Scalar and vector backpropagation applied to shape identification from experimental data: Recent results and open problems

The inverse problem considered herewith consists of determining the shape of a two dimensional, perfectly conducting obstacle from knowledge of the incident monochromatic plane electromagnetic wave and of a set of laboratory measurements related to the scattered wave: the Ipswich data. The solution algorithm and its numerical implementation rely on approximate back propagation (ABP). The latter is an algebraic transformation, which is deduced from the properties of complete families of base functions and relates the far field scattering coefficients to those on the obstacle boundary. The features presented herewith for the first time are: a) a consistency result for a forward propagator, which transforms least squares boundary coefficients into far field scattering coefficients; b) the development and testing of a scalar ABP method for two dimensional electromagnetic problems, where polarization is vertical; c) the outline and the implementation of a vector ABP method for horizontal polarization; d) the application of both methods to the inversion of experimental data. Since general conditions for the well posedness of ABP methods are still unknown, the main related open problems are also stated.