Stable source reconstruction from a finite number of measurements in the Multi-frequency Inverse Source Problem
This work provides a theoretical and numerical framework for stable source reconstruction in inverse problems, which is relevant for applications like antenna design and medical imaging, but the results are incremental as they extend existing methods to a specific finite-dimensional setting.
The paper addresses the multi-frequency inverse source problem for the scalar Helmholtz equation, showing that sources in a finite-dimensional subspace spanned by Fourier-Bessel functions can be uniquely and stably reconstructed from measurements at a finite set of frequencies. They provide a constructive criterion for minimal frequency sets and validate their method numerically.
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.