2 7 M ay 2 01 5 Solution of the nonlinear inverse scattering problem by T - matrix completion . I . Theory

We propose a conceptually new method for solving nonlinear inverse scattering problems (ISPs). The method is inspired by the theory of nonlocality of physical interactions and utilizes the relevant mathematical formalism. We formulate the ISP as a problem whose goal is to determine an unknown and, generally, nonlocal interaction potential V from external scattering data. We then utilize the one-to-one correspondence between V and the T-matrix of the problem, T . An iterative algorithm is proposed in which we seek T that is (i) compatible with the data and (ii) corresponds to an interaction potential V that is as diagonally-dominated as possible. We refer to this algorithm as to the data-compatible T-matrix completion (DCTMC). This paper is Part I in a two-part series and contains theory only. Numerical examples are given in Part II (arXiv:1505.06777). Submitted to: Inverse Problems