Uncertainty Principles for Three-Dimensional Inverse Source Problems

Abstract

Novel uncertainty principles for far fields of time-harmonic acoustic or electromagnetic waves radiated by collections of well-separated localized sources in two-dimensional free space have recently been established in [R. Griesmaier and J. Sylvester, SIAM J. Appl. Math., 77 (2017), pp. 154--180]. These uncertainty principles have been utilized to develop stability estimates and reconstruction algorithms for the recovery of the far field radiated by each of the individual sources, and the simultaneous restoration of missing data segments. In this paper, we present generalizations of these uncertainty principles for a relevant three-dimensional setting. We consider extensions of the reconstruction schemes for far field splitting and data completion, including their stability analysis, and we discuss the sharpness of our results in the three-dimensional case. We briefly comment on the numerical implementation of the reconstruction algorithms and include a numerical example.