Numerical Optimization of a Walk-on-Spheres Solver for the Linear Poisson-Boltzmann Equation

Stochastic walk-on-spheres (WOS) algorithms for solving the linearized Poisson-Boltzmann equation (LPBE) provide several attractive features not available in traditional deterministic solvers: Gaussian error bars can be computed easily, the algorithm is readily parallelized and requires minimal memory and multiple solvent environments can be accounted for by reweighting trajectories. However, previously-reported computational times of these Monte Carlo methods were not competitive with existing deterministic numerical methods. The present paper demonstrates a series of numerical optimizations that collectively make the computational time of these Monte Carlo LPBE solvers competitive with deterministic methods. The optimization techniques used are to ensure that each atom’s contribution to the variance of the electrostatic solvation free energy is the same, to optimize the bias-generating parameters in the algorithm and to use an epsilon-approximate rather than exact nearest-neighbor search when determining the size of the next step in the Brownian motion when outside the molecule.

Bregman Hyperplane Trees for Fast Approximate Nearest Neighbor Search

This article presents a new approximate index structure, the Bregman hyperplane tree, for indexing the Bregman divergence, aiming to decrease the number of distance computations required at query processing time, by sacrificing some accuracy in the result. The experimental results on various high-dimensional data sets demonstrate that the proposed index structure performs comparably to the state-of-the-art Bregman ball tree in terms of search performance and result quality. Moreover, this method results in a speedup of well over an order of magnitude for index construction. The authors also apply their space partitioning principle to the Bregman ball tree and obtain a new index structure for exact nearest neighbor search that is faster to build and a slightly slower at query processing than the original.