Algorithms for Direct and Inverse Scattering

We present some numerical methods to compute the solution of direct and inverse scattering problems. We consider a time dependent three dimensional direct acoustic scattering problem. That is given a bounded simply connected obstacle of known impedance find the scattered field generated by an incoming wave packet that hits the obstacle. We obtain the scattered field as superposition of time harmonic acoustic waves. The time harmonic waves are computed using the operator expansion technique. We consider also some three dimensional time harmonic acoustic and electromagnetic inverse obstacle scattering problems. That is from some knowledge of the scattered field corresponding to known incoming fields and from the knowledge of some a-priori information about the obstacle that scatters the incoming fields reconstruct the shape and eventually the impedance of the obstacle. We show how to reformulate the inverse problem as two different non linear optimization problems. These optimization problems translate the so called Herglotz function method to solve the inverse problem. Numerical results on test problems obtained with the numerical methods proposed here using synthetic data are shown. Moreover we report the results obtained with the Herglotz function method to solve the inverse problem using the Ipswich Data, an experimental data set of electromagnetic scattering data provided by the Rome Laboratory of the U.S. Air Force