Reconstruction via the intrinsic geometric structures of interior transmission eigenfunctions

TL;DR

This paper introduces a novel inverse scattering method leveraging the intrinsic geometric properties of interior transmission eigenfunctions to efficiently reconstruct the shape and singularities of unknown inhomogeneous media from acoustic far-field data.

Contribution

The paper develops a new inverse scattering scheme based on geometric properties of interior transmission eigenfunctions, enabling shape recovery and cusp singularity detection.

Findings

Method effectively captures cusp singularities.

Theoretical analysis supports the method's validity.

Numerical experiments demonstrate practical success.

Abstract

We are concerned with the inverse scattering problem of extracting the geometric structures of an unknown/inaccessible inhomogeneous medium by using the corresponding acoustic far-field measurement. Using the intrinsic geometric properties of the so-called interior transmission eigenfunctions, we develop a novel inverse scattering scheme. The proposed method can efficiently capture the cusp singularities of the support of the inhomogeneous medium. If further a priori information is available on the support of the medium, say, it is a convex polyhedron, then one can actually recover its shape. Both theoretical analysis and numerical experiments are provided. Our reconstruction method is new to the literature and opens up a new direction in the study of inverse scattering problems.