{
  "abstract": "Stochastic dynamic mode decomposition (SDMD) approximates the Koopman semigroup of an Itô diffusion by a Galerkin generator matrix on a dictionary of observables, computed from samples of the invariant law and evaluations of the generator, and advances observables with a first-order stochastic Taylor step (SDMD-1). The second-order Taylor step (SDMD-2) also needs the matrix of second generator actions, hence a dictionary with four derivatives, evaluations of a fourth-order differential operator at every sample, and a third matrix. We show that the Richardson extrapolation of two SDMD-1 half steps (SDMD-R) is second-order accurate with the data, the matrices, and the regularity of SDMD-1. One error bound covers exact and empirical Galerkin matrices. It consists of the Galerkin error of the dictionary, which is common to all schemes, a time-stepping term, and a data term of order $m^{-1/2}$ in the sample size $m$. The time-stepping term is $O(\\dt^2)$ for SDMD-R and $O(Θ_N\\dt+\\dt^2)$ for SDMD-2, where $Θ_N$ is the norm of the second-order Galerkin residual $\\Proj_N\\cA(I-\\Proj_N)\\cA|_{\\cV_N}$. The first-order term is attained and, for self-adjoint generators, vanishes only for generator-invariant dictionaries. For reversible periodic diffusions and trigonometric dictionaries with cutoff $K$, the step restriction $\\dt\\lesssim K^{-2}$ makes all three schemes contractions, and the error against the Koopman semigroup on $H^\\ell$ observables is $O(K^{-(\\ell-2)}+\\dt^p)$ plus a sampling term of order $K^{d+2}m^{-1/2}$ in dimension $d$, with $p=1$ for SDMD-1 and $p=2$ for SDMD-R and SDMD-2. Numerical experiments confirm that SDMD-R matches or improves on the accuracy of SDMD-2 at the cost of SDMD-1.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00793",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuanchao Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/egk37v73dfed9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00793",
  "fulltext_url": null,
  "id": "egk37v73dfed9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00793",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00793",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/egk37v73dfed9.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00793",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00793"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Second-Order Stochastic Dynamic Mode Decomposition by Richardson Extrapolation",
  "title_text": "A Second-Order Stochastic Dynamic Mode Decomposition by Richardson Extrapolation",
  "type": "preprint",
  "updated_at": "2026-11-04T15:21:05Z",
  "url": "https://research.draconitic.com/paper/egk37v73dfed9",
  "venue": "arXiv",
  "versions": []
}