On the series solutions of integral equations in scattering

TL;DR

This paper analyzes the convergence of Neumann series solutions for the Helmholtz equation in inverse scattering, deriving optimal conditions and proposing an interpolation method to extend applicability, supported by numerical validation.

Contribution

It establishes the necessary and sufficient conditions for strong convergence of the series and introduces an interpolation technique to broaden its use in scattering problems.

Findings

Derived optimal convergence conditions for the series

Proposed an interpolation method for wider applicability

Validated results through numerical experiments

Abstract

We study the validity of the Neumann or Born series approach in solving the Helmholtz equation and coefficient identification in related inverse scattering problems. Precisely, we derive a sufficient and necessary condition under which the series is strongly convergent. We also investigate the rate of convergence of the series. The obtained condition is optimal and it can be much weaker than the traditional requirement for the convergence of the series. Our approach makes use of reduction space techniques proposed by Suzuki \cite{Suzuki-1976}. Furthermore we propose an interpolation method that allows the use of the Neumann series in all cases. Finally, we provide several numerical tests with different medium functions and frequency values to validate our theoretical results.