In the past thirty years the field of inverse scattering theory has become a major theme of applied mathematics with applications to such diverse areas as medical imaging, geophysical exploration and nondestructive testing. The growth of this field has been characterized by the realization that the inverse scattering problem is both nonlinear and ill-posed, thus presenting particular problems in the development of efficient inversion algorithms. Although linearized models continue to play an important role in many applications, the increased need to focus on problems in which multiple scattering effects can no longer be ignored has led to the nonlinearity of the inverse scattering problem playing a central role. In addition, the possibility of collecting large amounts of data over limited regions of space has led to the situation where the ill-posed nature of the inverse scattering problem becomes a problem of central importance. Initial efforts to deal with the nonlinear and ill-posed nature of the inverse scattering problem focused on the use of nonlinear optimization methods. Although efficient in many situations, their use suffers from the need for strong a priori information in order to implement such an approach. In addition, recent advances in material science and nanostructure fabrications have introduced new exotic materials for which full reconstruction of the constitutive parameters from scattering data is challenging or even impossible. In order to circumvent these difficulties, a recent trend in inverse scattering theory has focused on the development of a new approach, in which the amount of a priori information needed is drastically reduced but at the expense of obtaining only limited information of the scatterers.

In this talk we present an overview of the aforementioned new ideas in the context of solving the inverse scattering problem for anisotropic inhomogeneous media. Our approach exploits properties of the linear scattering operator to decode non-linear information about the scattering medium, yielding mathematically justified and computationally simple reconstruction algorithms. We first show that the support of the scattering media can be rigorously characterized from the range of the scattering operator. The mathematical study of this operator is based on the analysis of a non-selfadjoint eigenvalue problem known as the transmission eigenvalue problem. Transmission eigenvalues relate to interrogating frequencies for which there is an incident field that does not scatter. We show that the transmission eigenvalues can be determined from the scattering data and provide information on the constitutive properties of the scattering media. Numerical examples will be presented to confirm the viability of our reconstruction algorithms.