Random-batch method for multi-species stochastic interacting particle systems

TL;DR

This paper introduces a random-batch algorithm for multi-species stochastic particle systems that significantly reduces computational costs while maintaining accuracy, with proven convergence and practical applications demonstrated through numerical simulations.

Contribution

The paper extends existing random-batch methods to multi-species systems, providing a new efficient algorithm with proven convergence and real-world application examples.

Findings

Reduces computational cost by an order of magnitude.

Proves $L^2$ error behaves like the square root of time step size.

Demonstrates effectiveness through opinion dynamics simulations.

Abstract

A random-batch method for multi-species interacting particle systems is proposed, extending the method of S. Jin, L. Li, and J.-G. Liu [J. Comput. Phys. 400 (2020), 108877]. The idea of the algorithmus is to randomly divide, at each time step, the ensemble of particles into small batches and then to evolve the interaction of each particle within the batches until the next time step. This reduces the computational cost by one order of magnitude, while keeping a certain accuracy. It is proved that the $L^2$ error of the error process behaves like the square root of the time step size, uniformly in time, thus providing the convergence of the scheme. The numerical efficiency is tested for some examples, and numerical simulations of the opinion dynamics in a hierarchical company, consisting of workers, managers, and CEOs, are presented.