Inverse wave-number-dependent source problems for the Helmholtz equation

TL;DR

This paper develops a multi-frequency factorization method to reconstruct the support of wave-number-dependent sources in the Helmholtz equation, using far-field data at fixed observation directions, applicable in both near-field and far-field scenarios.

Contribution

It introduces a novel computational criterion for support reconstruction from sparse multi-frequency data, including near-field data, and establishes the method's validity through numerical tests.

Findings

Effective support recovery in 2D and 3D cases.

Validates approach with numerical experiments.

Applicable to sparse observation directions and near-field data.

Abstract

This paper is concerned with the multi-frequency factorization method for imaging the support of a wave-number-dependent source function. It is supposed that the source function is given by the inverse Fourier transform of some time-dependent source with a priori given radiating period. Using the multi-frequency far-field data at a fixed observation direction, we provide a computational criterion for characterizing the smallest strip containing the support and perpendicular to the observation direction. The far-field data from sparse observation directions can be used to recover a $\Theta$-convex polygon of the support. The inversion algorithm is proven valid even with multi-frequency near-field data in three dimensions. The connections to time-dependent inverse source problems are discussed in the near-field case. Numerical tests in both two and three dimensions are implemented to show effectiveness and feasibility of the approach. This paper provides numerical analysis for a frequency-domain approach to recover the support of an admissible class of time-dependent sources.