Dynamical Low-Rank Approximations for Kalman Filtering

Overview

Abstract 

Data Assimilation consists in combining model knowledge with a stream of observations to improve the prediction of a system's state. Two prominent instances of this approach are the Kalman-Bucy filter (KBF) and its particle-based analog, the Ensemble Kalman filter (ENKF). While the former describes the exact filtering density evolution under linear-Gaussian dynamics, the latter is widely used in real-world applications such as climate and geosciences due to its computational tractability. However, using a small ensemble may lead to significant Monte Carlo error and stochastic fluctuations.

We propose a principled model order reduction of the KBF via Dynamical Low-Rank (DLR) Approximation, evolving the filtering density in a dynamically adapting low-dimensional subspace at reduced computational cost. Under certain assumptions, our framework preserves key properties of the KBF, including mean and covariance characterisation, and we establish error bounds between the true and reduced-order filter. We then introduce the DLR-ENKF, a DLR formulation of the Ensemble Kalman filter, and establish its well-posedness and a propagation of chaos property to its rank-reduced mean-field limit. To make this framework computationally viable for nonlinear systems, we develop compatible time-integration schemes and hyperreduction techniques, and apply them to joint state/parameter estimation problems. Numerical experiments show that the DLR reductions effectively capture the intrinsic rank of filtered systems, yielding accurate approximations at reduced cost.

This talk is based on joint work with Fabio Nobile and Sébastien Riffaud.