The numerics of phase retrieval

Phase retrieval, i.e. the problem of recovering a function from the squared magnitude of its Fourier transform, arises in many applications, such as X-ray crystallography, diffraction imaging, optics, quantum mechanics and astronomy. This problem has confounded engineers, physicists, and mathematicians for many decades. Recently, phase retrieval has seen a resurgence in research activity, ignited by new imaging modalities and novel mathematical concepts. As our scientific experiments produce larger and larger datasets and we aim for faster and faster throughput, it is becoming increasingly important to study the involved numerical algorithms in a systematic and principled manner. Indeed, the past decade has witnessed a surge in the systematic study of computational algorithms for phase retrieval. In this paper we will review these recent advances from a numerical viewpoint.