Translator Disclaimer
April 2018 On the stability and the uniform propagation of chaos properties of Ensemble Kalman–Bucy filters
P. Del Moral, J. Tugaut
Ann. Appl. Probab. 28(2): 790-850 (April 2018). DOI: 10.1214/17-AAP1317

Abstract

The ensemble Kalman filter is a sophisticated and powerful data assimilation method for filtering high dimensional problems arising in fluid mechanics and geophysical sciences. This Monte Carlo method can be interpreted as a mean-field McKean–Vlasov-type particle interpretation of the Kalman–Bucy diffusions. In contrast to more conventional particle filters and nonlinear Markov processes, these models are designed in terms of a diffusion process with a diffusion matrix that depends on particle covariance matrices.

Besides some recent advances on the stability of nonlinear Langevin-type diffusions with drift interactions, the long-time behaviour of models with interacting diffusion matrices and conditional distribution interaction functions has never been discussed in the literature. One of the main contributions of the article is to initiate the study of this new class of models. The article presents a series of new functional inequalities to quantify the stability of these nonlinear diffusion processes.

In the same vein, despite some recent contributions on the convergence of the ensemble Kalman filter when the number of sample tends to infinity very little is known on stability and the long-time behaviour of these mean-field interacting type particle filters. The second contribution of this article is to provide uniform propagation of chaos properties as well as $\mathbb{L}_{n}$-mean error estimates w.r.t. to the time horizon. Our regularity condition is also shown to be sufficient and necessary for the uniform convergence of the ensemble Kalman filter.

The stochastic analysis developed in this article is based on an original combination of functional inequalities and Foster–Lyapunov techniques with coupling, martingale techniques, random matrices and spectral analysis theory.

Citation

Download Citation

P. Del Moral. J. Tugaut. "On the stability and the uniform propagation of chaos properties of Ensemble Kalman–Bucy filters." Ann. Appl. Probab. 28 (2) 790 - 850, April 2018. https://doi.org/10.1214/17-AAP1317

Information

Received: 1 May 2016; Revised: 1 January 2017; Published: April 2018
First available in Project Euclid: 11 April 2018

zbMATH: 1356.60065
MathSciNet: MR3784489
Digital Object Identifier: 10.1214/17-AAP1317

Subjects:
Primary: 35Q84, 60G25, 60J22, 60J60, 60M20, 93E11

Rights: Copyright © 2018 Institute of Mathematical Statistics

JOURNAL ARTICLE
61 PAGES


SHARE
Vol.28 • No. 2 • April 2018
Back to Top