PDE Real Analysis Seminar Inverse Obstacle Recovery when the boundary condition is also unknown

2008-06-04 16:00 - 2008-06-04 17:00
Graduate School of Mathematical Sciences the University of Tokyo, Room #056
Professor William Rundell ( Texas A&M University )
We consider the inverse problem of recovering the shape, location and surface properties of an object where the surrounding medium is both conductive and homogeneous. It is assumed that the physical situation is modeled by either harmonic functions or solutions of the Helmholtz equation and that the boundary condition on the obstacle is one of impedance type. We measure either Cauchy data, on an accessible part of the exterior boundary or the far field pattern resulting from a plane wave. Given sets of Cauchy data pairs we wish to recover both the shape and location of the unknown obstacle together with its impedance. It turns out this adds considerable complexity to the analysis. We give a local injectivity result and use two different algorithms to investigate numerical reconstructions. The setting is in two space dimensions, but indications of possible extensions (and difficulties) to three dimensions are provided. We also look at the case of a nonlinear impedance function.