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Mean field error estimate of the random batch method for large interacting particle system

Zhenyu Huang, Shi Jin, Lei Li

Abstract

The random batch method (RBM) proposed in [Jin et al., J. Comput. Phys., 400(2020), 108877] for large interacting particle systems is an efficient with linear complexity in particle numbers and highly scalable algorithm for $N$-particle interacting systems and their mean-field limits when $N$ is large. We consider in this work the quantitative error estimate of RBM toward its mean-field limit, the Fokker-Planck equation. Under mild assumptions, we obtain a uniform-in-time $O(τ^2 + 1/N)$ bound on the scaled relative entropy between the joint law of the random batch particles and the tensorized law at the mean-field limit, where $τ$ is the time step size and $N$ is the number of particles. Therefore, we improve the existing rate in discretization step size from $O(\sqrtτ)$ to $O(τ)$ in terms of the Wasserstein distance.

Mean field error estimate of the random batch method for large interacting particle system

Abstract

The random batch method (RBM) proposed in [Jin et al., J. Comput. Phys., 400(2020), 108877] for large interacting particle systems is an efficient with linear complexity in particle numbers and highly scalable algorithm for -particle interacting systems and their mean-field limits when is large. We consider in this work the quantitative error estimate of RBM toward its mean-field limit, the Fokker-Planck equation. Under mild assumptions, we obtain a uniform-in-time bound on the scaled relative entropy between the joint law of the random batch particles and the tensorized law at the mean-field limit, where is the time step size and is the number of particles. Therefore, we improve the existing rate in discretization step size from to in terms of the Wasserstein distance.
Paper Structure (15 sections, 13 theorems, 126 equations, 1 figure, 1 algorithm)

This paper contains 15 sections, 13 theorems, 126 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discrete random batch
  4. An analogue of the Liouville equation
  5. Assumptions
  6. Some auxiliary results
  7. The Log-Sobolev inequality
  8. Moment control
  9. Estimate of the Fisher information
  10. The main results
  11. Conclusion
  12. Proof of Lemma \ref{['Lioulemma']}
  13. Omitted details in Section \ref{['mainthm']}
  14. The Radon-Nikodym derivatives of path measures
  15. The Law of Large Numbers at exponential scale

Key Result

Lemma 2.1

Denote by $\bar{\varrho}_t^{N, \boldsymbol{\xi}}$ the probability density function of $\bar{\textbf{X}}_t = \left(\bar{X}_t^1, \cdots, \bar{X}_t^N\right)$ defined in (disrbm) for $t\in\left[T_k, T_{k+1}\right)$. Then the following Liouville equation holds: where and Here, $\textbf{x}=(x_1, \cdots, x_n) \in \mathbb{R}^{Nd}$.

Figures (1)

  • Figure 1: Illustration of the various equations.

Theorems & Definitions (21)

  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Theorem 4.1
  • Corollary 4.1
  • proof : Proof of Theorem \ref{['mainthm']}
  • proof : Proof of Lemma \ref{['Lioulemma']}
  • Lemma B.1
  • ...and 11 more