The hardware upgrades at the Advanced Photon Source (APS) over the next few years will bring with them the capability to probe materials with X-rays in ways that were unfeasible so far. The new experiments will result in increased volumes and various complex representations of materials data, with tremendous scope for the development of new analysis techniques. These methods, whose initial development is currently underway, potentially call for a variety of intersecting expertise in physics and applied mathematics. In this talk I will describe some of this current work in our group, largely focused on inverse problems in materials imaging, which include signal processing for X-ray scattering data and new methods of phase retrieval. In addition to some related past work that touches upon robust optimization, I will talk about future computational prospects in the problem of polycrystal imaging. This problem, whose success is heavily invested in the APS upgrade, will potential ly require the development of optimization and machine learning methods (particularly deep learning) in conjunction with accurate physical modeling wherever required along its data acquisition/processing pipeline. In summary, I will provide a broad view of the potential avenues of computational development in this area of inverse problems in coherent X-ray imaging.