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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[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.

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

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[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.

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

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[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.

[1]

Kleber Carrapatoso. Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules. Kinetic and Related Models, 2016, 9 (1) : 1-49. doi: 10.3934/krm.2016.9.1

[2]

Maxime Hauray, Samir Salem. Propagation of chaos for the Vlasov-Poisson-Fokker-Planck system in 1D. Kinetic and Related Models, 2019, 12 (2) : 269-302. doi: 10.3934/krm.2019012

[3]

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246

[4]

Laurent Bernis, Laurent Desvillettes. Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 13-33. doi: 10.3934/dcds.2009.24.13

[5]

Samir Salem. A gradient flow approach of propagation of chaos. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5729-5754. doi: 10.3934/dcds.2020243

[6]

Hui Huang, Jian-Guo Liu. Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos. Kinetic and Related Models, 2016, 9 (4) : 715-748. doi: 10.3934/krm.2016013

[7]

Sie Long Kek, Kok Lay Teo, Mohd Ismail Abd Aziz. Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 207-222. doi: 10.3934/naco.2012.2.207

[8]

Kaijen Cheng, Kenneth Palmer, Yuh-Jenn Wu. Period 3 and chaos for unimodal maps. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1933-1949. doi: 10.3934/dcds.2014.34.1933

[9]

Costică Moroşanu. Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1567-1587. doi: 10.3934/dcdss.2020089

[10]

Jorge A. Esquivel-Avila. Qualitative analysis of a nonlinear wave equation. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 787-804. doi: 10.3934/dcds.2004.10.787

[11]

J. Alberto Conejero, Francisco Rodenas, Macarena Trujillo. Chaos for the Hyperbolic Bioheat Equation. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 653-668. doi: 10.3934/dcds.2015.35.653

[12]

Kang-Ling Liao, Chih-Wen Shih, Chi-Jer Yu. The snapback repellers for chaos in multi-dimensional maps. Journal of Computational Dynamics, 2018, 5 (1&2) : 81-92. doi: 10.3934/jcd.2018004

[13]

Louis-Pierre Chaintron, Antoine Diez. Propagation of chaos: A review of models, methods and applications. Ⅱ. Applications. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022018

[14]

H. Thomas Banks, Shuhua Hu, Zackary R. Kenz, Hien T. Tran. A comparison of nonlinear filtering approaches in the context of an HIV model. Mathematical Biosciences & Engineering, 2010, 7 (2) : 213-236. doi: 10.3934/mbe.2010.7.213

[15]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285

[16]

Mario Lefebvre. A stochastic model for computer virus propagation. Journal of Dynamics and Games, 2020, 7 (2) : 163-174. doi: 10.3934/jdg.2020010

[17]

Tianliang Yang, J. M. McDonough. Solution filtering technique for solving Burgers' equation. Conference Publications, 2003, 2003 (Special) : 951-959. doi: 10.3934/proc.2003.2003.951

[18]

Shangbing Ai, Wenzhang Huang, Zhi-An Wang. Reaction, diffusion and chemotaxis in wave propagation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 1-21. doi: 10.3934/dcdsb.2015.20.1

[19]

Pao-Liu Chow. Asymptotic solutions of a nonlinear stochastic beam equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 735-749. doi: 10.3934/dcdsb.2006.6.735

[20]

Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717

 Impact Factor: 

Metrics

  • PDF downloads (122)
  • HTML views (384)
  • Cited by (0)

Other articles
by authors

[Back to Top]