A modified well-conditioned iterative solution to the 1D inverse scattering problem using the Born iterative method

In this paper we present a natural modification to a previously proposed well-conditioned algorithm for solving the 1D inverse scattering problem using adaptive whole-domain basis functions and the iterative Born method. we make use of the moment method in conjunction with a whole-domain, adaptive basis function expansion of the permittivity distribution. By considering a multitude of scattering experiments at specially selected frequencies (which effectively changes the kernel of the integral operator) we are capable of producing a well-conditioned linear system of equations for computing the basis function coefficients thereby eliminating the need for regularization. We refer to this type of kernel-altering moment method as a generalized method of moments. Herein we discuss the natural extension to an expansion where each basis function is assigned a unique parameter. We show that such a technique eliminates the need for evaluating the condition number of the matrix directly, decreasing computational cost. For simplicity we develop the appropriate theory in 1D