Mediation of Ill-posedness in Multiparameter Inverse Scattering

It is advantageous to formulate the Inverse Scattering problem as a nonlinear integro-differential equation. However, intrinsic to the nature of these inverse problems is their potential for ill-posedness. We review our experience in overcoming ill-posedness of inverse scattering problems and some of the theoretical and practical results we have achieved in our pursuit of numerically efficient and robust algorithms, i.e. fast algorithms that also mediate the ill-posed nature of some inverse problems. This is commensurate with the stated goal of the Center for Inverse Problems, Imaging, and Tomography, which is the development of imaging algorithms that produce quantitative maps of material parameters. The incident (electromagnetic or acoustic) energy is diffracting in the cases we study, so that standard Computed Tomographic (CT) algorithms are inadequate for the kind of resolution required for quantitative maps. The need to solve the wave equation exactly, dramatically increases the complexity of the inversion problem.