Two Methods for Profile Reconstruction in Scattering Theory

Abstract

In this paper we discuss the problem of finding a profile for the scattering of electromagnetic waves with fixed frequencies. This type of problem is one of the most important and interesting problems arising in mathematical physics and has applications in geophysics, technology, medicine and nondestructive testing (cf. Rfs. 1–5). The difficulty in solving such problems is due to two unpleasant facts: They are nonlinear and, more seriously, they fall in a group of problems called ill-posed that is the solution-if it exists at all-does not depend on the data. Many authors suggested methods for overcoming these difficulties. The methods can be broadly divided into two classes: Optimization and iterative methods. In the previous works most algorithms were for TM wave illuminations whereas much less was reported for the most complicated TE case. The TE incident field is complicated compared to the TM case because of the strong nonlinearity and singularity of the kernel of the integral equation. In our work we consider one optimization (the dual space) method and one iterative (the simplified Newton) method. We shall consider the two scattering problems: the TM and the TE polarizations. It is well known that a combination of the two cases improves the quality of reconstruction. Our objective is to implement numerically the two methods to both problems and discuss the advantages and disadvantages in each case. We shall emphasize on the frequency and the type of profiles for the reconstructions. The simplified Newton method we shall develop has many advantages over other iterative methods: It is computationally inexpensive and the forward problem does not need to be solved at each iteration step. We shall compare that results with the implementation of the dual space method for different profiles and frequencies.