The Inverse Scattering Problem for Time-Harmonic Acoustic Waves

Abstract

The inverse scattering problem we are considering in this paper is to determine the shape of a sound-soft, bounded, connected obstacle from a knowledge of the time-harmonic incident wave with frequency in the resonance region and the far field pattern of the scattered wave. After some introductory remarks, we begin by describing the method of integral equations and the null-field method for solving the direct scattering problem. This enables us to define an operator ${\bf T}$ mapping the boundary of the scattering obstacle and the incident field onto the far field pattern. The inverse scattering problem is to invert this operator. In order to do this, we first must examine the range of ${\bf T}$, i.e. to characterize the class of far field patterns, and to establish the existence of ${\bf T}^{ - 1} $ on the range of ${\bf T}$, i.e. to show the uniqueness of the solution to the inverse scattering problem. This analysis shows that for a given measured far field pattern, in general no solution exists to the inverse scattering problem, and if a solution does exist, it does not depend continuously on the measured data, i.e. the problem is improperly posed. This motivates us to consider a linearized model and to examine various methods for studying linearized improperly posed problems based on the ideas of a priori assumptions and compactness arguments. We then consider a simple model problem that focuses on the nonlinear character of the inverse scattering problem and, motivated by our study of the linearized model, reformulate the inverse scattering problem as a problem in constrained optimization. We conclude by considering the numerical solution of this nonlinear optimization problem.