Deterministic Mean-field Ensemble Kalman Filtering

TL;DR

This paper extends the convergence proof of the ensemble Kalman filter to non-Gaussian models and introduces a deterministic approximation method that outperforms standard EnKF under certain conditions.

Contribution

It proposes a density-based deterministic mean-field EnKF approximation using PDEs and quadrature, extending convergence results to non-Gaussian models.

Findings

Deterministic filter approximation is asymptotically superior to standard EnKF when dimension < 2κ.

Fidelity of the approximation is established using an extended total variation metric.

Numerical results support the theoretical convergence and performance improvements.

Abstract

The proof of convergence of the standard ensemble Kalman filter (EnKF) from Legland etal. (2011) is extended to non-Gaussian state space models. A density-based deterministic approximation of the mean-field limit EnKF (DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given a certain minimal order of convergence $\kappa$ between the two, this extends to the deterministic filter approximation, which is therefore asymptotically superior to standard EnKF when the dimension $d<2\kappa$. The fidelity of approximation of the true distribution is also established using an extension of total variation metric to random measures. This is limited by a Gaussian bias term arising from non-linearity/non-Gaussianity of the model, which exists for both DMFEnKF and standard EnKF. Numerical results support and extend the theory.