A Fast Newton Algorithm for Entropy Maximization in Phase Determination

Abstract

A long-standing issue in the Bayesian statistical approach to the phase problem in X-ray crystallography is to solve an entropy maximization subproblem efficiently in every iteration of phase estimation. The entropy maximization problem is a semi-infinite convex program and can be solved in a finite dual space by using a standard Newton method. However, the Newton method is too expensive for this application since it requires O(n3) floating point operations per iteration, where n corresponds to the number of phases to be estimated. Other less expensive methods have been used, but they cannot guarantee fast convergence. In this paper, we present a fast Newton algorithm for the entropy maximization problem, which uses the Sherman--Morrison--Woodbury formula and the fast Fourier transform to compute the Newton step and requires only O(n log n) floating point operations per iteration, yet has the same convergence rate as the standard Newton. We describe the algorithm and discuss related computational issues. Numerical results on simple test cases will also be presented to demonstrate the behavior of the algorithm.