Speaker: Gang Bao
Title: Inverse Boundary Value Problems
Affiliation: Michigan State University.
Date: Friday, September 21, 2007.
Place and Time: Room 101, Love Building, 3:35-4:30 pm.
Refreshments: Room 204, Love Building, 3:00 pm.

Abstract. 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 new 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. Roughly speaking, in order to obtain one digit numerical reconstruction of the coefficient function, at least ten digit accurate boundary data would be required.
In this talk, progress of our research group over the past several years in mathematical analysis and computational studies of the inverse boundary value problems for the Helmholtz and Maxwell equations will be reported. I will present a continuation approach based on the uncertainty principle. 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. I will also discuss convergence issues for the continuation algorithm and highlight ongoing projects in limited aperture imaging, breast cancer imaging (dispersive medium), and nano optics modeling.