Multilevel summation for the fast evaluation of forces for the simulation of biomolecules

The multilevel summation method computes an approximation to the pairwise electrostatic interaction potential and respective forces. The scalar potential is smoothly split into a short-range part computed exactly and a slowly varying long-range part approximated from a hierarchy of grids. Multilevel summation is especially appropriate for the dynamical simulation of biomolecules, because it computes continuous forces that are the gradient of a scalar potential. It provides a unified approach to computing electrostatics, in which the same method can be used for periodic and nonperiodic boundary conditions, with an amount of work that scales linearly as the size of the system. Multilevel summation is also flexible enough to be applied to other pairwise potentials. This thesis provides the most thorough investigation to date of the multilevel summation method and its use for computing electrostatic interactions. The mathematical and algorithmic details are presented along with a precise operation count. The approximation error from the method is analyzed, with error bounds formulated in terms of the fundamental method parameters. The cost and error analyses enable the determination of optimal method parameters for a desired error tolerance. Various interpolation schemes for the approximation are considered, and several alternative approaches to smoothing the electrostatic potential are examined. The use of the method with different boundary conditions is discussed, and it is shown that the application of multilevel summation to the periodic potential yields a finite sum, with the truncation expressed as bounded approximation error. The performance of multilevel summation is demonstrated to be superior to other commonly used fast methods for electrostatics, while providing comparable accuracy. The method is also shown to produce stable dynamics for cheaper, lower accuracy approximation.