Asymptotic uniform estimate of random batch method with replacement for the Cucker-Smale model

TL;DR

This work analyzes the Random Batch Method (RBM) applied to the Cucker–Smale flocking model, focusing on two variants: RBM-r (with replacement) and RBM-1 (without replacement). It proves that both RBM schemes exhibit asymptotic flocking with velocity alignment and uniformly bounded spatial spread, with decay rates that depend on batch size $p$, time step $\tau$, and the kernel bounds but are independent of the particle number $N$. A central contribution is a uniform-in-time error estimate between the RBM-r (and RBM-1) dynamics and the original CS system, established via an auxiliary dynamics IPS' that links stochastic batch interactions to full all-to-all interactions. The results are complemented by numerical simulations confirming the theoretical rates and demonstrating momentum conservation and the practical efficacy of RBM-r and RBM-1 in large-scale particle simulations. The approach advances understanding of stochastic batch-approximation methods for nonlinear multi-agent systems and suggests avenues for extending to broader consensus models.

Abstract

The Random Batch Method (RBM) [S. Jin, L. Li and J.-G. Liu, Random Batch Methods (RBM) for interacting particle systems, J. Comput. Phys. 400 (2020) 108877] is not only an efficient algorithm for simulating interacting particle systems, but also a randomly switching networked model for interacting particle system. This work investigates two RBM variants (RBM-r and RBM-1) applied to the Cucker-Smale flocking model. We establish the asymptotic emergence of global flocking and derive corresponding error estimates. By introducing a crucial auxiliary system and leveraging the intrinsic characteristics of the Cucker-Smale model, and under suitable conditions on the force, our estimates are uniform in both time and particle numbers. In the case of RBM-1, our estimates are sharper than those in Ha et al. (2021). Additionally, we provide numerical simulations to validate our analytical results.

Figures (9)

• Figure 1: A simulation on trajectories along time on the original system (left), the RBM-1 system (middle) and the RBM-r system (right) with $p=2$.
• Figure 2: (a): The SSD of V (velocities) from 100 simulations at $\tau =0.1$. (b): The SSD of X (positions) from 100 simulations at $\tau =0.1$.
• Figure 3: (a): The SSD of V (velocities) from 100 simulations at $\tau =0.01$. (b): The SSD of X (positions) from 100 simulations at $\tau =0.01$.
• Figure 4: (RBM-r)(a): The $\ell^2$-errors of velocities from 1000 simulations, computed with different $p$. (b): Scaled error by the term $\sqrt{1- \frac{p}{N} + \frac{1}{p-1}-\frac{1}{N-1}}$. The scaled errors from different $p$ show similar values along time.
• Figure 5: (RBM-1)(a): The $\ell^2$-errors of velocities from 1000 simulations, computed with different $p$. (b): Scaled error by the term $\sqrt{ \ \frac{1}{p-1}-\frac{1}{N-1}}$. The scaled errors from different $p$ show similar values along time.

Theorems & Definitions (45)

• Proposition 1
• proof
• Definition 1
• Proposition 2: ha2021uniform, Lemma 2.2, Proposition 2.1
• Proposition 3
• Proposition 4
• proof
• Definition 2
• Theorem 1: Stochastic flocking dynamics of the RBM-r
• Theorem 2: Stochastic flocking dynamics of the RBM-1