×

Deterministic mean-field ensemble Kalman filtering. (English) Zbl 1351.60047

Summary: The proof of convergence of the standard ensemble Kalman filter (EnKF) from F. Le Gland et al. [in: The Oxford handbook of nonlinear filtering. Oxford: Oxford University Press. 598–631 (2011; Zbl 1225.93108)] is extended to non-Gaussian state-space models. A density-based deterministic approximation of the mean-field limit EnKF (DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given a certain minimal order of convergence \(\kappa\) between the two, this extends to the deterministic filter approximation, which is therefore asymptotically superior to standard EnKF for dimensions \(d<2\kappa\). The fidelity of approximation of the true distribution is also established using an extension of the total variation metric to random measures. This is limited by a Gaussian bias term arising from nonlinearity/non-Gaussianity of the model, which arises in both deterministic and standard EnKF. Numerical results support and extend the theory.

MSC:

60G35 Signal detection and filtering (aspects of stochastic processes)
60F05 Central limit and other weak theorems
60J05 Discrete-time Markov processes on general state spaces
62M20 Inference from stochastic processes and prediction
93E11 Filtering in stochastic control theory
60G15 Gaussian processes
60G57 Random measures
65C35 Stochastic particle methods
65C20 Probabilistic models, generic numerical methods in probability and statistics
65C50 Other computational problems in probability (MSC2010)

Citations:

Zbl 1225.93108
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] M. Ades and P. J. van Leeuwen, {\it An exploration of the equivalent weights particle filter}, Q. J. R. Meteorol. Soc., 139 (2013), pp. 820-840.
[2] A. Apte, C. K. R. T. Jones, A. M. Stuart, and J. Voss, {\it Data assimilation: Mathematical and statistical perspectives}, Internat. J. Numer. Methods Fluids, 56 (2008), pp. 1033-1046. · Zbl 1384.62300
[3] F. Augustin, A. Gilg, M. Paffrath, P. Rentrop, and U. Wever, {\it Polynomial chaos for the approximation of uncertainties: Chances and limits}, European J. Appl. Math., 19 (2008), pp. 149-190. · Zbl 1148.65004
[4] A. Bain and D. Crisan, {\it Fundamentals of Stochastic Filtering}, Springer, New York, 2009. · Zbl 1176.62091
[5] F. Bao, Y. Cao, C. Webster, and G. Zhang, {\it A hybrid sparse-grid approach for nonlinear filtering problems based on adaptive-domain of the Zakai equation approximations}, SIAM/ASA J. Uncertain. Quantif., 2 (2014), pp. 784-804. · Zbl 1343.60041
[6] P. Bickel, B. Li, and T. Bengtsson, {\it Sharp failure rates for the bootstrap particle filter in high dimensions}, in Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, Inst. Math. Stat. Collect. 3, Institute of Mathematical Statistics, Beachwood, OH, 2008, pp. 318-329.
[7] D. Blömker, K. Law, A. M. Stuart, and K. C. Zygalakis, {\it Accuracy and stability of the continuous-time \textup3DVAR filter for the Navier-Stokes equation}, Nonlinearity, 26 (2013), pp. 2193-2219. · Zbl 1271.34013
[8] A. Bobrowski, {\it Functional Analysis for Probability and Stochastic Processes: An Introduction}, Cambridge University Press, Cambridge, UK, 2005. · Zbl 1092.46001
[9] N. Bou-Rabee and E. Vanden-Eijnden, {\it Pathwise accuracy and ergodicity of metropolized integrators for SDEs}, Comm. Pure Appl. Math., 63 (2010), pp. 655-696. · Zbl 1214.60031
[10] M. Branicki and A. J. Majda, {\it Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities}, Commun. Math. Sci., 11 (2013), pp. 55-103. · Zbl 1281.60054
[11] C. E. A. Brett, K. F. Lam, K. J. H. Law, D. S. McCormick, M. R. Scott, and A. M. Stuart, {\it Accuracy and stability of filters for dissipative PDEs}, Phys. D, 245 (2013), pp. 34-45. · Zbl 1350.65102
[12] H.-J. Bungartz and M. Griebel, {\it Sparse grids}, Acta Numer., 13 (2004), pp. 147-269. · Zbl 1118.65388
[13] G. Burgers, P. J. van Leeuwen, and G. Evensen, {\it Analysis scheme in the ensemble Kalman filter}, Mon. Wea. Rev., 126 (1998), pp. 1719-1724.
[14] O. Cappé, E. Moulines, and T. Rydén, {\it Inference in Hidden Markov Models}, Springer, New York, 2005. · Zbl 1080.62065
[15] A. Chorin, M. Morzfeld, and X. Tu, {\it Implicit particle filters for data assimilation}, Commun. Appl. Math. Comput. Sci., 5 (2010), pp. 221-240. · Zbl 1229.60047
[16] A. J. Chorin and P. Krause, {\it Dimensional reduction for a Bayesian filter}, Proc. Natl. Acad. Sci. USA, 101 (2004), pp. 15013-15017. · Zbl 1135.93377
[17] P. Del Moral and A. Guionnet, {\it On the stability of interacting processes with applications to filtering and genetic algorithms}, Ann. Inst. H. Poincaré Probab. Statist., 37 (2001), pp. 155-194. · Zbl 0990.60005
[18] R. L. Dobrushin, {\it Central limit theorem for nonstationary Markov chains. I}, Theory Probab. Appl., 1 (1956), pp. 65-80.
[19] R. L. Dobrushin, {\it Central limit theorem for nonstationary Markov chains. II}, Theory Probab. Appl., 1 (1956), pp. 329-383.
[20] A. Doucet, N. de Frietas, and N. Gordon, {\it Sequential Monte Carlo Methods in Practice}, Springer-Verlag, New York, 2001. · Zbl 0967.00022
[21] A. Doucet, S. Godsill, and C. Andrieu, {\it On sequential Monte Carlo sampling methods for Bayesian filtering}, Stat. Comput., 10 (2000), pp. 197-208.
[22] T. A. El Moselhy and Y. M. Marzouk, {\it Bayesian inference with optimal maps}, J. Comput. Phys., 231 (2012), pp. 7815-7850. · Zbl 1318.62087
[23] T. A. El Moselhy and Y. M. Marzouk, {\it private communication}, MIT, Cambridge, MA, 2014.
[24] O. G. Ernst, B. Sprungk, and H.-J. Starkloff, {\it Bayesian inverse problems and Kalman filters}, in Extraction of Quantifiable Information from Complex Systems, Springer, Cham, Switzerland, 2014, pp. 133-159. · Zbl 1328.93260
[25] L. C. Evans, {\it Partial Differential Equations}, AMS, Providence, RI, 1998. · Zbl 0902.35002
[26] L. C. Evans, {\it An Introduction to Stochastic Differential Equations}, American Mathematical Society, Providence, RI, 2013. · Zbl 1416.60002
[27] G. Evensen, {\it Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics}, J. Geophys. Res. Oceans, 99 (1994), pp. 10143-10162.
[28] G. L. Eyink, J. M. Restrepo, and F. J. Alexander, {\it A mean field approximation in data assimilation for nonlinear dynamics}, Phys. D, 195 (2004), pp. 347-368. · Zbl 1081.82019
[29] K. Hayden, E. Olson, and E. S. Titi, {\it Discrete data assimilation in the Lorenz and 2D Navier-Stokes equations}, Phys. D, 240 (2011), pp. 1416-1425. · Zbl 1302.76048
[30] I. Hoteit, X. Luo, and D.-T. Pham, {\it Particle Kalman filtering: A nonlinear Bayesian framework for ensemble Kalman filters}, Mon. Wea. Rev., 140 (2012), pp. 528-542.
[31] A. H. Jazwinski, {\it Stochastic Processes and Filtering Theory}, Math. Sci. Eng. 64, Academic Press, New York, 1970. · Zbl 0203.50101
[32] J. P. Kaipio and E. Somersalo, {\it Statistical and Computational Inverse Problems}, Springer-Verlag, New York, 2005. · Zbl 1068.65022
[33] R. E. Kalman, {\it A new approach to linear filtering and prediction problems}, Trans. ASME J. Basic Eng., 82 (1960), pp. 35-45.
[34] E. Kalnay, {\it Atmospheric Modeling, Data Assimilation and Predictability}, Cambridge University Press, Cambridge, UK, 2003.
[35] D. T. B. Kelly, K. J. H. Law, and A. M. Stuart, {\it Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time}, Nonlinearity, 27 (2014), pp. 2579-2603. · Zbl 1305.62323
[36] E. Kwiatkowski and J. Mandel, {\it Convergence of the square root ensemble Kalman filter in the large ensemble limit}, SIAM/ASA J. Uncertain. Quantif., 3 (2015), pp. 1-17. · Zbl 1329.60083
[37] P. S. Laplace, {\it Memoir on the probability of the causes of events}, Statist. Sci., 1 (1986), pp. 364-378.
[38] K. J. H. Law and A. M. Stuart, {\it Evaluating data assimilation algorithms}, Mon. Wea. Rev., 140 (2012), pp. 3757-3782.
[39] K. J. H. Law, A. M. Stuart, and K. C. Zygalakis, {\it Data Assimilation: A Mathematical Introduction}, Texts Appl. Math. 62, Springer, Cham, Switzerland, 2015. · Zbl 1353.60002
[40] F. Le Gland, V. Monbet, and V.-D. Tran, {\it Large sample asymptotics for the ensemble Kalman filter}, in The Oxford Handbook of Nonlinear Filtering, Oxford University Press, Oxford, UK, 2011, pp. 598-631. · Zbl 1225.93108
[41] J. Li and D. Xiu, {\it A generalized polynomial chaos based ensemble Kalman filter with high accuracy}, J. Comput. Phys., 228 (2009), pp. 5454-5469. · Zbl 1280.93084
[42] D. G. Luenberger, {\it Optimization by Vector Space Methods}, John Wiley & Sons, New York, London, Sydney, 1969. · Zbl 0176.12701
[43] J. Mandel and J. D. Beezley, {\it An ensemble Kalman-particle predictor-corrector filter for non-Gaussian data assimilation}, in Computational Science–ICCS 2009, Springer-Verlag, Berlin, Heidelberg, 2009, pp. 470-478.
[44] J. Mandel, L. Cobb, and J. D. Beezley, {\it On the convergence of the ensemble Kalman filter}, Appl. Math., 56 (2011), pp. 533-541. · Zbl 1248.62164
[45] P. A. Markowich and C. Villani, {\it On the trend to equilibrium for the Fokker-Planck equation: An interplay between physics and functional analysis}, Mat. Contemp., 19 (2000), pp. 1-29. · Zbl 1139.82326
[46] J. C. Mattingly, A. M. Stuart, and D. J. Higham, {\it Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise}, Stochastic Process. Appl., 101 (2002), pp. 185-232. · Zbl 1075.60072
[47] B. Øksendal, {\it Stochastic Differential Equations. An Introduction with Applications}, Universitext, 6th ed., Springer-Verlag, Berlin, 2003.
[48] O. Pajonk, {\it Stochastic Spectral Methods for Linear Bayesian Inference}, Ph.D. thesis, Technische Universität Braunschweig, Braunschweig, Germany, 2012.
[49] O. Pajonk, B. V. Rosić, A. Litvinenko, and H. G. Matthies, {\it A deterministic filter for non-Gaussian Bayesian estimation applications to dynamical system estimation with noisy measurements}, Phys. D, 241 (2012), pp. 775-788. · Zbl 1237.62129
[50] O. Pajonk, B. V. Rosić, and H. G. Matthies, {\it Sampling-free linear Bayesian updating of model state and parameters using a square root approach}, Comput. Geosci., 55 (2013), pp. 70-83.
[51] P. Rebeschini and R. van Handel, {\it Can Local Particle Filters Beat the Curse of Dimensionality?}, preprint, http://arxiv.org/abs/1301.6585 arXiv:1301.6585v1 [math.ST], 2013. · Zbl 1325.60058
[52] S. Reich, {\it A Gaussian-mixture ensemble transform filter}, Q. J. R. Meteorol. Soc., 138 (2012), pp. 222-233.
[53] H. Risken, {\it The Fokker-Planck Equation. Methods of Solution and Applications}, 2nd ed., Springer Ser. Synergetics 18, Springer-Verlag, Berlin, 1989. · Zbl 0665.60084
[54] H. Salman, {\it A hybrid grid/particle filter for Lagrangian data assimilation. \textupI: Formulating the passive scalar approximation}, Q. J. R. Meteorol. Soc., 134 (2008), pp. 1539-1550.
[55] N. Santitissadeekorn and C. Jones, {\it Two-state filtering for joint state-parameter estimation}, Mon. Wea. Rev., 143 (2015), pp. 2028-2042.
[56] T. P. Sapsis and A. J. Majda, {\it Blended reduced subspace algorithms for uncertainty quantification of quadratic systems with a stable mean state}, Phys. D, 258 (2013), pp. 61-76. · Zbl 1284.93216
[57] A. Solonen, H. Haario, J. Hakkarainen, H. Auvinen, I. Amour, and T. Kauranne, {\it Variational ensemble Kalman filtering using limited memory BFGS}, Electron. Trans. Numer. Anal., 39 (2012), pp. 271-285. · Zbl 1321.93060
[58] A. M. Stuart, {\it Inverse problems: A Bayesian approach}, Acta Numer., 19 (2010), pp. 451-559. · Zbl 1242.65142
[59] F. Uboldi, A. Trevisan, and A. Carrassi, {\it Developing a dynamically based assimilation method for targeted and standard observations}, Nonlinear Process. Geophys., 12 (2005), pp. 149-156.
[60] E. Vanden-Eijnden and J. Weare, {\it Data assimilation in the low noise, accurate observation regime with application to the Kuroshio}, Mon. Wea. Rev., 141 (2013), pp. 1822-1841.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.