Reconstruction of the Support of an Inhomogeneity for the Fractional Helmholtz Equation

TL;DR

This paper develops a factorization method for reconstructing the support of inhomogeneities governed by the fractional s-Helmholtz equation, providing a new approach with proven uniqueness and numerical validation.

Contribution

It introduces a novel factorization method for the fractional Helmholtz inverse problem, including a new transmission eigenvalue analysis and a practical reconstruction algorithm.

Findings

The method successfully reconstructs inhomogeneity support in 2D numerical examples.

Eigenvalues of the transmission problem are shown to be discrete with no finite accumulation points.

The approach guarantees uniqueness for the support determination under certain conditions.

Abstract

We consider the inverse scattering problem for inhomogeneous media of compact support governed by the fractional s-Helmholtz equation, with $0<s<1$, in dimensions $d=1,2,3$. In particular, we study the determination of the support of the inhomogeneity from the far-field pattern of the scattered field generated by plane waves for all incident directions at a fixed frequency. The far-field pattern is defined as the principal term in the asymptotic expansion of the scattered field at infinity. It is shown in \cite{zilberberg2026limiting} that, up to a multiplicative constant, this coincides with the far-field pattern corresponding to the classical Helmholtz equation with the same inhomogeneity. Our approach is based on the development of the factorization method, which not only leads to an efficient and easily implementable reconstruction algorithm, but also provides a uniqueness result for determining the support of an admissible set of inhomogeneities. A fundamental ingredient in the analysis is a new transmission eigenvalue problem, whose eigenvalues must be excluded. Therefore, we prove that they are discrete with no finite accumulation points. We present numerical examples in dimension $d=2$ for both the direct and inverse problems.