Hierarchical O(N) Computation of Small-Angle Scattering Profiles and their Associated Derivatives

Fast algorithms for Debye summation, which arises in computations performed in crystallography, small/wide-angle X-ray scattering (SAXS/WAXS) and small-angle neutron scattering (SANS), were recently presented in Gumerov et al. (J. Comput. Chem., 2012, 33:1981). The use of these algorithms can speed up computation of scattering profiles in macromolecular structure refinement protocols. However, these protocols often employ an iterative gradient-based optimization procedure, which then requires derivatives of the profile with respect to atomic coordinates. An extension to one of the algorithms is presented which allows accurate, O(N) cost computation of the derivatives along with the scattering profile. Computational results show orders of magnitude improvement in computational efficiency, while maintaining prescribed accuracy. This opens the possibility to efficiently integrate small-angle scattering data into structure determination and refinement of macromolecular systems