A novel quantitative inverse scattering scheme using interior resonant modes

TL;DR

This paper introduces a two-phase quantitative imaging method for identifying obstacles in acoustic scattering, combining interior eigenvalue determination with a Newton-type iteration to accurately reconstruct obstacle boundaries efficiently.

Contribution

The paper presents a novel two-phase scheme that determines interior eigenvalues and eigenfunctions from far-field data, enabling high-accuracy obstacle boundary reconstruction without direct scattering computations.

Findings

Accurate obstacle boundary reconstructions in 2D and 3D.

Efficient method avoiding repeated scattering problem solutions.

Theoretical justification and numerical validation of the scheme.

Abstract

This paper is devoted to a novel quantitative imaging scheme of identifying impenetrable obstacles in time-harmonic acoustic scattering from the associated far-field data. The proposed method consists of two phases. In the first phase, we determine the interior eigenvalues of the underlying unknown obstacle from the far-field data via the indicating behaviour of the linear sampling method. Then we further determine the associated interior eigenfunctions by solving a constrained optimization problem, again only involving the far-field data. In the second phase, we propose a novel iteration scheme of Newton's type to identify the boundary surface of the obstacle. By using the interior eigenfunctions determined in the first phase, we can avoid computing any direct scattering problem at each Newton's iteration. The proposed method is particularly valuable for recovering a sound-hard obstacle, where the Newton's formula involves the geometric quantities of the unknown boundary surface in a natural way. We provide rigorous theoretical justifications of the proposed method. Numerical experiments in both 2D and 3D are conducted, which confirm the promising features of the proposed imaging scheme. In particular, it can produce quantitative reconstructions of high accuracy in a very efficient manner.