{
  "id": "2206.02166",
  "version": "v1",
  "published": "2022-06-05T12:45:48.000Z",
  "updated": "2022-06-05T12:45:48.000Z",
  "title": "Error Analysis of Time-Discrete Random Batch Method for Interacting Particle Systems and Associated Mean-Field Limits",
  "authors": [
    "Xuda Ye",
    "Zhennan Zhou"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "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 $O(\\sqrt{\\tau} + e^{-\\beta t})$, where $\\tau$ is the time step and $\\beta$ is the convergence rate which does not depend on the time step $\\tau$ 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 $N\\rightarrow\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-05T12:45:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65C20",
      "37M05"
    ],
    "keywords": [
      "time-discrete random batch method",
      "interacting particle system",
      "associated mean-field limits",
      "error analysis",
      "fully discrete random batch method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}