Exact and approximate reconstruction methods for a multi-channel inverse problem
In this talk some reconstruction results for the multi-channel Gel'fand-Calderon inverse problem in two dimensions are presented. This is the problem of the recovery of a matrix-valued potential in the Schrodinger equation from boundary data (Dirichlet-to-Neumann map) at fixed energy. The principal motivation for studying the multi-channel 2D problem is the approximation of the 3D scalar problem, which is formally overdetermined. We will first present an exact reconstruction procedure, based on modified Faddeev-type functions inspired by some results of Bukhgeim. This exact reconstruction yields directly an uniqueness result, but it satisfies a log-type stability estimates which is too weak for practical application. The next result is an approximate reconstruction algorithm for the same problem, which is Lipschitz stable and rapidly converging at high energies. The algorithm admits practical applications, namely for ocean acoustic tomography. It is based on the theory of inverse quantum scattering and it exploits a stabilizing phenomena when the energy is sufficiently large. Stability estimates, explicitly depending on the energy and the regularity of coefficients, will be presented as well. This talk is based, in particular, on the works [R. Novikov, M. Santacesaria, 2011, 2013] and [M. Santacesaria, 2012, 2013].