Error analysis of time-discrete random batch method for interacting particle systems and associated mean-field limits

• Published in: IMA journal of numerical analysis 2024-06, Vol.44 (3), p.1660-1698
• Main Authors: Ye, Xuda, Zhou, Zhennan
• Format: Article
• Language: English
• ISSN: 0272-4979; 1464-3642
• DOI: 10.1093/imanum/drad043

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. The triangle inequality framework employed in this paper is a convenient approach to estimate the long-time sampling error of the numerical methods. Using the triangle inequality framework, we show that the long-time error of the discrete random batch 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 $.