The inverse scattering problem arises in diverse areas of industrial and military applications, such as nondestructive testing, seismic imaging, submarine detections, near-field or subsurface imaging, near-field and nano optical imaging, and medical imaging. A model problem is concerned with a time-harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain. Given the incident field, the direct problem is to determine the scattered field for the known scatterer. The inverse scattering problem is to determine the scatterer from the boundary measurements of near field currents densities. Although this is a classical problem in mathematical physics, mathematical issues and numerical solution of the inverse problems remain to be challenging since the problems are nonlinear, large-scale, and most of all ill-posed! The severe ill-posedness has thus far limited in many ways the scope of inverse problem methods in practical applications.

Our progress over the past ten years in mathematical analysis and computational studies of the inverse boundary value problems for the Helmholtz and Maxwell equations will be reported. Several classes of inverse scattering problems will be studied, namely inverse medium problems, inverse source problems, inverse obstacle problems, and inverse waveguide problems. A novel stable continuation approach based on the uncertainty principle will be presented. By using multi-frequency or multi-spatial frequency boundary data, our approach is shown to overcome the ill-posedness for the inverse scattering problems. Convergence issues for the continuation algorithm will be examined. New stability and uniqueness results for the inverse problem will be presented. In particular, our most recent stability result on inverse problems for the related time-domain wave equation with possible caustics will be highlighted. The speaker will also discuss our recent efforts in multi-scale/multi-physics nano optics modeling and inverse problems in nano science.