Mathematical analysis and numerical methods for inverse scattering problems

Abstract

Inverse scattering problems arise in diverse application areas, such as geophysical prospecting, submarine detection, near-field and nanooptical imaging, and medical imaging. For a given wave incident on a medium enclosed by a bounded domain, the scattering (direct) problem is to determine the scattered field or the energy distribution for the known scatterer. An inverse scattering problem is to determine the scatterer from the boundary measurements of the fields. Although significant recent progress has been made in solving the inverse problems, many challenging mathematical and computational issues remain unresolved. In particular, the severe ill-posedness has thus far limited the scope of inverse problem methods in practical applications. This paper is concerned with mathematical analysis and numerical methods for solving inverse scattering problems of broad interest. Based on multifrequency data, effective computational and mathematical approaches are presented for overcoming the ill-posedness of the inverse problems. A brief overview of these approaches and results is provided. Particular attention is paid to inverse medium, inverse obstacle, and inverse source scattering problems. Related topics and open problems are also discussed.