Numerical Methods for Dynamical Low-Rank Approximations of Stochastic Differential Equations -- Part II: Stochastic discretization
Abstract
This paper is the follow up (Part II) of arXiv:2601.21428, which proposed and analyzed time discretization schemes for the Dynamical Low-Rank Approximation (DLRA) of high-dimensional systems of Stochastic Differential Equations (SDEs). In this work, we discuss and analyze the corresponding stochastic discretization. Specifically, we consider Monte Carlo type methods to approximate the stochastic space and all related quantities of interest, leading to produce noisy interacting particle systems whose error analysis is non standard. Assuming subgaussian tails of the initial condition, in the case of elliptic diffusion, we provide convergence results for the Monte Carlo approximation of the DLR-PS-SDE integrator presented in Part I. The latter requires inversion of the Gram matrix associated to the stochastic modes. When the corresponding empirical Gramian in the Monte Carlo approximation remains full rank over the evolution, we prove a convergence rate close to the usual Monte Carlo one, i.e. O(1/√M), where M is the sample size. If a regularization of this matrix is employed at each time-step, the rate is close to O(1/√[3]M). We also investigate intermediate options where the regularization is activated only when the smallest eigenvalue of the empirical covariance falls below a prescribed tolerance, showing convergence rates in between the one mentioned above.
Related
- Convergence and error analysis for numerical approximations of stochastic Leray-Lions equations
- Structure-Preserving Dynamical Low-Rank Approximations for Stochastic Vlasov--Poisson Equations
- Multilevel Picard approximations for McKean-Vlasov stochastic differential equations with jumps
- Strong Convergence of Fully Discrete Tamed Euler Approximations for Time-Changed Stochastic Delay Differential Equations
- Numerical methods for nonlinear stochastic differential equations driven by subdiffusion
BibTeX
@misc{kazashi2026numerical,
title = {{Numerical Methods for Dynamical Low-Rank Approximations of Stochastic Differential Equations -- Part II: Stochastic discretization}},
author = {Yoshihito Kazashi and Fabio Nobile and Fabio Zoccolan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.20317},
eprint = {2609.20317},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20317},
}
RIS
TY - UNPB
TI - Numerical Methods for Dynamical Low-Rank Approximations of Stochastic Differential Equations -- Part II: Stochastic discretization
AU - Yoshihito Kazashi
AU - Fabio Nobile
AU - Fabio Zoccolan
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2609.20317
UR - https://arxiv.org/abs/2609.20317
AB - This paper is the follow up (Part II) of arXiv:2601.21428, which proposed and analyzed time discretization schemes for the Dynamical Low-Rank Approximation (DLRA) of high-dimensional systems of Stochastic Differential Equations (SDEs). In this work, we discuss and analyze the corresponding stochastic discretization. Specifically, we consider Monte Carlo type methods to approximate the stochastic space and all related quantities of interest, leading to produce noisy interacting particle systems whose error analysis is non standard. Assuming subgaussian tails of the initial condition, in the case of elliptic diffusion, we provide convergence results for the Monte Carlo approximation of the DLR-PS-SDE integrator presented in Part I. The latter requires inversion of the Gram matrix associated to the stochastic modes. When the corresponding empirical Gramian in the Monte Carlo approximation remains full rank over the evolution, we prove a convergence rate close to the usual Monte Carlo one, i.e. O(1/√M), where M is the sample size. If a regularization of this matrix is employed at each time-step, the rate is close to O(1/√[3]M). We also investigate intermediate options where the regularization is activated only when the smallest eigenvalue of the empirical covariance falls below a prescribed tolerance, showing convergence rates in between the one mentioned above.
ER -