@article{7b835857089b4f798baac614569c46cc,
title = "Dynamical low-rank approximation for stochastic differential equations",
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.",
keywords = "dynamically orthogonal approximation, dynamical low-rank approximation, stochastic differential equations, non-linear evolution equation",
author = "Yoshihito Kazashi and Fabio Nobile and Fabio Zoccolan",
year = "2024",
month = aug,
day = "22",
doi = "10.1090/mcom/3999",
language = "English",
journal = "Mathematics of Computation",
issn = "0025-5718",
}