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

TL;DR

This paper analyzes three time-discretization methods for dynamical low-rank approximations of high-dimensional stochastic differential equations, establishing convergence conditions and stability properties supported by computational experiments.

Contribution

It introduces and compares three time-discretization algorithms for DLRA of SDEs, providing convergence analysis and stability results, especially for staggered schemes.

Findings

Forward discretization requires a time-step restriction based on the smallest singular value.

Staggered schemes are more stable and do not require the same time-step restriction.

Computational experiments confirm theoretical convergence and stability results.

Abstract

In this work (Part I), we study three time-discretization procedures of the Dynamical Low-Rank Approximation (DLRA) of high-dimensional stochastic differential equations (SDEs). Specifically, we consider the Dynamically Orthogonal (DO) method for DLRA proposed and analyzed in arXiv:2308.11581v4, which consists of a linear combination of products between deterministic orthonormal modes and stochastic modes, both time-dependent. The first strategy we consider for numerical time-integration is very standard, consisting in a forward discretization in time of both deterministic and stochastic components. Its convergence is proven subject to a time-step restriction dependent on the smallest singular value of the Gram matrix associated to the stochastic modes. Under the same condition on the time-step, this smallest singular value is shown to be always positive, provided that the SDE under study is driven by a non-degenerate noise. The second and the third algorithms, on the other hand, are staggered ones, in which we alternately update the deterministic and the stochastic modes in half steps. These approaches are shown to be more stable than the first one and allow us to obtain convergence results without the aforementioned restriction on the time-step. Computational experiments support theoretical results. In this work we do not consider the discretization in probability, which will be the topic of Part II.