A Second-Order Stochastic Dynamic Mode Decomposition by Richardson Extrapolation
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(dt2) for SDMD-R and O(ΘNdt+dt2) for SDMD-2, where ΘN is the norm of the second-order Galerkin residual ProjNcA(I−ProjN)cA|cVN. 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≲ K−2 makes all three schemes contractions, and the error against the Koopman semigroup on Hℓ observables is O(K−(ℓ−2)+dtp) plus a sampling term of order Kd+2m−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.
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BibTeX
@misc{xu2026secondorder,
title = {{A Second-Order Stochastic Dynamic Mode Decomposition by Richardson Extrapolation}},
author = {Yuanchao Xu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00793},
eprint = {2611.00793},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00793},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Second-Order Stochastic Dynamic Mode Decomposition by Richardson Extrapolation
AU - Yuanchao Xu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00793
UR - https://arxiv.org/abs/2611.00793
AB - 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_Ndt+dt^2) for SDMD-2, where Θ_N is the norm of the second-order Galerkin residual Proj_NcA(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≲ K^−2 makes all three schemes contractions, and the error against the Koopman semigroup on H^ℓ observables is O(K^−(ℓ−2)+dt^p) plus a sampling term of order K^d+2m^−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.
ER -