A novel multi-frequency regularization method for the 2D inverse scattering problem under the born iterative method

摘要

In this paper, we consider the scalar transverse magnetic (TM), two-dimensional, time-harmonic, lossless inverse scattering problem. The goal of this problem is to determine an unknown permittivity contrast within some domain from field measurements taken outside that domain. It is well known that this problem is both non-linear and ill-posed in the classical sense, i.e., the solution is non-unique and small changes in the measured field data may cause arbitrarily large changes in the solution [1]. At a single frequency, the illposedness of the problem remains even under linearizing assumptions e.g., the Born approximation [2]. Under such an approximation, one obtains a Fredholm integral equation of the first kind and a discretization of the monochromatic integral equation yields an ill-conditioned system as a result of the illposedness of the underlying continuous problem [1]. The result of this ill-conditioning is that one must utilize some form of regularization, i.e., a method which selects a particular solution from within a class of possible solutions by imposing additional constraints.