ITR: Phase Retrieval Algorithms

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X-ray crystallography is a fundamental technique for determining the structure of proteins. However, the physics of this process means that information about the relative phases of the thousands of x-ray beams used in the process is lost. The recovery of phases is referred to as "phase retrieval." Essentially all modern algorithms for phase retrieval are iterative, including the well-known "Shake and Bake" algorithm. However, a general problem with iterative approaches is that the iterates are drawn to attractors that are not true solutions. This project will investigate a significantly different, non-iterative approach that does not share this drawback. If successful, the result will be a computing environment that will benefit a variety of disciplines that engage in phase retrieval.The new algorithm exploits techniques in optimization theory that have never been used in phase retrieval before. One key idea is to use a different objective function from past approaches: the zero-frequency component, or "charge" Q. A key property of Q is linearity, which avoids the many local minima that stagnate iterative methods for other objective functions. In the language of optimization theory, the objective function Q permits a direct translation of the phase retrieval problem into a mixed-integer program (MIP). Decades of research on solving MIPs can thus be applied to solve difficult instances of phase retrieval encountered in crystallography.