The Annals of Applied Probability

On the stability and the uniform propagation of chaos properties of Ensemble Kalman–Bucy filters

P. Del Moral and J. Tugaut

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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.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 2 (2018), 790-850.

Dates
Received: May 2016
Revised: January 2017
First available in Project Euclid: 11 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1523433625

Digital Object Identifier
doi:10.1214/17-AAP1317

Mathematical Reviews number (MathSciNet)
MR3784489

Zentralblatt MATH identifier
1356.60065

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 60J22: Computational methods in Markov chains [See also 65C40] 35Q84: Fokker-Planck equations 93E11: Filtering [See also 60G35] 60M20 60G25: Prediction theory [See also 62M20]

Keywords
Ensemble Kalman filter Kalman–Bucy filter Riccati equations ill-conditioned systems mean-field particle models sequential Monte Carlo methods interacting particle systems random covariance matrices nonlinear Markov processes

Citation

Del Moral, P.; Tugaut, J. On the stability and the uniform propagation of chaos properties of Ensemble Kalman–Bucy filters. Ann. Appl. Probab. 28 (2018), no. 2, 790--850. doi:10.1214/17-AAP1317. https://projecteuclid.org/euclid.aoap/1523433625


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