Dynamical Low-Rank Approximation for Stochastic Differential Equations
Yoshihito Kazashi, Fabio Nobile, Fabio Zoccolan
TL;DR
This work provides a rigorous dynamical low‑rank framework for stochastic differential equations by formulating a DO representation $X=oldsymbol{U}^{ op}oldsymbol{Y}$ with rank $R$, and derives a parameter‑free DLRA equation on the low‑rank manifold. It establishes local existence and uniqueness for both the DO and DLRA formulations, proves their equivalence (up to a rotation), and characterizes the explosion time via a loss of linear independence of the stochastic basis. The paper also demonstrates how the DLRA can extend beyond the DO explosion time and presents a practical global existence condition under uniformly nondegenerate diffusion. These results lay a solid theoretical foundation for DLRA in SDEs and inform stable numerical schemes and extension strategies for high‑dimensional stochastic systems.
Abstract
In this paper, we set the mathematical foundations of the Dynamical Low-Rank Approximation (DLRA) method for stochastic differential equations (SDEs). DLRA aims at approximating the solution as a linear combination of a small number of basis vectors with random coefficients (low rank format) with the peculiarity that both the basis vectors and the random coefficients vary in time. While the formulation and properties of DLRA are now well understood for random/parametric equations, the same cannot be said for SDEs and this work aims to fill this gap. We start by rigorously formulating a Dynamically Orthogonal (DO) approximation (an instance of DLRA successfully used in applications) for SDEs, which we then generalize to define a parametrization independent DLRA for SDEs. We show local well-posedness of the DO equations and their equivalence with the DLRA formulation. We also characterize the explosion time of the DO solution by a loss of linear independence of the random coefficients defining the solution expansion and give sufficient conditions for global existence.