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On the continuous time limit of the ensemble Kalman filter


Authors: Theresa Lange and Wilhelm Stannat
Journal: Math. Comp. 90 (2021), 233-265
MSC (2010): Primary 60H35, 93E11, 60F99
DOI: https://doi.org/10.1090/mcom/3588
Published electronically: October 6, 2020
MathSciNet review: 4166460
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Abstract: We present recent results on the existence of a continuous time limit for Ensemble Kalman Filter algorithms. In the setting of continuous signal and observation processes, we apply the original Ensemble Kalman Filter algorithm proposed by Burgers, van Leeuwen, and Evensen [Monthly Weather Review 126 (1998), pp. 1719-1724] as well as a recent variant of de Wiljes, Reich, and Stannat [SIAM J. Appl. Dyn. Syst. 17 (2018), no. 2, pp. 1152-1181] to the respective discretizations and show that in the limit of decreasing stepsize the filter equations converge to an ensemble of interacting (stochastic) differential equations in the ensemble-mean-square sense. Our analysis also allows for the derivation of convergence rates with respect to the stepsize.

An application of our analysis is the rigorous derivation of continuous-time ensemble filtering algorithms consistent with discrete-time approximation schemes. Conversely, the continuous time limit allows for a better qualitative and quantitative analysis of the discrete-time counterparts using the rich theory of dynamical systems in continuous time.


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Additional Information

Theresa Lange
Affiliation: Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany
Email: tlange@math.tu-berlin.de

Wilhelm Stannat
Affiliation: Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany; and Bernstein Center for Computational Neuroscience, Philippstr. 13, D-10115 Berlin, Germany
Email: stannat@math.tu-berlin.de

DOI: https://doi.org/10.1090/mcom/3588
Keywords: Ensemble Kalman filter, continuous time limit
Received by editor(s): December 14, 2018
Received by editor(s) in revised form: January 13, 2020
Published electronically: October 6, 2020
Additional Notes: The research of both authors was partially funded by Deutsche Forschungsgemeinschaft (DFG) - SFB1294/1 - 318763901.
Article copyright: © Copyright 2020 American Mathematical Society