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. $O(\frac{1}{\sqrt{M}})$, where $M$ is the sample size. If a regularization of this matrix is employed at each time-step, the rate is close to $O(\frac{1}{\sqrt[3]{M}})$. 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.

arXiv comment65 pages, 6 figures