Dynamical low-rank approximation for stochastic differential equations
Kazashi, Yoshihito and Nobile, Fabio and Zoccolan, Fabio (2024) Dynamical low-rank approximation for stochastic differential equations. Mathematics of Computation. ISSN 0025-5718 (https://doi.org/10.1090/mcom/3999)
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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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Item type: Article ID code: 90360 Dates: DateEvent22 August 2024Published22 August 2024Published Online17 June 2024Accepted22 August 2023SubmittedSubjects: Science > Mathematics
Science > Mathematics > Electronic computers. Computer scienceDepartment: Faculty of Science > Mathematics and Statistics Depositing user: Pure Administrator Date deposited: 26 Aug 2024 15:10 Last modified: 01 Sep 2024 01:49 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/90360