Remarks on an optimization method in profile inversion

We discuss the problem of finding a profile or its location for the scattering of electromagnetic waves with fixed frequencies. We consider optimization methods, in particular, a dual space method (DSM) which is known to be very efficient. It has the advantage of reducing the number of unknowns and the disadvantage that the data has to be measured for many angles. Since we are mainly motivated by medical applications, the disadvantage is not a problem because the scattering data can be measured for many angles. Our goal is to try and see how much information we need from the frequency in order to get a reasonable reconstruction using DSM. We extend the DSM to a more general formulation of the TE-mode and consider also the TM-mode. Moreover, we consider the reconstruction of the refractive index as well as the boundary of the scatterer. Our results suggest that small wave numbers can lead to a satisfactory reconstruction for finding the index of the TM-mode and the boundary for the TE-mode and TM-mode. Big wave numbers lead to poor reconstructions, regardless of mode and of what we try to reconstruct. For the case of TE-mode when the wave number is neither too small nor too big, we may be able to obtain satisfactory results.