On the reconstruction of Dirichlet-to-Neumann map in inverse scattering problems with stability estimates

Consider the determination of Dirichlet-to-Neumann (D-to-N) map from the far-field pattern in inverse scattering problems, which is the key step in some recently developed inversion schemes such as probe method. Essentially, this problem is related to the reconstruction of the scattered wave from its far-field data. We firstly prove the well-known uniqueness result of the D-to-N map from the far-field pattern using a new scheme based on the mixed reciprocity principle. The advantage of this new proof scheme is that it provides an efficient algorithm for computing the D-to-N map, avoiding the numerical differentiation for the scattered wave. Then combining with the classical potential theory, a simple and feasible regularizing reconstruction scheme for the D-to-N map is proposed. Finally the stability estimate for the reconstruction with noisy input data is rigorously analyzed.