective function is computationally expensive to evaluate (since a large number of Helmholtz boundary value problems need to be solved at every iteration in the optimization loop). The recursive linearization approach (RLA) proposed by Chen has been successful in addressing these issues in the context of recovering the sound speed of an inhomogeneous object or the shape of a sound-soft obstacle. We present an extension of the RLA for the recovery of both the shape and impedance functions of the object. The RLA is, in essence, a continuation method in frequency where a sequence of single frequency inverse problems is solved. At each higher frequency, one attempts to recover incrementally higher resolution features using a step assumed to be small enough to ensure that the initial guess obtained at the preceding frequency lies in the basin of attraction for Newton's method at the new frequency. We demonstrate the effectiveness of this approach with several numerical examples. Surprisingly, we find that one can recover the shape with high accuracy even when the measurements are generated by sound-hard or sound-soft objects, eliminating the need to know the precise boundary conditions appropriate for modeling the object under consideration. While the method is effective in obtaining high quality reconstructions for many complicated geometries and impedance functions, a number of interesting open questions remain regarding the convergence behavior of the approach. We present numerical experiments that suggest underlying mechanisms of success and failure, pointing out areas where improvements could help lead to robust and automatic tools for the solution of inverse obstacle scattering problems.