Fast divide-and-conquer algorithm for evaluating polarization in classical force fields

Evaluation of the self-consistent polarization energy forms a major computational bottleneck in polarizable force fields. In large systems, the linear polarization equations are typically solved iteratively with techniques based on Jacobi iterations (JI) or preconditioned conjugate gradients (PCG). Two new variants of JI are proposed here that exploit domain decomposition to accelerate the convergence of the induced dipoles. The first, divide-and-conquer JI (DC-JI), is a block Jacobi algorithm which solves the polarization equations within non-overlapping sub-clusters of atoms directly via Cholesky decomposition, and iterates to capture interactions between sub-clusters. The second, fuzzy DC-JI, achieves further acceleration by employing overlapping blocks. Fuzzy DC-JI is analogous to an additive Schwarz method, but with distance-based weighting when averaging the fuzzy dipoles from different blocks. Key to the success of these algorithms is the use of K-means clustering to identify natural atomic sub-clusters automatically for both algorithms and to determine the appropriate weights in fuzzy DC-JI. The algorithm employs knowledge of the 3-D spatial interactions to group important elements in the 2-D polarization matrix. When coupled with direct inversion in the iterative subspace (DIIS) extrapolation, fuzzy DC-JI/DIIS in particular converges in a comparable number of iterations as PCG, but with lower computational cost per iteration. In the end, the new algorithms demonstrated here accelerate the evaluation of the polarization energy by 2-3 fold compared to existing implementations of PCG or JI/DIIS.