September  2021, 3(3): 615-645. doi: 10.3934/fods.2021023

Analysis of the feedback particle filter with diffusion map based approximation of the gain

1. 

University of Potsdam, Institute of Mathematics, Karl-Liebknecht Str. 24/25, D-14476 Potsdam, Germany

2. 

TU Berlin, Institute of Mathematics, Str. des 17. Juni 136, D-10623 Berlin, Germany

3. 

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

* Corresponding author: Sahani Pathiraja

Received  March 2021 Revised  July 2021 Published  September 2021 Early access  September 2021

Fund Project: This research has been partially funded by the Deutsche Forschungsgemeinschaft (DFG)-Project-ID 318763901 - SFB1294.

Control-type particle filters have been receiving increasing attention over the last decade as a means of obtaining sample based approximations to the sequential Bayesian filtering problem in the nonlinear setting. Here we analyse one such type, namely the feedback particle filter and a recently proposed approximation of the associated gain function based on diffusion maps. The key purpose is to provide analytic insights on the form of the approximate gain, which are of interest in their own right. These are then used to establish a roadmap to obtaining well-posedness and convergence of the finite $ N $ system to its mean field limit. A number of possible future research directions are also discussed.

Citation: Sahani Pathiraja, Wilhelm Stannat. Analysis of the feedback particle filter with diffusion map based approximation of the gain. Foundations of Data Science, 2021, 3 (3) : 615-645. doi: 10.3934/fods.2021023
References:
[1]

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

[2]

K. Berntorp and P. Grover, Data-driven gain computation in the feedback particle filter, Proceedings of the American Control Conference, (2016), 2711–2716. doi: 10.1109/ACC.2016.7525328.  Google Scholar

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T. Berry and J. Harlim, Variable bandwidth diffusion kernels, Applied and Computational Harmonic Analysis, 40 (2016), 68-96.  doi: 10.1016/j.acha.2015.01.001.  Google Scholar

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A. N. BishopP. del Moral and S. D. Pathiraja, Perturbations and projections of Kalman-Bucy semigroups, Stochastic Processes and their Applications, 128 (2018), 2857-2904.  doi: 10.1016/j.spa.2017.10.006.  Google Scholar

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F. BolleyA. Guillin and F. Malrieu, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2010), 867-884.  doi: 10.1051/m2an/2010045.  Google Scholar

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H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, Journal of Functional Analysis, 22 (1976), 366-389.  doi: 10.1016/0022-1236(76)90004-5.  Google Scholar

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E. A. CarlenD. Cordero-erausquin and E. H. Lieb, Asymmetric covariance estimates of Brascamp-Lieb type and related inequalities for log-concave measures, Ann. Inst. Henri Poincaré Probab. Stat., 49 (2013), 1-12.  doi: 10.1214/11-AIHP462.  Google Scholar

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P. Cattiaux and A. Guillin, On the Poincaré constant of log-concave measures, in Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2017-2019 Volume I (eds. B. Klartag), Springer International Publishing, (2020), 171–217. doi: 10.1007/978-3-030-36020-7_9.  Google Scholar

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R. R. Coifman and S. Lafon, Diffusion maps, Applied and Computational Harmonic Analysis, 21 (2006), 5-30.  doi: 10.1016/j.acha.2006.04.006.  Google Scholar

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D. Crisan and J. Xiong, Approximate McKean-Vlasov representations for a class of SPDEs, Stochastics, 82 (2010), 53-68.  doi: 10.1080/17442500902723575.  Google Scholar

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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. Applied Dynamical Systems, 17 (2018), 1152-1181.  doi: 10.1137/17M1119056.  Google Scholar

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J. de Wiljes and X. Tong, Analysis of a localised nonlinear Ensemble Kalman Bucy Filter with complete and accurate observations, Nonlinearity, 33 (2020), 4752-4782.  doi: 10.1088/1361-6544/ab8d14.  Google Scholar

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P. del MoralA. Kurtzmann and J. Tugaut, On the stability and the uniform propagation of chaos of a class of extended ensemble Kalman–Bucy filters, SIAM Journal on Control and Optimization, 55 (2017), 119-155.  doi: 10.1137/16M1087497.  Google Scholar

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G. Evensen, Data Assimilation. The Ensemble Kalman Filter, Springer-Verlag, New York, 2009. doi: 10.1007/978-3-642-03711-5.  Google Scholar

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G. Evensen, The ensemble Kalman filter: Theoretical formulation and practical implementation, Ocean Dynamics, 53 (2003), 343-367.  doi: 10.1007/s10236-003-0036-9.  Google Scholar

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G. Evensen and P. J. van Leeuwen, An ensemble Kalman smoother for nonlinear dynamics, Monthly Weather Review, 128 (2000), 1852-1867.  doi: 10.1175/1520-0493(2000)128<1852:AEKSFN>2.0.CO;2.  Google Scholar

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T. Lange and W. Stannat, Mean field limit of ensemble square root filters - discrete and continuous time, Foundations of Data Science, (2021). doi: 10.3934/fods.2021003.  Google Scholar

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T. Lange and W. Stannat, On the continuous time limit of the ensemble Kalman filter, Mathematics of Computation, 90 (2021), 233-265.  doi: 10.1090/mcom/3588.  Google Scholar

[20]

R. S. LaugesenP. G. MehtaS. P. Meyn and M. Raginsky, Poissons equation in nonlinear filtering, SIAM Journal on Control and Optimization, 53 (2015), 501-525.  doi: 10.1137/13094743X.  Google Scholar

[21]

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

[22]

A. J. Majda and X. T. Tong, Robustness and accuracy of finite ensemble Kalman filters in large dimensions, Comm. Pure Appl. Math., 71 (2018), 892–937, arXiv: 1606.0932. doi: 10.1002/cpa.21722.  Google Scholar

[23]

S. Y. Olmez, A. Taghvaei and P. G. Mehta, Deep FPF : Gain function approximation in high-dimensional setting, arXiv: 2010.01183v1. Google Scholar

[24]

S. Pathiraja, L2 convergence of smooth approximations of stochastic differential equations with unbounded coefficients, preprint, arXiv: 2011.13009. Google Scholar

[25]

S. Pathiraja, S. Reich and W. Stannat, McKean-Vlasov SDEs in nonlinear filtering, SIAM Journal on Control and Optimization, (accepted), (2021) Google Scholar

[26]

A. Radhakrishnan, A. Devraj and S. P. Meyn, Learning techniques for feedback particle filter design, Proceedings of the IEEE 55th Conference on Decision and Control, (2016), 5453–5459. doi: 10.1109/CDC.2016.7799106.  Google Scholar

[27]

S. Reich, A dynamical systems framework for intermittent data assimilation, BIT Numerical Mathematics, 51 (2010), 235-249.  doi: 10.1007/s10543-010-0302-4.  Google Scholar

[28]

C. Schillings and A. M. Stuart, Analysis of the ensemble Kalman filter for inverse problems, SIAM Journal on Numerical Analysis, 55 (2016), 1264-1290.  doi: 10.1137/16M105959X.  Google Scholar

[29]

A. TaghvaeiP. G. Mehta and S. P. Meyn, Diffusion map-based algorithm for gain function approximation in the feedback particle filter, SIAM-ASA Journal on Uncertainty Quantification, 8 (2020), 1090-1117.  doi: 10.1137/19M124513X.  Google Scholar

[30]

X. TongA. J. Majda and D. Kelly, Nonlinear stability and ergodicity of ensemble based Kalman filters, Nonlinearity, 29 (2016), 657-691.  doi: 10.1088/0951-7715/29/2/657.  Google Scholar

[31]

J. Touboul, Propagation of chaos in neural fields, Annals of Applied Probability, 24 (2014), 1298-1328.  doi: 10.1214/13-AAP950.  Google Scholar

[32]

C. L. Wormell and S. Reich, Spectral convergence of diffusion maps: Improved error bounds and an alternative normalization, SIAM Journal on Numerical Analysis, 59 (2021), 1687-1734.  doi: 10.1137/20M1344093.  Google Scholar

[33]

T. YangP. G. Mehta and S. P. Meyn, Feedback particle filter, IEEE Transactions on Automatic Control, 58 (2013), 2465-2480.  doi: 10.1109/TAC.2013.2258825.  Google Scholar

[34]

T. Yang, P. G. Mehta and S. P. Meyn, A mean-field control-oriented approach to particle filtering, Proceedings of the American Control Conference, (2011), 2037–2043. doi: 10.1109/ACC.2011.5991422.  Google Scholar

show all references

References:
[1]

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

[2]

K. Berntorp and P. Grover, Data-driven gain computation in the feedback particle filter, Proceedings of the American Control Conference, (2016), 2711–2716. doi: 10.1109/ACC.2016.7525328.  Google Scholar

[3]

T. Berry and J. Harlim, Variable bandwidth diffusion kernels, Applied and Computational Harmonic Analysis, 40 (2016), 68-96.  doi: 10.1016/j.acha.2015.01.001.  Google Scholar

[4]

A. N. BishopP. del Moral and S. D. Pathiraja, Perturbations and projections of Kalman-Bucy semigroups, Stochastic Processes and their Applications, 128 (2018), 2857-2904.  doi: 10.1016/j.spa.2017.10.006.  Google Scholar

[5]

F. BolleyA. Guillin and F. Malrieu, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2010), 867-884.  doi: 10.1051/m2an/2010045.  Google Scholar

[6]

H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, Journal of Functional Analysis, 22 (1976), 366-389.  doi: 10.1016/0022-1236(76)90004-5.  Google Scholar

[7]

E. A. CarlenD. Cordero-erausquin and E. H. Lieb, Asymmetric covariance estimates of Brascamp-Lieb type and related inequalities for log-concave measures, Ann. Inst. Henri Poincaré Probab. Stat., 49 (2013), 1-12.  doi: 10.1214/11-AIHP462.  Google Scholar

[8]

P. Cattiaux and A. Guillin, On the Poincaré constant of log-concave measures, in Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2017-2019 Volume I (eds. B. Klartag), Springer International Publishing, (2020), 171–217. doi: 10.1007/978-3-030-36020-7_9.  Google Scholar

[9]

R. R. Coifman and S. Lafon, Diffusion maps, Applied and Computational Harmonic Analysis, 21 (2006), 5-30.  doi: 10.1016/j.acha.2006.04.006.  Google Scholar

[10]

D. Crisan and J. Xiong, Approximate McKean-Vlasov representations for a class of SPDEs, Stochastics, 82 (2010), 53-68.  doi: 10.1080/17442500902723575.  Google Scholar

[11]

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. Applied Dynamical Systems, 17 (2018), 1152-1181.  doi: 10.1137/17M1119056.  Google Scholar

[12]

J. de Wiljes and X. Tong, Analysis of a localised nonlinear Ensemble Kalman Bucy Filter with complete and accurate observations, Nonlinearity, 33 (2020), 4752-4782.  doi: 10.1088/1361-6544/ab8d14.  Google Scholar

[13]

P. del MoralA. Kurtzmann and J. Tugaut, On the stability and the uniform propagation of chaos of a class of extended ensemble Kalman–Bucy filters, SIAM Journal on Control and Optimization, 55 (2017), 119-155.  doi: 10.1137/16M1087497.  Google Scholar

[14]

G. Evensen, Data Assimilation. The Ensemble Kalman Filter, Springer-Verlag, New York, 2009. doi: 10.1007/978-3-642-03711-5.  Google Scholar

[15]

G. Evensen, The ensemble Kalman filter: Theoretical formulation and practical implementation, Ocean Dynamics, 53 (2003), 343-367.  doi: 10.1007/s10236-003-0036-9.  Google Scholar

[16]

G. Evensen and P. J. van Leeuwen, An ensemble Kalman smoother for nonlinear dynamics, Monthly Weather Review, 128 (2000), 1852-1867.  doi: 10.1175/1520-0493(2000)128<1852:AEKSFN>2.0.CO;2.  Google Scholar

[17] A. M. Kulik, Introduction to Ergodic Rates for Markov Chains and Processes, with Applications to Limit Theorems, Potsdam University Press, Germany, 2015.   Google Scholar
[18]

T. Lange and W. Stannat, Mean field limit of ensemble square root filters - discrete and continuous time, Foundations of Data Science, (2021). doi: 10.3934/fods.2021003.  Google Scholar

[19]

T. Lange and W. Stannat, On the continuous time limit of the ensemble Kalman filter, Mathematics of Computation, 90 (2021), 233-265.  doi: 10.1090/mcom/3588.  Google Scholar

[20]

R. S. LaugesenP. G. MehtaS. P. Meyn and M. Raginsky, Poissons equation in nonlinear filtering, SIAM Journal on Control and Optimization, 53 (2015), 501-525.  doi: 10.1137/13094743X.  Google Scholar

[21]

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

[22]

A. J. Majda and X. T. Tong, Robustness and accuracy of finite ensemble Kalman filters in large dimensions, Comm. Pure Appl. Math., 71 (2018), 892–937, arXiv: 1606.0932. doi: 10.1002/cpa.21722.  Google Scholar

[23]

S. Y. Olmez, A. Taghvaei and P. G. Mehta, Deep FPF : Gain function approximation in high-dimensional setting, arXiv: 2010.01183v1. Google Scholar

[24]

S. Pathiraja, L2 convergence of smooth approximations of stochastic differential equations with unbounded coefficients, preprint, arXiv: 2011.13009. Google Scholar

[25]

S. Pathiraja, S. Reich and W. Stannat, McKean-Vlasov SDEs in nonlinear filtering, SIAM Journal on Control and Optimization, (accepted), (2021) Google Scholar

[26]

A. Radhakrishnan, A. Devraj and S. P. Meyn, Learning techniques for feedback particle filter design, Proceedings of the IEEE 55th Conference on Decision and Control, (2016), 5453–5459. doi: 10.1109/CDC.2016.7799106.  Google Scholar

[27]

S. Reich, A dynamical systems framework for intermittent data assimilation, BIT Numerical Mathematics, 51 (2010), 235-249.  doi: 10.1007/s10543-010-0302-4.  Google Scholar

[28]

C. Schillings and A. M. Stuart, Analysis of the ensemble Kalman filter for inverse problems, SIAM Journal on Numerical Analysis, 55 (2016), 1264-1290.  doi: 10.1137/16M105959X.  Google Scholar

[29]

A. TaghvaeiP. G. Mehta and S. P. Meyn, Diffusion map-based algorithm for gain function approximation in the feedback particle filter, SIAM-ASA Journal on Uncertainty Quantification, 8 (2020), 1090-1117.  doi: 10.1137/19M124513X.  Google Scholar

[30]

X. TongA. J. Majda and D. Kelly, Nonlinear stability and ergodicity of ensemble based Kalman filters, Nonlinearity, 29 (2016), 657-691.  doi: 10.1088/0951-7715/29/2/657.  Google Scholar

[31]

J. Touboul, Propagation of chaos in neural fields, Annals of Applied Probability, 24 (2014), 1298-1328.  doi: 10.1214/13-AAP950.  Google Scholar

[32]

C. L. Wormell and S. Reich, Spectral convergence of diffusion maps: Improved error bounds and an alternative normalization, SIAM Journal on Numerical Analysis, 59 (2021), 1687-1734.  doi: 10.1137/20M1344093.  Google Scholar

[33]

T. YangP. G. Mehta and S. P. Meyn, Feedback particle filter, IEEE Transactions on Automatic Control, 58 (2013), 2465-2480.  doi: 10.1109/TAC.2013.2258825.  Google Scholar

[34]

T. Yang, P. G. Mehta and S. P. Meyn, A mean-field control-oriented approach to particle filtering, Proceedings of the American Control Conference, (2011), 2037–2043. doi: 10.1109/ACC.2011.5991422.  Google Scholar

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