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Propagation of chaos and approximation error of random batch particle system in the mean field regime

Lei Li, Yuelin Wang, Shi Jin

TL;DR

The paper investigates propagation of chaos for the Random Batch Method (RBM) in the mean-field regime and provides explicit, sharp error bounds between the RBM distribution and the tensorized mean-field law. It develops time-marginal BBGKY-type equations, introduces auxiliary random batch divisions to decouple interactions, and employs entropy methods to derive an ODE hierarchy for the k-particle relative entropies. The main result is a bound $H_t^k \le C t e^{Ct} (\frac{k^2}{N^2} + k\tau^2)$, together with a total-variation estimate, establishing precise convergence rates in both $N$ and the time-step $\tau$. These findings justify the RBM as an efficient and reliable simulator in the mean-field limit and quantify the impact of random batching on approximation error.

Abstract

The random batch method [J. Comput. Phys. 400 (2020) 108877] is not only an efficient algorithm for simulation of classical $N$-particle systems and their mean-field limit, but also a new model for interacting particle system that could be more physical in some applications. In this work, we establish the propagation of chaos for the random batch particle system and at the same time obtain its sharp approximation error to the classical mean field limit of $N$-particle systems. The proof leverages the BBGKY hierarchy and achieves a sharp bound both in the particle number $N$ and the time step $τ$. In particular, by introducing a coupling of the division of the random batches to resolve the $N$-dependence, we derive an $\mathcal{O}(k^2/N^2 + kτ^2)$ bound on the $k$-particle relative entropy between the law of the system and the tensorized law of the mean-field limit. This result provides a useful understanding of the convergence properties of the random batch system in the mean field regime.

Propagation of chaos and approximation error of random batch particle system in the mean field regime

TL;DR

The paper investigates propagation of chaos for the Random Batch Method (RBM) in the mean-field regime and provides explicit, sharp error bounds between the RBM distribution and the tensorized mean-field law. It develops time-marginal BBGKY-type equations, introduces auxiliary random batch divisions to decouple interactions, and employs entropy methods to derive an ODE hierarchy for the k-particle relative entropies. The main result is a bound , together with a total-variation estimate, establishing precise convergence rates in both and the time-step . These findings justify the RBM as an efficient and reliable simulator in the mean-field limit and quantify the impact of random batching on approximation error.

Abstract

The random batch method [J. Comput. Phys. 400 (2020) 108877] is not only an efficient algorithm for simulation of classical -particle systems and their mean-field limit, but also a new model for interacting particle system that could be more physical in some applications. In this work, we establish the propagation of chaos for the random batch particle system and at the same time obtain its sharp approximation error to the classical mean field limit of -particle systems. The proof leverages the BBGKY hierarchy and achieves a sharp bound both in the particle number and the time step . In particular, by introducing a coupling of the division of the random batches to resolve the -dependence, we derive an bound on the -particle relative entropy between the law of the system and the tensorized law of the mean-field limit. This result provides a useful understanding of the convergence properties of the random batch system in the mean field regime.
Paper Structure (16 sections, 14 theorems, 134 equations, 1 algorithm)

This paper contains 16 sections, 14 theorems, 134 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setup and main result
  3. The dynamics of the RBM-driven distribution
  4. Derivation of the dynamics of $k$-particle marginal distribution
  5. Introduction of auxiliary random batch divisions
  6. The discrepancy between the coupled random divisions
  7. Proof of Theorem \ref{['thm:1']}
  8. Derivation of the ODEs for the local relative entropy
  9. Boundedness of $H_t^N$
  10. Improvement of the result
  11. Auxiliary lemmas
  12. Moment bounds
  13. Estimates of the density for the RBM system
  14. Auxiliary inequalities
  15. Estimates of iterated exponential integrals
  16. ...and 1 more sections

Key Result

Theorem 2.1

Suppose that $b_0$ and $b_1$ satisfy Assumption ass:b0b1, the initial law $\mu_0$ satisfies Assumption ass:ini, and $\sigma>0$. Consider the joint law $\mu_t^N$ for the $N$-particle system eq:rbm1d and the law $\bar{\mu}_t$ for system eq:mckean. Then, for any $t>0$, there exists a constant $C$ indep

Theorems & Definitions (25)

  • Remark 2.1
  • Theorem 2.1
  • Corollary 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Remark 3.1
  • Proposition 3.2
  • proof
  • ...and 15 more