The Linear Sampling Method in Inverse Electromagnetic Scattering

Excerpt

The inverse scattering problem for electromagnetic waves is an area of major importance in applied mathematics. In particular, one can argue that the invention of radar is one of the most important inventions of the twentieth century. However, since radar is based on a weak scattering approximation and typically ignores polarization effects, it is of limited use for many target identification problems involving complex environments in which multiple scattering and/or polarization effects can no longer be ignored. For this reason considerable effort has been made in recent years to avoid the incorrect models inherent in the use of weak scattering approximations and instead to develop target identification algorithms without invoking such approximations. Initial efforts in this direction focused on nonlinear optimization techniques. However, although these techniques were successful in certain applications, it soon became apparent that they relied too heavily on strong a priori information about the scatterer and were numerically expensive as well. This then led to the search for target identification algorithms that, while avoiding incorrect model assumptions, were nevertheless easy to implement and required little a priori information. One result of this search has been the introduction of a class of methods collectively known as qualitative methods in inverse scattering theory (cf. [22]).

Qualitative methods in inverse scattering theory are characterized by the fact that, although they avoid the problems inherent in the use of weak scattering approximations or nonlinear optimization techniques, they typically recover less information than the latter two methods. In particular, with essentially no a priori assumptions about the material properties or geometry of the scatterer, the qualitative approach to the inverse scattering problem typically recovers the support of the scatterer as well as partial information on the scatterer's material properties. Furthermore, since the inversion algorithm is linear (even though the inverse scattering problem itself is nonlinear), the implementation of a given qualitative method is very rapid and easy to carry out (however, the implementation of a given qualitative method typically requires more data than the use of a nonlinear optimization scheme).

The oldest and most developed of the qualitative methods in inverse scattering theory is the linear sampling method (LSM), first introduced by Colton and Kirsch [49] in 1996 for the scalar case, and it is this approach (for the vector case) that will be the main focus of this book. For qualitative methods in electromagnetic inverse scattering theory other than the LSM, we refer the reader to Chapter 5 of the recent monograph by Kirsch and Grinberg [77] as well as to the article [69].