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Published November 15, 2024 | Version Published
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The Mean-Field Ensemble Kalman Filter: Near-Gaussian Setting

Creators

  • 1. ROR icon University of Oxford
  • 2. ROR icon California Institute of Technology
  • 3. ROR icon French Institute for Research in Computer Science and Automation

Abstract

The ensemble Kalman filter is widely used in applications because, for high-dimensional filtering problems, it has a robustness that is not shared, for example, by the particle filter; in particular, it does not suffer from weight collapse. However, there is no theory which quantifies its accuracy as an approximation of the true filtering distribution, except in the Gaussian setting. To address this issue, we provide the first analysis of the accuracy of the ensemble Kalman filter beyond the Gaussian setting. We prove two types of results: The first type comprises a stability estimate controlling the error made by the ensemble Kalman filter in terms of the difference between the true filtering distribution and a nearby Gaussian, and the second type uses this stability result to show that, in a neighborhood of Gaussian problems, the ensemble Kalman filter makes a small error in comparison with the true filtering distribution. Our analysis is developed for the mean-field ensemble Kalman filter. We rewrite the update equations for this filter and for the true filtering distribution in terms of maps on probability measures. We introduce a weighted total variation metric to estimate the distance between the two filters, and we prove various stability estimates for the maps defining the evolution of the two filters in this metric. Using these stability estimates, we prove results of the first and second types in the weighted total variation metric. We also provide a generalization of these results to the Gaussian projected filter, which can be viewed as a mean-field description of the unscented Kalman filter.

Copyright and License

© 2024 Society for Industrial and Applied Mathematics.

Acknowledgement

We are grateful to the reviewers for very useful comments on a previous version of this paper. In particular, we thank the reviewers for pointing out the elementary inequality in Lemma A.3 and for showing us how this inequality could be used to significantly improve the proofs of Lemmas A.4, A.6, and C.1. The constructive proof of Lemma A.4, in particular, was proposed by a reviewer.

Funding

The first author was supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns, and Synchronization) of the European Research Council (ERC) Executive Agency under the European Union's Horizon 2020 research and innovation program (grant agreement 883363). The first author was also partially supported by the Engineering and Physical Sciences Research Council (EPSRC) under grants
EP/T022132/1 and EP/V051121/1. The second author was supported by start-up funds at the California Institute of Technology, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via project 390685813-GZ 2047/1-HCM, and also by NSF CAREER Award 2340762. The work of the third author was supported by a Department of Defense Vannevar Bush Faculty Fellowship and by the SciAI Center, funded by the Office of Naval Research (ONR), under grant N00014-23-1-2729. The fourth author was partially supported by the ERC under the European Union's Horizon 2020 research and innovation program (grant agreement 810367) and by the
Agence Nationale de la Recherche under grants ANR-21-CE40-0006 (SINEQ) and ANR-23-CE40-0027 (IPSO).

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Additional details

Additional titles

Alternative title
The Ensemble Kalman Filter in the Near-Gaussian Setting

Related works

Is new version of
Discussion Paper: arXiv:2212.13239 (arXiv)

Funding

European Research Council
883363
Engineering and Physical Sciences Research Council
EP/T022132/1
Engineering and Physical Sciences Research Council
EP/V051121/1
California Institute of Technology
Deutsche Forschungsgemeinschaft
390685813 - GZ 2047/1 - HCM
National Science Foundation
DMS-2340762
United States Department of Defense
Vannevar Bush Faculty Fellowship
Office of Naval Research
N00014-23-1-272
European Research Council
810367
Agence Nationale de la Recherche
ANR-21-CE40-0006 (SINEQ)
Agence Nationale de la Recherche
ANR-23-CE40-0027 (IPSO)

Dates

Accepted
2024-08-26
Accepted

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Publication Status
Published