Analytische und numerische Untersuchungen bei inversen Transmissionsproblemen zur zeitharmonischen Wellengleichung

Abstract

Scattering theory deals with a scattering object´s effect that on an incident wave. In this context the time-harmonic wave equation is a mathematical model for the scattering of acoustic and electromagnetic fields.Basically two kinds of scattering problems can be distinguished. The direct problem consists of calculating the scattered field from the knowledge of the incident field, the scattering object and further problem parameters (for example material constants of the scattering object). The aim of the inverse problem is to reconstruct the scattering object (and possibly further parameters) from the knowledge of the incident and the scattered field.Whereas the direct problem has a unique solution that depends continuously on the data, this does not hold for the inverse problem. Therefore it is much harder to deal with from a theoretical as well as a numerical point of view.Modelling different physical conditions leads mathematically to different boundary value problems. In this context the transmission problem describes scattering by a penetrable object. This means that an incident field produces scattered fields in the exterior as well as the interior of the scattering objects.This thesis has three main parts. The first one deals with the direct transmission problem which is examined for a wide range of scattering objects. Furthermore algorithms for solving the direct problem for several objects are given.In the next part it is proven that the scattered fields depend analytically on the boundaries of the scattering objects and on the problem parameters. This justifies the use of a Newton method for the reconstruction procedure.The third main part is concerned with the numerical implementation of the theoretical results for the two dimensional transmission problem. Numerical examples are given for the direct and the inverse problem. In particular several scattering objects together with problem parameters are reconstructed.