Abstract
We consider the problem of discrete time filtering (intermittent data assimilation) for differential equation models and discuss methods for its numerical approximation. The focus is on methods based on ensemble/particle techniques and on the ensemble Kalman filter technique in particular. We summarize as well as extend recent work on continuous ensemble Kalman filter formulations, which provide a concise dynamical systems formulation of the combined dynamics-assimilation problem. Possible extensions to fully nonlinear ensemble/particle based filters are also outlined using the framework of optimal transportation theory.
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Anderson, J., Anderson, S.: A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Weather Rev. 127, 2741–2758 (1999)
Ascher, U., Huang, H., van den Doel, K.: Artificial time integration. BIT 47, 3–25 (2007)
Ascher, U., van den Doel, K., Huang, H., Svaiter, B.: Gradient descent and fast artificial time integration. Modél. Math. Anal. Numér. 43, 689–708 (2009)
Bain, A., Crisan, D.: Fundamentals of stochastic filtering. Stochastic modelling and applied probability, vol. 60. Springer, New York (2009)
Bergemann, K., Reich, S.: A localization technique for ensemble Kalman filters. Q. J. R. Meteorol. Soc. 136, 701–707 (2010)
Bergemann, K., Reich, S.: A mollified ensemble Kalman filter. Q. J. R. Meteorol. Soc. 136, 1636–1643 (2010)
Bloom, S., Takacs, L., Silva, A.D., Ledvina, D.: Data assimilation using incremental analysis updates. Q. J. R. Meteorol. Soc. 124, 1256–1271 (1996)
Burgers, G., van Leeuwen, P., Evensen, G.: On the analysis scheme in the ensemble Kalman filter. Mon. Weather Rev. 126, 1719–1724 (1998)
Crisan, D., Xiong, J.: Approximate McKean-Vlasov representation for a class of SPDEs. Stochastics 82, 53–68 (2010)
Dempster, A., Laird, N., Rubin, D.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. B 39B, 1–38 (1977)
Evensen, G.: Data Assimilation. The Ensemble Kalman Filter. Springer, New York (2006)
Friedrichs, K.: The identity of weak and strong extensions of differential operators. Trans. Am. Math. Soc. 55, 132–151 (1944)
Gardiner, C.: Handbook on Stochastic Methods, 3rd edn. Springer, New York (2004)
Gaspari, G., Cohn, S.: Construction of correlation functions in two and three dimensions. Q. J. R. Meteorol. Soc. 125, 723–757 (1999)
Hamill, T., Whitaker, J., Snyder, C.: Distance-dependent filtering of background covariance estimates in an ensemble Kalman filter. Mon. Weather Rev. 129, 2776–2790 (2001)
Houtekamer, P., Mitchell, H.: A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Weather Rev. 129, 123–136 (2001)
Houtekamer, P., Mitchell, H.: Ensemble Kalman filtering. Q. J. R. Meteorol. Soc. 131, 3269–3289 (2005)
Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems. Springer, New York (2005)
Kepert, J.: Covariance localisation and balance in an ensemble Kalman Filter. Q. J. R. Meteorol. Soc. 135, 1157–1176 (2009)
Leeuwen, P.V.: Particle filtering in geophysical systems. Mon. Weather Rev. 137, 4089–4114 (2009)
Lei, L., Stauffer, D.: A hybrid ensemble Kalman filter approach to data assimilation in a two-dimensional shallow-water model. In: 23rd Conference on Weather Analysis and Forecasting/19th Conference on Numerical Weather Prediction, AMS Conference Proceedings, p. 9A.4. American Meteorological Society, Omaha (2009)
Lewis, J., Lakshmivarahan, S., Dhall, S.: Dynamic Data Assimilation: A Least Squares Approach. Cambridge University Press, Cambridge (2006)
Øksendal, B.: Stochastic Differential Equations, 5th edn. Springer, Berlin (2000)
Simon, D.: Optimal State Estimation. Wiley, New York (2006)
Smith, K.: Cluster ensemble Kalman filter. Tellus 59A, 749–757 (2007)
Tippett, M., Anderson, J., Bishop, G., Hamill, T., Whitaker, J.: Ensemble square root filters. Mon. Weather Rev. 131, 1485–1490 (2003)
Villani, C.: Topics in Optimal Transportation. Am. Math. Soc., Providence (2003)
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Communicated by Uri Ascher.
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Reich, S. A dynamical systems framework for intermittent data assimilation. Bit Numer Math 51, 235–249 (2011). https://doi.org/10.1007/s10543-010-0302-4
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DOI: https://doi.org/10.1007/s10543-010-0302-4