Penalized Ensemble Kalman Filters for High Dimensional Non-linear Systems

TL;DR

This paper introduces PEnKF, a penalized ensemble Kalman filter that improves high-dimensional, non-linear system state estimation with fewer ensemble members, learning covariance structures and ensuring convergence.

Contribution

It proposes a novel penalized EnKF method that learns covariance structures and guarantees convergence with fewer ensemble members than state dimensions.

Findings

PEnKF accurately estimates states in high-dimensional non-linear systems.

Theoretical proof of convergence under certain conditions.

Supported by simulations demonstrating effectiveness.

Abstract

The ensemble Kalman filter (EnKF) is a data assimilation technique that uses an ensemble of models, updated with data, to track the time evolution of a usually non-linear system. It does so by using an empirical approximation to the well-known Kalman filter. However, its performance can suffer when the ensemble size is smaller than the state space, as is often necessary for computationally burdensome models. This scenario means that the empirical estimate of the state covariance is not full rank and possibly quite noisy. To solve this problem in this high dimensional regime, we propose a computationally fast and easy to implement algorithm called the penalized ensemble Kalman filter (PEnKF). Under certain conditions, it can be theoretically proven that the PEnKF will be accurate (the estimation error will converge to zero) despite having fewer ensemble members than state dimensions. Further, as contrasted to localization methods, the proposed approach learns the covariance structure associated with the dynamical system. These theoretical results are supported with simulations of several non-linear and high dimensional systems.