doi: 10.3934/fods.2021003

Mean field limit of Ensemble Square Root filters - discrete and continuous time

1. 

Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany

2. 

Bernstein Center for Computational Neuroscience, Philippstr. 13, 10115 Berlin, Germany

* Corresponding author: Theresa Lange

Received  November 2020 Revised  January 2021 Published  January 2021

Fund Project: The research of Theresa Lange and Wilhelm Stannat has been partially funded by Deutsche Forschungsgemeinschaft (DFG) - SFB1294/1 - 318763901

Consider the class of Ensemble Square Root filtering algorithms for the numerical approximation of the posterior distribution of nonlinear Markovian signals, partially observed with linear observations corrupted with independent measurement noise. We analyze the asymptotic behavior of these algorithms in the large ensemble limit both in discrete and continuous time. We identify limiting mean-field processes on the level of the ensemble members, prove corresponding propagation of chaos results and derive associated convergence rates in terms of the ensemble size. In continuous time we also identify the stochastic partial differential equation driving the distribution of the mean-field process and perform a comparison with the Kushner-Stratonovich equation.

Citation: Theresa Lange, Wilhelm Stannat. Mean field limit of Ensemble Square Root filters - discrete and continuous time. Foundations of Data Science, doi: 10.3934/fods.2021003
References:
[1]

J. L. Anderson, An ensemble adjustment Kalman filter for data assimilation, Monthly Weather Rev., 129 (2001), 2884-2903.  doi: 10.1175/1520-0493(2001)129<2884:AEAKFF>2.0.CO;2.  Google Scholar

[2]

K. Bergemann and S. Reich, An ensemble Kalman-Bucy filter for continuous data assimilation, Meteorologische Zeitschrift, 21 (2012), 213-219.  doi: 10.1127/0941-2948/2012/0307.  Google Scholar

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A. N. Bishop and P. Del Moral, On the mathematical theory of ensemble (linear-Gaussian) Kalman-Bucy filtering, preprint, arXiv: 2006.08843v1. Google Scholar

[5]

A. N. Bishop and P. Del Moral, On the stability of matrix-valued Riccati diffusions, Electron. J. Probab., 24 (2019), 40pp. doi: 10.1214/19-ejp342.  Google Scholar

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A. N. BishopP. Del MoralK. Kamatani and B. Rémillard, On one-dimensional Riccati diffusions, Ann. Appl. Probab., 29 (2019), 1127-1187.  doi: 10.1214/18-AAP1431.  Google Scholar

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A. N. BishopP. Del Moral and A. Niclas, A perturbation analysis of stochastic matrix Riccati diffusions, Ann. Inst. Henri Poincaré Probab. Stat., 56 (2020), 884-916.  doi: 10.1214/19-AIHP987.  Google Scholar

[8]

C. H. BishopB. J. Etherton and S. J. Majumdar, Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects, Monthly Weather Rev., 129 (2001), 420-436.  doi: 10.1175/1520-0493(2001)129<0420:ASWTET>2.0.CO;2.  Google Scholar

[9]

J. de WiljesS. Reich and W. 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), 1152-1181.  doi: 10.1137/17M1119056.  Google Scholar

[10]

R. E. Kalman, A new approach to linear filtering and prediction problems, Trans. ASME Ser. D. J. Basic Engrg., 82 (1960), 35-45.  doi: 10.1115/1.3662552.  Google Scholar

[11]

E. Kwiatkowski and J. Mandel, Convergence of the square root ensemble Kalman filter in the Large Ensemble Limit, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 1-17.  doi: 10.1137/140965363.  Google Scholar

[12]

T. Lange and W. Stannat, On the continuous time limit of the Ensemble Square Root Filters, preprint, arXiv: 1910.12493v1. Google Scholar

[13]

F. Le Gland, V. Monbet and V.-D. Tran, Large sample asymptotics for the ensemble Kalman filter, in The Oxford Handbook of Nonlinear Filtering, Oxford Univ. Press, Oxford, 2011,598–631.  Google Scholar

[14]

O. LeuuwenburghG. Evensen and L. Bertino, The impact of ensemble filter definition on the assimilation of temperature profiles in the tropical Pacific, Quart. J. Roy. Meteorological Soc., 131 (2005), 3291-3300.  doi: 10.1256/qj.05.90.  Google Scholar

[15]

D. M. LivingsS. L. Dance and N. K. Nichols, Unbiased ensemble square root filters, Phys. D, 237 (2008), 1021-1028.  doi: 10.1016/j.physd.2008.01.005.  Google Scholar

[16]

J. MandelL. Cobb and J. D. Beezley, On the convergence of the ensemble Kalman filter, Appl. Math., 56 (2011), 533-541.  doi: 10.1007/s10492-011-0031-2.  Google Scholar

[17]

T. L. O'Kane and J. S. Frederiksen, Comparison of statistical dynamical, square root and ensemble Kalman filters, Entropy, 10 (2008), 684-721.  doi: 10.3390/e10040684.  Google Scholar

[18]

M. Scheutzow, A stochastic Gronwall lemma, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 16 (2013), 4pp. doi: 10.1142/S0219025713500197.  Google Scholar

[19]

M. K. TippettJ. L. AndersonC. H. BishopT. M. Hamill and J. S. Whitaker, Ensemble Square Root Filters, Monthly Weather Rev., 131 (2003), 1485-1490.  doi: 10.1175/1520-0493(2003)131<1485:ESRF>2.0.CO;2.  Google Scholar

[20]

J. L. van Hemmen and T. Ando, An inequality for trace ideals, Comm. Math. Phys., 76 (1980), 143-148.  doi: 10.1007/BF01212822.  Google Scholar

[21]

X. WangC. H. Bishop and S. J. Julier, Which is better, an ensemble of positive-negative pairs or centered spherical simplex ensemble?, Monthly Weather Rev., 132 (2004), 1590-1605.  doi: 10.1175/1520-0493(2004)132<1590:WIBAEO>2.0.CO;2.  Google Scholar

[22]

J. S. Whitaker and T. M. Hamill, Ensemble data assimilation without perturbed observations, Monthly Weather Rev., 130 (2002), 1913-1924.  doi: 10.1175/1520-0493(2002)130<1913:EDAWPO>2.0.CO;2.  Google Scholar

show all references

References:
[1]

J. L. Anderson, An ensemble adjustment Kalman filter for data assimilation, Monthly Weather Rev., 129 (2001), 2884-2903.  doi: 10.1175/1520-0493(2001)129<2884:AEAKFF>2.0.CO;2.  Google Scholar

[2]

K. Bergemann and S. Reich, An ensemble Kalman-Bucy filter for continuous data assimilation, Meteorologische Zeitschrift, 21 (2012), 213-219.  doi: 10.1127/0941-2948/2012/0307.  Google Scholar

[3]

S. Beyou, A. Cuzol, S. Subrahmanyam Gorthi and E. Mémin, Weighted ensemble transform Kalman filter for image assimilation, Tellus A: Dynamic Meteorology and Oceanography, 65 (2013). doi: 10.3402/tellusa.v65i0.18803.  Google Scholar

[4]

A. N. Bishop and P. Del Moral, On the mathematical theory of ensemble (linear-Gaussian) Kalman-Bucy filtering, preprint, arXiv: 2006.08843v1. Google Scholar

[5]

A. N. Bishop and P. Del Moral, On the stability of matrix-valued Riccati diffusions, Electron. J. Probab., 24 (2019), 40pp. doi: 10.1214/19-ejp342.  Google Scholar

[6]

A. N. BishopP. Del MoralK. Kamatani and B. Rémillard, On one-dimensional Riccati diffusions, Ann. Appl. Probab., 29 (2019), 1127-1187.  doi: 10.1214/18-AAP1431.  Google Scholar

[7]

A. N. BishopP. Del Moral and A. Niclas, A perturbation analysis of stochastic matrix Riccati diffusions, Ann. Inst. Henri Poincaré Probab. Stat., 56 (2020), 884-916.  doi: 10.1214/19-AIHP987.  Google Scholar

[8]

C. H. BishopB. J. Etherton and S. J. Majumdar, Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects, Monthly Weather Rev., 129 (2001), 420-436.  doi: 10.1175/1520-0493(2001)129<0420:ASWTET>2.0.CO;2.  Google Scholar

[9]

J. de WiljesS. Reich and W. 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), 1152-1181.  doi: 10.1137/17M1119056.  Google Scholar

[10]

R. E. Kalman, A new approach to linear filtering and prediction problems, Trans. ASME Ser. D. J. Basic Engrg., 82 (1960), 35-45.  doi: 10.1115/1.3662552.  Google Scholar

[11]

E. Kwiatkowski and J. Mandel, Convergence of the square root ensemble Kalman filter in the Large Ensemble Limit, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 1-17.  doi: 10.1137/140965363.  Google Scholar

[12]

T. Lange and W. Stannat, On the continuous time limit of the Ensemble Square Root Filters, preprint, arXiv: 1910.12493v1. Google Scholar

[13]

F. Le Gland, V. Monbet and V.-D. Tran, Large sample asymptotics for the ensemble Kalman filter, in The Oxford Handbook of Nonlinear Filtering, Oxford Univ. Press, Oxford, 2011,598–631.  Google Scholar

[14]

O. LeuuwenburghG. Evensen and L. Bertino, The impact of ensemble filter definition on the assimilation of temperature profiles in the tropical Pacific, Quart. J. Roy. Meteorological Soc., 131 (2005), 3291-3300.  doi: 10.1256/qj.05.90.  Google Scholar

[15]

D. M. LivingsS. L. Dance and N. K. Nichols, Unbiased ensemble square root filters, Phys. D, 237 (2008), 1021-1028.  doi: 10.1016/j.physd.2008.01.005.  Google Scholar

[16]

J. MandelL. Cobb and J. D. Beezley, On the convergence of the ensemble Kalman filter, Appl. Math., 56 (2011), 533-541.  doi: 10.1007/s10492-011-0031-2.  Google Scholar

[17]

T. L. O'Kane and J. S. Frederiksen, Comparison of statistical dynamical, square root and ensemble Kalman filters, Entropy, 10 (2008), 684-721.  doi: 10.3390/e10040684.  Google Scholar

[18]

M. Scheutzow, A stochastic Gronwall lemma, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 16 (2013), 4pp. doi: 10.1142/S0219025713500197.  Google Scholar

[19]

M. K. TippettJ. L. AndersonC. H. BishopT. M. Hamill and J. S. Whitaker, Ensemble Square Root Filters, Monthly Weather Rev., 131 (2003), 1485-1490.  doi: 10.1175/1520-0493(2003)131<1485:ESRF>2.0.CO;2.  Google Scholar

[20]

J. L. van Hemmen and T. Ando, An inequality for trace ideals, Comm. Math. Phys., 76 (1980), 143-148.  doi: 10.1007/BF01212822.  Google Scholar

[21]

X. WangC. H. Bishop and S. J. Julier, Which is better, an ensemble of positive-negative pairs or centered spherical simplex ensemble?, Monthly Weather Rev., 132 (2004), 1590-1605.  doi: 10.1175/1520-0493(2004)132<1590:WIBAEO>2.0.CO;2.  Google Scholar

[22]

J. S. Whitaker and T. M. Hamill, Ensemble data assimilation without perturbed observations, Monthly Weather Rev., 130 (2002), 1913-1924.  doi: 10.1175/1520-0493(2002)130<1913:EDAWPO>2.0.CO;2.  Google Scholar

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