Uncertainty principles for inverse source problems, far field splitting and data completion

TL;DR

This paper establishes criteria and algorithms for stable recovery of individual source components and missing data in inverse far field problems for acoustic and electromagnetic waves, considering physical and geometric constraints.

Contribution

It provides new precise conditions relating physical parameters for stable reconstruction and introduces algorithms validated by analytic and numerical examples.

Findings

Stable recovery is possible under specific wavelength and support conditions.

The methods effectively restore missing data segments.

Results demonstrate improved stability and resolution in inverse source problems.

Abstract

Starting with far field data of time-harmonic acoustic or electromagnetic waves radiated by a collection of compactly supported sources in two-dimensional free space, we develop criteria and algorithms for the recovery of the far field components radiated by each of the individual sources, and the simultaneous restoration of missing data segments. Although both parts of this inverse problem are severely ill-conditioned in general, we give precise conditions relating the wavelength, the diameters of the supports of the individual source components and the distances between them, and the size of the missing data segments, which guarantee that stable recovery in presence of noise is possible. The only additional requirement is that a priori information on the approximate location of the individual sources is available. We give analytic and numerical examples to confirm the sharpness of our results and to illustrate the performance of corresponding reconstruction algorithms, and we discuss consequences for stability and resolution in inverse source and inverse scattering problems.