Scattering and Inverse Scattering
David Colton, Peter Monk
The field of inverse scattering theory has been a particularly active field in applied mathematics for the past thirty years. The aim of research in this field has been to not only detect but also to identify unknown objects through the use of acoustic, electromagnetic or elastic waves. Although the success of such techniques as ultrasound and x-ray tomography in medical imaging has been truly spectacular, progress has lagged in other areas of application which are forced to rely on different modalities using limited data in complex environments. Indeed it is often said that "Target identification is the great unsolved problem.We detect almost everything, we identify nothing". Until a few years ago, all existing algorithms for target identification were based on either a linearizing weak scattering approximation or on the use of nonlinear optimization techniques. However, as the demands of imaging increased, it became clear that incorrect model assumptions inherent in weak scattering approximations imposed severe limitations on when reliable reconstructions were possible. On the other hand, it was also realized that for many practical applications nonlinear optimization techniques required information that is in general not available. Hence, in recent years, alternative methods for imaging have been developed that avoid incorrect model assumptions but, as opposed to nonlinear optimization techniques, also avoid strong a priori assumptions about the scattering object. Such methods come under the general title of qualitative methods in inverse scattering theory and are based on the development and use of linear sampling methods and transmission eigenvalues, both of which were discovered and developed here at Delaware in collaboration with researchers in Germany and France. In particular, such methods are based on solving a linear integral equation for a range of "sampling points" and frequencies which are not "transmission eigenvalues" and lead to an approximation to the shape of the scattering object together with limited information about the material properties of the scatterer. Such an approach is remarkable since the inverse scattering problem itself is nonlinear but no linearizing assumptions have been made in the derivation of the above mentioned linear integral equation. This research project with AFOSR is devoted to the further development and application of this new approach in inverse scattering theory.