Inverse Boundary Value Problems for Maxwell's Equations
Gang Bao, Michigan Center for Industrial and Applied Mathematics (MCIAM), Department of Mathematics, Michigan State University

Since A. P. Calderon's ground-breaking paper in 1980, inverse boundary value problems have received ever growing attentions because of  broad industrial, medical, and military applications, such as nondestructive testing, seismic imaging, submarine detections, near-field or subsurface imaging, and medical imaging. Lots of exciting theorems have been proved about the uniqueness, stability, and range of the inverse problems. However, numerical solution of the inverse problems remains 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. For instance, on the best mathematically studied inverse conductivity problem, the optimal stability result is a logarithm type estimate.

In this talk, our recent progress in mathematical analysis and computational studies of the inverse boundary value problems for the Helmholtz and Maxwell equations will be reported. A novel 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 medium scattering problems. Convergence issues for the continuation algorithm will be examined. The speaker will also highlight ongoing projects on related topics.