Error Analysis of Time-Discrete Random Batch Method for Interacting Particle Systems and Associated Mean-Field Limits

TL;DR

This paper analyzes the error of the fully discrete random batch method for interacting particle systems, showing it converges with a specific rate and applies to mean-field limits like the McKean-Vlasov process.

Contribution

It provides the first rigorous error estimates for the long-time behavior of the fully discrete random batch method, including invariant distribution approximation.

Findings

Long-time error is $O(\sqrt{ au} + e^{-\lambda t})$

Error bounds are independent of particle number $N$

Results extend to McKean-Vlasov mean-field processes

Abstract

The random batch method provides an efficient algorithm for computing statistical properties of a canonical ensemble of interacting particles. In this work, we study the error estimates of the fully discrete random batch method, especially in terms of approximating the invariant distribution. Using a triangle inequality framework, we show that the long-time error of the method is $O(\sqrt{\tau} + e^{-\lambda t})$, where $\tau$ is the time step and $\lambda$ is the convergence rate which does not depend on the time step $\tau$ or the number of particles $N$. Our results also apply to the McKean-Vlasov process, which is the mean-field limit of the interacting particle system as the number of particles $N\rightarrow\infty$.