Uniform error estimates for the random batch method to the first-order consensus models with antisymmetric interaction kernels

Abstract

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 P and the time step τ, uniformly in particle number and time, namely, L2‐error is of O(τ/P). Thus the computational cost per time step is O(NP), where N is the number of particles and one typically chooses P≪N, while the direct summation would cost O(N2). Our analytical error estimate is further verified by numerical simulations.