Iterative Regularization Methods in Inverse Scattering

We consider the problem of reconstructing the shape of an acoustic or electromagnetic scatterer from far eld measurements of the scattered wave corresponding to one incident time harmonic wave in the resonance region. This problem is di cult to solve since it is nonlinear and severely ill-posed. The characterization of the Fr echet derivatives, which has been accomplished some years ago by Kirsch and others, has paved the way for the application of iterative regularization methods to solve these problems numerically. Recently iterative regularization methods have been intensively investigated, and convergence results have been obtained under some conditions on the nonlinearity of the operator. Examples include Landweber iteration (Hanke, Neubauer, Scherzer), the Iteratively Regularized Gauss-Newton Method (IRGNM) (Bakushinskii, Blaschke/Kaltenbacher, Hohage) and inexact Newton methods such as the Levenberg-Marquardt and the Newton-CG Algorithm (Hanke) or most recently a second degree method (Hettlich, Rundell). We compare the numerical performance of these methods applied to inverse scattering problems and suggest a new, more e ective modi cation of the IRGNM. While the numerical solution of some direct model problems is quite fast on modern computers, computation time becomes a central issue in large scale and 3-dimensional problems. Hence we address the problem of minimizing computation time. Roughly speaking, it does not pay o to pay too much e ort in an accurate evaluation of the operator as long as one is still far away from the solution. We show how various discretization parameters have to be increased during the iteration such that the order of convergence established in the in nite dimensional setting is maintained. 1 AN INVERSE SCATTERING PROBLEM We consider the following problem. Let D IR be a star-shaped smooth domain describing the cross section of a long cylindrical scattering obstacle. For an incident plane wave ui(x) := e , jdj = 1; k > 0 the scattered eld us and the total eld u := ui + us satisfy u+ ku = 0 in IRn D (1) p r @u @r iku ! 0; r = jxj ! 1 (2)