Monday, February 25, 2008
Mathematical Sciences Colloquium
Speaker: Gang Bao (Michigan State University) Direct and Inverse Scattering Problems for Electromagnetic and Optical Wave Propagation

The inverse medium scattering problem arises in diverse areas of industrial and military applications, such as nondestructive testing, seismic imaging, submarine detections, near-field or subsurface imaging, and medical imaging. The 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 medium 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, 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.

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 in the modeling, analysis, computation of micro diffractive optics, near-field optics, and nano optics.