Acceleration of Born Series by Change of Variables

Abstract

In this work, we propose a method to enhance the convergence of the Born series. The Born series is widely used in scattering theory, but its convergence is only guaranteed under certain restrictive conditions which limit the cases where this formulation can be applied. The proposed method, based on modifying the singularities of the resolvent operator by a change of variables, accelerates the convergence of the series or even achieves convergence in otherwise divergent cases. Due to its low computational cost and ease of implementation, the method preserves the advantages of the Born series while it extends its range of application. It can also be applied to a variety of methods relying on the same mathematical principles as the Born series. Numerical examples are provided to show that a significant improvement can be achieved with simple techniques and with a limited knowledge about the operator spectrum.

A Monte Carlo method for solving the electromagnetic scattering problem in dielectric bodies

“…where the summation runs over the k-tuples of non-negative integers j i that fulfill the condition k i=1 j i = l. For the case l < k, since j i are non-negative integers, then at least one of the elements j i should be zero to fulfill k i=1 j i = l < k. This implies that, for l < k, the zero order derivative (the function t −1 itself) will appear at least once on the product on the right side of (16). Recalling that t −1 (0) = 0, we conclude (16) will be zero when evaluated at μ = 0, due to the presence of one or more zero elements on the product. Thus we only need to consider the l ≥ k elements in (15).…”

“…An equation of the form u = Hu + c, where H is a linear operator, only can be solved by a Neumann series if the the spectral radius of H is smaller than one [15]. For this reason, we developed a modified Born series with enhanced convergence [16]. In this work, we propose a particular implementation of this method where the coefficients are obtained by solving a conformal mapping problem.…”