Some Reconstruction Methods for Inverse Scattering Problems

Inverse scattering problems are one of the main research areas in the optimization techniques. The main purpose of inverse scattering problems is to detect the physical properties of an obstacle from some information related to the scattered waves of the obstacle for given incident wave. Generally, if the incident plane waves are given from the finite number of directions, which are indeed the practical situations, there is no uniqueness for reconstructing the obstacle properties such as the boundary shape. In these cases, the optimization techniques can be applied to reconstructing the obstacle approximately. That is, the obstacle shape is approximated by a minimizer of some cost functional which measures the defect between the measurement data of the scattered wave and the computational scattered wave related to the approximate obstacle. Of course, for this optimization problems in infinite dimensional space, some regularizing term should be introduced to the cost functional. Although these general optimization techniques have been applied widely in the last century, they also suffer from many disadvantages theoretically and numerically. From the theoretical point of view, the lack of uniqueness makes the obtained approximate obstacle from the optimization procedure ambiguous, namely, we do not know whether or not the computational result indeed approximates the true one. From the numerical respects, such an optimization procedure needs to solve the direct scattering problem at each iteration step, which entails huge time cost. Moreover, to get the convergence of the iteration, a good initial guess is required. Even if in the case of convergence, the approximate sequence generally approaches to some local minimizer. These shortcomings of the direct optimizations used in inverse scattering problems cause some problems for the numerical reconstruction of the obstacles. In recent years, some modified version of optimization algorithms based on the potential methods for inverse scattering problems are proposed, in the sense that the direct problems are not required to solve in the iteration procedures. On the other hand, some new schemes combining the advantages of optimizations with the exact reconstruction formulas have been developed for inverse scattering problems. In this chapter, we give an overview on these two directions. We begin with a brief introduction to the physical background to acoustic scattering problems as well as some well-known inverse scattering models. Then we review some classical and recently developed inversion methods for detecting the information about the unknown scatterer from a knowledge of the far-field pattern u ∞ for one or several incident plane waves. For each method, the basic idea is described, and some main results are presented. To test their validity, the numerical implementation of all inversion methods needs to be studied. So we finally focus on the numerical realizations of these existing methods, pointing out the main difficulties encountered in numerical realization. In addition, the advantages and disadvantages of these methods are also analyzed.