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

Xuda Ye, Zhennan Zhou

Published 2022-06-05Version 1

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 $O(\sqrt{\tau} + e^{-\beta t})$, where $\tau$ is the time step and $\beta$ 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 $N\rightarrow\infty$.