We propose a random batch method (RBM) for a contractive interacting particle system on a network, which can be formulated as a first‐order consensus model with heterogeneous intrinsic dynamics and convolution‐type consensus interactions. The RBM was proposed and analyzed recently in a series of work by the third author and his collaborators for a general interacting particle system with a conservative external force, with particle‐number independent error estimate established under suitable regularity assumptions on the external force and interacting kernel. Unlike the aforementioned original RBM, our consensus model has two competing dynamics, namely “dispersion” (generated by heterogeneous intrinsic dynamics) and “concentration” (generated by consensus forcing). In a close‐to‐consensus regime, we present a uniform error estimate for a modified RBM in which a random batch algorithm is also applied to the part of intrinsic dynamics, not only to the interaction terms. We prove that the obtained error depends on the batch size and the time step , uniformly in particle number and time, namely, ‐error is of . Thus the computational cost per time step is , where is the number of particles and one typically chooses , while the direct summation would cost . Our analytical error estimate is further verified by numerical simulations.

中文翻译：

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具有反对称交互核的一阶共识模型的随机批处理方法的均匀误差估计

我们为网络上的压缩相互作用粒子系统提出了一种随机批处理方法（RBM），可以将其表述为具有异质内在动力学和卷积型共识相互作用的一阶共识模型。第三作者和他的合作者最近在一系列工作中提出了RBM，并对其进行了分析，以研究具有保守外力的一般相互作用粒子系统，并在适当的外力和相互作用核规律性假设下建立了独立于粒子数的误差估计。与前面提到的原始RBM不同，我们的共识模型具有两个相互竞争的动力学，即“分散”（由异质内在动力学生成）和“集中””（通过共识强制生成）。在一个接近共识的体制中，我们提出了一种改进的RBM的统一误差估计，其中随机批处理算法也应用于内在动力学部分，而不仅应用于交互项。我们证明所获得的误差取决于批次大小和时间步长，在颗粒数量和时间上均一，即－误差为。因此，每时间步长的计算成本为，其中是粒子的数量，通常选择一个，而直接求和则需要成本。我们的分析误差估计值已通过数值模拟进一步验证。