Computational Inverse Boundary Value Problems

Gang Bao

Michigan Center for Industrial and Applied Mathematics
Department of Mathematics
Michigan State University


ABSTRACT:


Since A. P. Calderon's ground-breaking paper in 1980, inverse boundary
value problems have received ever growing attention because of broad
industrial, medical, and military applications. Exciting theorems have
been proved about the uniqueness, stability, and range of the inverse
problems. However, numerical solution of the inverse problems continues
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, I report on 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. 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, and breast cancer imaging
(dispersive medium).