随机替换批次方法应用于Cucker-Smale模型的一致时间误差估计 (Uniform-in-time error estimate of random batch method with replacement for the Cucker-Smale model)

from arxiv, We found a critical mistake in the proof of the main text. To modify this mistake, we almost totally rewrite the paper and increase many results

The Random Batch Method (RBM), proposed by Jin et al. in 2020, is an efficient algorithm for simulating interacting particle systems. The uniform-in-time error estimates of the RBM without replacement have been obtained for various interacting particle systems, while the analysis of the RBM with replacement is just considered in (Cai et al., 2024) recently for the first-order systems governed by Langevin dynamics. In this work, we present the error estimate for the RBM with replacement applied to a second-order system known as the Cucker-Smale model. By introducing a crucial auxiliary system and leveraging the intrinsic characteristics of the Cucker-Smale model, we derive an estimate that is uniform in both time and particle numbers. Additionally, we provide numerical simulations to validate the analytical results.

翻译：随机批次方法（RBM）由Jin等人于2020年提出，是一种用于模拟相互作用粒子系统的高效算法。对于无替换的RBM，已在多种相互作用粒子系统中获得了一致时间误差估计；而针对有替换RBM的分析，近期才由Cai等人在2024年首次针对朗之万动力学控制的一阶系统展开研究。本工作中，我们提出了有替换RBM应用于二阶系统（即Cucker-Smale模型）的误差估计。通过引入关键辅助系统并利用Cucker-Smale模型的内在特性，我们推导出了在时间和粒子数上均保持一致的估计结果。此外，我们还提供了数值模拟以验证理论分析结果。