Random Batch Methods (RBM) for interacting particle systems

Abstract

We develop Random Batch Methods for interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from O(N 2 ) per time step to O(N ), for a system with N particles with binary interactions. On one hand, these methods are efficient Asymptotic-Preserving schemes for the underlying particle systems, allowing N -independent time steps and also capture, in the N → ∞ limit, the solution of the mean field limit which are nonlinear Fokker-Planck equations; on the other hand, the stochastic processes generated by the algorithms can also be regarded as new models for the underlying problems. For one of the methods, we give a particle number independent error estimate under some special interactions. Then, we apply these methods to some representative problems in mathematics, physics, social and data sciences, including the Dyson Brownian motion from random matrix theory, Thomson's problem, distribution of wealth, opinion dynamics and clustering. Numerical results show that the methods can capture both the transient solutions and the global equilibrium in these problems.

Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives

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Random Batch Algorithms for Quantum Monte Carlo Simulations

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“…We now turn to the random batch algorithm (2.16) with replacement [23]. The convergence property has recently been proved in [22]: Theorem 2.1.…”

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