Dynamical low-rank approximation for stochastic differential equations

Kazashi, Yoshihito; Nobile, Fabio; Zoccolan, Fabio

doi:10.1090/mcom/3999

Dynamical low-rank approximation for stochastic differential equations
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by Yoshihito Kazashi, Fabio Nobile and Fabio Zoccolan;

Math. Comp. 94 (2025), 1335-1375

DOI: https://doi.org/10.1090/mcom/3999

Published electronically: August 22, 2024

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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.

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