Dynamical Low-Rank Smoothing

2026/07/29 by Youssef Marzouk, Fabio Nobile, Fabio Zoccolan
Mathematics · Computer Science · #math.NA #cs.NA #msc:62M20 #msc:60G35 #msc:60H35 #msc:65C35 #msc:65F55

paper · pdf

11 pages

arxiv created 2026/07/29 · arxiv updated 2026/07/31

Abstract

Computational costs often make smoothing procedures prohibitive for high-dimensional data assimilation problems. To address this challenge, we propose a dynamical low-rank approximation (DLRA) methodology for smoothing concerning frameworks based on stochastic differential equations. We extend the previously developed joint mean-and-covariance optimization (JMCO) filtering setting to derive a reduced-order smoother via the Rauch--Tung--Striebel recursion and establish the corresponding Kalman--Bucy smoothing for affine drift dynamics. The resulting algorithms retain the adaptive nature of DLRA while significantly reducing the computational time and storage of the whole smoothing procedure.

Citations

• Dynamical Low-Rank Filters for Data Assimilation
• Numerical Methods for Dynamical Low-Rank Approximations of Stochastic Differential Equations -- Part I: Time discretization
• Dynamical Low-Rank Approximations for Kalman Filtering
• Sequential Filtering Techniques for Simultaneous Tracking and Parameter Estimation
• Interpolatory dynamical low-rank approximation for the 3+3d Boltzmann-BGK equation
• A Bayesian Approach to Atmospheric Circulation Regime Assignment
• Dynamically orthogonal field equations for continuous stochastic dynamical systems
• Maximum likelihood estimates of linear dynamic systems

Related

• Dynamical Low-Rank Filters for Data Assimilation
• The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensions
• Generalized Kalman Smoothing: Modeling and Algorithms
• Dynamical Low-Rank Approximations for Kalman Filtering
• Data assimilation and parameter estimation for a multiscale stochastic system with alpha-stable Levy noise