  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 becomes increasingly important to study the involved
numerical algorithms in a systematic and principled manner. Indeed, the last
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.
