Random Batch Method with Momentum Correction

TL;DR

This paper introduces an improved Random Batch Method with Momentum Correction (RBM-M) that enhances convergence and accuracy for stochastic differential equations with singular interactions, supported by theoretical analysis and numerical validation.

Contribution

The paper develops a momentum-corrected version of RBM that extends its applicability to systems with singular kernels and provides theoretical error bounds.

Findings

RBM-M achieves smaller errors than original RBM.

Numerical experiments confirm the effectiveness of RBM-M.

Theoretical proof supports error control of the new method.

Abstract

The Random Batch Method (RBM) is an effective technique to reduce the computational complexity when solving certain stochastic differential problems (SDEs) involving interacting particles. It can transform the computational complexity from O(N^2) to O(N), where N represents the number of particles. However, the traditional RBM can only be effectively applied to interacting particle systems with relatively smooth kernel functions to achieve satisfactory results. To address the issue of non-convergence of the RBM in particle interaction systems with significant singularities, we propose some enhanced methods to make the modified algorithm more applicable. The idea for improvement primarily revolves around a momentum-like correction, and we refer to the enhanced algorithm as the Random Batch Method with Momentum Correction ( RBM-M). We provide a theoretical proof to control the error of the algorithm, which ensures that under ideal conditions it has a smaller error than the original algorithm. Finally, numerical experiments have demonstrated the effectiveness of the RBM-M algorithm.