On the inverse scattering problem for radially-symmetric domains in two dimensions

TL;DR

This paper introduces a method for solving inverse Helmholtz problems in radially-symmetric domains using trace formulas to relate multi-frequency data to potential derivatives, enabling stable reconstruction demonstrated through numerical examples.

Contribution

It presents a novel trace formula-based approach for stable potential reconstruction in radially-symmetric inverse scattering problems.

Findings

Successful reconstruction of potentials in numerical tests

Stable solution method for inverse Helmholtz problems

Effective use of multi-frequency data for inverse problems

Abstract

In the present paper we describe a method for solving inverse problems for the Helmholtz equation in radially-symmetric domains given multi-frequency data. Our approach is based on the construction of suitable trace formulas which relate the impedance of the total field at multiple frequencies to derivatives of the potential. Using this trace formula we obtain a system of coupled differential equations which can be solved to obtain the potential in a stable manner. Finally, the performance of the reconstruction algorithm is illustrated with several numerical examples.