-----------------------------------------------------------

Applied Math Colloquium

Friday, April 5, 2013, 11:30 AM
Cullimore Lecture Hall, Lecture Hall II
New Jersey Institute of Technology

Inverse Scattering in Wave Propagation

Gang Bao

Michigan State University

Abstract

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. 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 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. Three classes of inverse scattering problems will be studied, namely inverse medium problems, inverse source problems, and inverse obstacle 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. Related results on uniqueness and stability for the inverse problems will be presented. The speaker will also discuss the effects of evanescent fields in achieving super-resolution via inverse scattering. Finally, he will highlight ongoing and future projects along these directions, particularly our recent work on multiscale modeling and computation of nano optics.