Dynamical Low-Rank Approximation for Stochastic Differential Equations

TL;DR

This paper establishes the mathematical foundations of Dynamical Low-Rank Approximation (DLRA) for stochastic differential equations, providing rigorous formulations, well-posedness results, and conditions for global existence of solutions.

Contribution

It introduces a rigorous formulation of DLRA for SDEs, including a parametrization-independent approach and analysis of solution existence and explosion times.

Findings

Formulated a Dynamically Orthogonal approximation for SDEs

Proved local well-posedness of the DO equations

Characterized explosion time and conditions for global existence

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.