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.
- J. L. Anderson, An ensemble adjustment Kalman filter for data assimilation, Monthly Weather Review, 129, (2001), no. 12, 2884–2903.
- K. Bergemann and S. Reich, A mollified ensemble Kalman filter, Quarterly Journal of the Royal Meteorological Society, 136 (2010), no. 651, 1636–1643.
- K. Bergemann and S. Reich, An ensemble Kalman-Bucy filter for continuous data assimilation, Meteorologische Zeitschrift, 21 (2012), no. 3, 213–219.
- C. H. Bishop, B. J. Etherton, and S. J. Majumdar, Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects, Monthly Weather Review, 129 (2001), no. 3, 420–436.
- A. N. Bishop, P. Del Moral, K. Kamatani, and B. Rémillard, On one-dimensional Riccati diffusions, Ann. Appl. Probab. 29 (2019), no. 2, 1127–1187. MR 3910025, DOI 10.1214/18-AAP1431
- Dirk Blömker, Claudia Schillings, and Philipp Wacker, A strongly convergent numerical scheme from ensemble Kalman inversion, SIAM J. Numer. Anal. 56 (2018), no. 4, 2537–2562. MR 3840898, DOI 10.1137/17M1132367
- G. Burgers, P. J. van Leeuwen, and G. Evensen, Analysis Scheme in the Ensemble Kalman Filter, Monthly Weather Review, 126 (1998), 1719–1724.
- Jana de Wiljes, Sebastian Reich, and Wilhelm Stannat, Long-time stability and accuracy of the ensemble Kalman-Bucy filter for fully observed processes and small measurement noise, SIAM J. Appl. Dyn. Syst. 17 (2018), no. 2, 1152–1181. MR 3787772, DOI 10.1137/17M1119056
- István Gyöngy and Miklós Rásonyi, A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients, Stochastic Process. Appl. 121 (2011), no. 10, 2189–2200. MR 2822773, DOI 10.1016/j.spa.2011.06.008
- Nikolaos Halidias and Peter E. Kloeden, A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient, BIT 48 (2008), no. 1, 51–59. MR 2386114, DOI 10.1007/s10543-008-0164-1
- Desmond J. Higham, Xuerong Mao, and Andrew M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40 (2002), no. 3, 1041–1063. MR 1949404, DOI 10.1137/S0036142901389530
- Martin Hutzenthaler, Arnulf Jentzen, and Peter E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab. 22 (2012), no. 4, 1611–1641. MR 2985171, DOI 10.1214/11-AAP803
- Martin Hutzenthaler and Arnulf Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Amer. Math. Soc. 236 (2015), no. 1112, v+99. MR 3364862, DOI 10.1090/memo/1112
- Marco A. Iglesias, Kody J. H. Law, and Andrew M. Stuart, Ensemble Kalman methods for inverse problems, Inverse Problems 29 (2013), no. 4, 045001, 20. MR 3041539, DOI 10.1088/0266-5611/29/4/045001
- D. T. B. Kelly, K. J. H. Law, and A. M. Stuart, Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time, Nonlinearity 27 (2014), no. 10, 2579–2604. MR 3265724, DOI 10.1088/0951-7715/27/10/2579
- Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992. MR 1214374, DOI 10.1007/978-3-662-12616-5
- Kody Law, Andrew Stuart, and Konstantinos Zygalakis, Data assimilation, Texts in Applied Mathematics, vol. 62, Springer, Cham, 2015. A mathematical introduction. MR 3363508, DOI 10.1007/978-3-319-20325-6
- Bernard Roynette, Mouvement brownien et espaces de Besov, Stochastics Stochastics Rep. 43 (1993), no. 3-4, 221–260 (French, with English summary). MR 1277166, DOI 10.1080/17442509308833837
- Claudia Schillings and Andrew M. Stuart, Analysis of the ensemble Kalman filter for inverse problems, SIAM J. Numer. Anal. 55 (2017), no. 3, 1264–1290. MR 3654885, DOI 10.1137/16M105959X
- J. S. Whitaker and T. M. Hamill, Ensemble Data Assimilation without Perturbed Observations, Monthly Weather Review, 130 (2002), no. 7, 1913–1924.
- Theresa Lange
- Affiliation: Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany
- Email: email@example.com
- 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
- MR Author ID: 357144
- Email: firstname.lastname@example.org
- 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.
- © Copyright 2020 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 233-265
- MSC (2010): Primary 60H35, 93E11, 60F99
- DOI: https://doi.org/10.1090/mcom/3588
- MathSciNet review: 4166460