Stable source reconstruction from a finite number of measurements in the multi-frequency inverse source problem

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. arXiv:1803.05010v1 [math.NA]

Stability of the inverse source problem for the Helmholtz equation in R3

“…As shown in [15], for m > −1 and n ∈ N it holds that µ m+1,n − µ m,n > 1. Recalling that λ m,n = (µ m,n /R 0 ) 2 , we have that…”

“…The result follows from Theorem 2 in [15]. We adapt the proof to the current setting, including all the relevant details for completeness.…”

“…In practical settings, it is impossible to perform infinitely many measurements, so we turn to reconstructing sources in finite-dimensional subspaces of L 2 (B 0 ). The analysis in this section is a natural extension of the methods in [15] from R 2 to R 3 , formulated in a simpler and more flexible way.…”

“…For M ∈ N, let Λ M = {λ α,β } be a set of any M eigenvalues in (15). We associate with Λ M the finite-dimensional source space…”

“…A key observation in [15] was that the eigenvalues become more densely spaced as one moves up the real line. An estimate for the expected number N λ (k, ∆k) of λ…”