Far Field Splitting

The aim of far field splitting for time-harmonic acoustic or electromagnetic waves is to decompose the far field of a wave radiated by an ensemble of several compactly supported sources into the individual far field components radiated by each of these sources separately. Without further assumptions this is an ill-posed inverse problem. Observing that far fields radiated by compactly supported sources have nearly sparse representations with respect to certain suitably transformed Fourier bases that depend on the approximate source locations, we develop an 1 characterization of these far fields and use it to reformulate the far field splitting problem as a weighted 1 minimization problem in the spirit of basis pursuit. To this end we assume that some a priori information on the locations of the individual source components is available. We prove that the unique solution to the weighted 1 minimization problem coincides with the solution to the far field splitting problem, and we discuss its numerical approximation. Furthermore, we propose an iterative strategy to successively improve the required a priori information by solving a sequence of these weighted 1 minimization problems, where estimates of the approximate locations of the individual source components that are used as a priori information for the next iteration are computed from the value of the current solution. This also gradually decreases the ill-posedness of the splitting problem, and it significantly improves the quality of the reconstructions. We present a series of numerical examples to demonstrate the performance of this algorithm.