Error Analysis of Time-Discrete Random Batch Method for Interacting Particle Systems and Associated Mean-Field Limits
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The random batch method provides an efficient algorithm for computing statistical properties of a canonical ensemble of interacting particles. In this work, we study the error estimates of the fully discrete random batch method, especially in terms of approximating the invariant distribution. Using a triangle inequality framework, we show that the long-time error of the method is O(√(τ) + e^-β t), where τ is the time step and β is the convergence rate which does not depend on the time step τ or the number of particles N. Our results also apply to the McKean-Vlasov process, which is the mean-field limit of the interacting particle system as the number of particles N→∞.
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