Numerical Methods for Dynamical Low-Rank Approximations of Stochastic Differential Equations -- Part II: Stochastic discretization

RESUMO

In this second article (Part II), we analyze the numerical algorithms for the Dynamical Low-Rank Approximation (DLRA) of Stochastic Differential Equations (SDEs) introduced in Part I arXiv:2601.21428 under the perspective of the stochastic discretization. Specifically, we employ a Monte Carlo method with $M$ samples to approximate the stochastic space and all the related quantities of interest. Consequently, these algorithms produce noisy interacting particle systems whose error analysis is not standard. Assuming high moments and subgaussian tails of the initial condition, in the case of elliptic diffusion, we provide convergence results for the DLR Projector Splitting for SDEs presented in Part I. When the fully discretized Gramian is of full rank for all the time evolution, then one observes a convergence rate close to the usual Monte Carlo one, i.e. $O(\frac{1}{\sqrt{M}})$, for large $M$. If a regularization of this matrix is employed at each time-step, the rate is close to $O(\frac{1}{\sqrt[3]{M}})$ for large $M$. On the other hand, in case the regularization occurs only when the smallest singular value of the Gramian is smaller than a prescribed positive threshold, the stochastic convergence rate is located between the aforementioned results. Furthermore, in the case of general diffusion, we provide an easy-to-implement convergent algorithm. Concerning the DLR Euler-Maruyama and the DLR Projector Splitting for Euler-Maruyama (EM), these results are not straightforward to derive and a brief discussion on why this is a challenging task is provided, too. Numerical simulations will complete this analysis.