Stable source reconstruction from a finite number of measurements in the Multi-frequency Inverse Source Problem

TL;DR

This paper addresses the inverse source problem for the Helmholtz equation, proposing a finite-frequency measurement approach for unique and stable source reconstruction within a specific subspace, supported by theoretical analysis and numerical validation.

Contribution

It introduces a constructive method for source reconstruction using finite measurements, including criteria for minimal frequency sets and stability conditions, in a finite-dimensional Fourier-Bessel subspace.

Findings

Unique reconstruction from finite frequency measurements.

Identification of minimal measurement frequency sets.

Stable reconstruction under mild assumptions.

Abstract

We consider the multi-frequency inverse source problem for the scalar Helmholtz equation in the plane. The goal is to reconstruct the source term in the equation from measurements of the solution on a surface outside the support of the source. We study the problem in a certain finite dimensional setting: From measurements made at a finite set of frequencies we uniquely determine and reconstruct sources in a subspace spanned by finitely many Fourier-Bessel functions. Further, we obtain a constructive criterion for identifying a minimal set of measurement frequencies sufficient for reconstruction, and under an additional, mild assumption, the reconstruction method is shown to be stable. Our analysis is based on a singular value decomposition of the source-to-measurement forward operators and the distribution of positive zeros of the Bessel functions of the first kind. The reconstruction method is implemented numerically and our theoretical findings are supported by numerical experiments.