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Ensemble Kalman methods: A mean-field perspective

Published online by Cambridge University Press:  01 July 2025

Edoardo Calvello ,
Sebastian Reich  and
Andrew M. Stuart Open the ORCID record for Andrew M. Stuart [Opens in a new window]
Edoardo Calvello
Affiliation:
California Institute of Technology, Pasadena, CA 91125, USA E-mail: e.calvello@caltech.edu
Sebastian Reich
Affiliation:
Institut für Mathematik, Universität Potsdam, D-14476 Potsdam, Germany E-mail: sebastian.reich@uni-potsdam.de
Andrew M. Stuart
Affiliation:
California Institute of Technology, Pasadena, CA 91125, USA E-mail: astuart@caltech.edu
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Abstract

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Ensemble Kalman methods, introduced in 1994 in the context of ocean state estimation, are now widely used for state estimation and parameter estimation (inverse problems) in many arenae. Their success stems from the fact that they take an underlying computational model as a black box to provide a systematic, derivative-free methodology for incorporating observations; furthermore the ensemble approach allows for sensitivities and uncertainties to be calculated. Analysis of the accuracy of ensemble Kalman methods, especially in terms of uncertainty quantification, is lagging behind empirical success; this paper provides a unifying mean-field-based framework for their analysis. Both state estimation and parameter estimation problems are considered, and formulations in both discrete and continuous time are employed. For state estimation problems, both the control and filtering approaches are considered; analogously for parameter estimation problems, the optimization and Bayesian perspectives are both studied. As well as providing an elegant framework, the mean-field perspective also allows for the derivation of a variety of methods used in practice. In addition it unifies a wide-ranging literature in the field and suggests open problems.

Information

Type
Research Article
Information
Acta Numerica , Volume 34 , July 2025 , pp. 123 - 291
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2025. Published by Cambridge University Press

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