Explicit tight bounds on the stably recoverable information for the inverse source problem

TL;DR

This paper establishes explicit bounds on the frequency cut-off for stable source recovery in the inverse source problem for the 2D Helmholtz equation, using spectral analysis of the forward operator.

Contribution

It provides the first explicit, tight bounds on the spectral cut-off for the inverse source problem, expressed via zeros of Bessel functions, supported by numerical validation.

Findings

Derived explicit lower bounds for the spectral cut-off.

Numerically validated the bounds and proposed a conjecture for the upper bound.

Connected spectral cut-off to zeros of Bessel functions.

Abstract

For the inverse source problem with the two-dimensional Helmholtz equation, the singular values of the 'source-to-near field' forward operator reveal a sharp frequency cut-off in the stably recoverable information on the source. We prove and numerically validate an explicit, tight lower bound for the spectral location of this cut-off. We also conjecture and support numerically a tight upper bound for the cut-off. The bounds are expressed in terms of zeros of Bessel functions of the first and second kind.