doi: 10.3934/dcdss.2021097
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A drift homotopy implicit particle filter method for nonlinear filtering problems

Department of Mathematics, Florida State University, Tallahassee, Florida

* Corresponding author

Received  February 2021 Revised  June 2021 Early access August 2021

In this paper, we develop a drift homotopy implicit particle filter method. The methodology of our approach is to adopt the concept of drift homotopy in the resampling procedure of the particle filter method for solving the nonlinear filtering problem, and we introduce an implicit particle filter method to improve the efficiency of the drift homotopy resampling procedure. Numerical experiments are carried out to demonstrate the effectiveness and efficiency of our drift homotopy implicit particle filter.

Citation: Xin Li, Feng Bao, Kyle Gallivan. A drift homotopy implicit particle filter method for nonlinear filtering problems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021097
References:
[1]

C. AndrieuA. Doucet and R. Holenstein, Particle markov chain monte carlo methods, J. R. Statist. Soc. B, 72 (2010), 269-342.  doi: 10.1111/j.1467-9868.2009.00736.x.  Google Scholar

[2]

C. Andrieu and G. O. Roberts, The pseudo-marginal approach for efficient monte carlo computations, Ann. Statist., 37 (2009), 697-725.  doi: 10.1214/07-AOS574.  Google Scholar

[3]

R. Archibald, F. Bao and X. Tu, A direct filter method for parameter estimation, J. Comput. Phys., 398 (2019), 108871, 17 pp. doi: 10.1016/j.jcp.2019.108871.  Google Scholar

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F. BaoR. Archibald and P. Maksymovych, Lévy backward SDE filter for jump diffusion processes and its applications in material sciences, Communications in Computational Physics, 27 (2020), 589-618.  doi: 10.4208/cicp.OA-2018-0238.  Google Scholar

[5]

F. BaoY. Cao and H. Chi, Adjoint forward backward stochastic differential equations driven by jump diffusion processes and its application to nonlinear filtering problems, Int. J. Uncertain. Quantif., 9 (2019), 143-159.  doi: 10.1615/Int.J.UncertaintyQuantification.2019028300.  Google Scholar

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[7]

F. BaoY. Cao and X. Han, Forward backward doubly stochastic differential equations and optimal filtering of diffusion processes, Communications in Mathematical Sciences, 18 (2020), 635-661.  doi: 10.4310/CMS.2020.v18.n3.a3.  Google Scholar

[8]

F. BaoY. CaoA. Meir and W. Zhao, A first order scheme for backward doubly stochastic differential equations, SIAM/ASA J. Uncertain. Quantif., 4 (2016), 413-445.  doi: 10.1137/14095546X.  Google Scholar

[9]

F. BaoY. CaoC. Webster and G. Zhang, 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), 784-804.  doi: 10.1137/140952910.  Google Scholar

[10]

F. BaoY. Cao and W. Zhao, Numerical solutions for forward backward doubly stochastic differential equations and zakai equations, International Journal for Uncertainty Quantification, 1 (2011), 351-367.  doi: 10.1615/Int.J.UncertaintyQuantification.2011003508.  Google Scholar

[11]

F. BaoY. Cao and W. Zhao, A first order semi-discrete algorithm for backward doubly stochastic differential equations, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), 1297-1313.  doi: 10.3934/dcdsb.2015.20.1297.  Google Scholar

[12]

F. BaoY. Cao and W. Zhao, A backward doubly stochastic differential equation approach for nonlinear filtering problems, Commun. Comput. Phys., 23 (2018), 1573-1601.  doi: 10.4208/cicp.oa-2017-0084.  Google Scholar

[13]

F. Bao and V. Maroulas, Adaptive meshfree backward SDE filter, SIAM J. Sci. Comput., 39 (2017), A2664–A2683. doi: 10.1137/16M1100277.  Google Scholar

[14]

A. J. Chorin and X. Tu, Implicit sampling for particle filters, Proc. Nat. Acad. Sc. USA, 106 (2009), 17249-17254.  doi: 10.1073/pnas.0909196106.  Google Scholar

[15]

D. Crisan, Exact rates of convergence for a branching particle approximation to the solution of the Zakai equation, Ann. Probab., 31 (2003), 693-718.  doi: 10.1214/aop/1048516533.  Google Scholar

[16]

D. Crisan and A. Doucet, A survey of convergence results on particle filtering methods for practitioners, IEEE Trans. Sig. Proc., 50 (2002), 736-746.  doi: 10.1109/78.984773.  Google Scholar

[17]

A. Doucet and A. M. Johansen, A tutorial on particle filtering and smoothing: Fifteen years later, The Oxford Handbook of Nonlinear Filtering, 2011,656–704.  Google Scholar

[18]

O. Dyck, M. Ziatdinov, S. Jesse, F. Bao, A. Yousefzadi Nobakht, A. Maksov, B. G. Sumpter, R. Archibald, K. J. H. Law and S. V. Kalinin, Probing potential energy landscapes via electron-beam-induced single atom dynamics, Acta Materialia, 203 (2021), 116508. Google Scholar

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

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G. Evensen, The ensemble Kalman filter for combined state and parameter estimation: Monte Carlo techniques for data assimilation in large systems, IEEE Control Syst. Mag., 29 (2009), 83-104.  doi: 10.1109/MCS.2009.932223.  Google Scholar

[21]

E. Gobet, G. Pagès, H. Pham and J. Printems, Discretization and simulation of the Zakai equation, SIAM J. Numer. Anal., 44 (2006), 2505–2538 (electronic). doi: 10.1137/050623140.  Google Scholar

[22]

N. J GordonD. J Salmond and A. F. M. Smith, Novel approach to nonlinear/non-gaussian bayesian state estimation, IEE Proceeding-F, 140 (1993), 107-113.  doi: 10.1049/ip-f-2.1993.0015.  Google Scholar

[23]

S. J. Julier and J. K. Uhlmann, Unscented filtering and nonlinear estimation, Proceedings of the IEEE, 92 (2004), 401-422.  doi: 10.1109/JPROC.2003.823141.  Google Scholar

[24]

K. KangV. MaroulasI. Schizas and F. Bao, Improved distributed particle filters for tracking in a wireless sensor network, Comput. Statist. Data Anal., 117 (2018), 90-108.  doi: 10.1016/j.csda.2017.07.009.  Google Scholar

[25]

H. R. Kunsch, Particle filters, Bernoulli, 19 (2013), 1391-1403.  doi: 10.3150/12-BEJSP07.  Google Scholar

[26]

F. Le Gland, Time discretization of nonlinear filtering equations, In Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 1–3 (Tampa, FL, 1989), 2601–2606, New York, 1989. IEEE.  Google Scholar

[27]

V. Maroulas and P. Stinis, Improved particle filters for multi-target tracking, Journal of Computational Physics, 231 (2012), 602-611.  doi: 10.1016/j.jcp.2011.09.023.  Google Scholar

[28]

M. MorzfeldX. TuE. Atkins and A. J. Chorin, A random map implementation of implicit filters, J. Comput. Phys., 231 (2012), 2049-2066.  doi: 10.1016/j.jcp.2011.11.022.  Google Scholar

[29]

M. K. Pitt and N. Shephard, Filtering via simulation: Auxiliary particle filters, J. Amer. Statist. Assoc., 94 (1999), 590-599.  doi: 10.1080/01621459.1999.10474153.  Google Scholar

[30]

C. SnyderT. BengtssonP. Bickel and J. Anderson, Obstacles to high-dimensional particle filtering, Mon. Wea. Rev., 136 (2008), 4629-4640.  doi: 10.1175/2008MWR2529.1.  Google Scholar

[31]

T. Song and J. Speyer, A stochastic analysis of a modified gain extended kalman filter with applications to estimation with bearings only measurements, IEEE Transactions on Automatic Control, 30 (1985), 940-949.   Google Scholar

[32]

X. T. 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

[33]

P. J. van Leeuwen, Nonlinear data assimilation in geosciences: An extremely efficient particle filter, Q. J. Roy. Meteor. Soc., 136 (2010), 1991-1999.  doi: 10.1002/qj.699.  Google Scholar

[34]

B. WangX. Zou and J. Zhu, Data assimilation and its applications, Proceedings of the National Academy of Sciences, 97 (2000), 11143-11144.  doi: 10.1073/pnas.97.21.11143.  Google Scholar

[35]

M. Zakai, On the optimal filtering of diffusion processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 11 (1969), 230-243.  doi: 10.1007/BF00536382.  Google Scholar

show all references

References:
[1]

C. AndrieuA. Doucet and R. Holenstein, Particle markov chain monte carlo methods, J. R. Statist. Soc. B, 72 (2010), 269-342.  doi: 10.1111/j.1467-9868.2009.00736.x.  Google Scholar

[2]

C. Andrieu and G. O. Roberts, The pseudo-marginal approach for efficient monte carlo computations, Ann. Statist., 37 (2009), 697-725.  doi: 10.1214/07-AOS574.  Google Scholar

[3]

R. Archibald, F. Bao and X. Tu, A direct filter method for parameter estimation, J. Comput. Phys., 398 (2019), 108871, 17 pp. doi: 10.1016/j.jcp.2019.108871.  Google Scholar

[4]

F. BaoR. Archibald and P. Maksymovych, Lévy backward SDE filter for jump diffusion processes and its applications in material sciences, Communications in Computational Physics, 27 (2020), 589-618.  doi: 10.4208/cicp.OA-2018-0238.  Google Scholar

[5]

F. BaoY. Cao and H. Chi, Adjoint forward backward stochastic differential equations driven by jump diffusion processes and its application to nonlinear filtering problems, Int. J. Uncertain. Quantif., 9 (2019), 143-159.  doi: 10.1615/Int.J.UncertaintyQuantification.2019028300.  Google Scholar

[6]

F. BaoY. Cao and X. Han, An implicit algorithm of solving nonlinear filtering problems, Communications in Computational Physics, 16 (2014), 382-402.  doi: 10.4208/cicp.180313.130214a.  Google Scholar

[7]

F. BaoY. Cao and X. Han, Forward backward doubly stochastic differential equations and optimal filtering of diffusion processes, Communications in Mathematical Sciences, 18 (2020), 635-661.  doi: 10.4310/CMS.2020.v18.n3.a3.  Google Scholar

[8]

F. BaoY. CaoA. Meir and W. Zhao, A first order scheme for backward doubly stochastic differential equations, SIAM/ASA J. Uncertain. Quantif., 4 (2016), 413-445.  doi: 10.1137/14095546X.  Google Scholar

[9]

F. BaoY. CaoC. Webster and G. Zhang, 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), 784-804.  doi: 10.1137/140952910.  Google Scholar

[10]

F. BaoY. Cao and W. Zhao, Numerical solutions for forward backward doubly stochastic differential equations and zakai equations, International Journal for Uncertainty Quantification, 1 (2011), 351-367.  doi: 10.1615/Int.J.UncertaintyQuantification.2011003508.  Google Scholar

[11]

F. BaoY. Cao and W. Zhao, A first order semi-discrete algorithm for backward doubly stochastic differential equations, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), 1297-1313.  doi: 10.3934/dcdsb.2015.20.1297.  Google Scholar

[12]

F. BaoY. Cao and W. Zhao, A backward doubly stochastic differential equation approach for nonlinear filtering problems, Commun. Comput. Phys., 23 (2018), 1573-1601.  doi: 10.4208/cicp.oa-2017-0084.  Google Scholar

[13]

F. Bao and V. Maroulas, Adaptive meshfree backward SDE filter, SIAM J. Sci. Comput., 39 (2017), A2664–A2683. doi: 10.1137/16M1100277.  Google Scholar

[14]

A. J. Chorin and X. Tu, Implicit sampling for particle filters, Proc. Nat. Acad. Sc. USA, 106 (2009), 17249-17254.  doi: 10.1073/pnas.0909196106.  Google Scholar

[15]

D. Crisan, Exact rates of convergence for a branching particle approximation to the solution of the Zakai equation, Ann. Probab., 31 (2003), 693-718.  doi: 10.1214/aop/1048516533.  Google Scholar

[16]

D. Crisan and A. Doucet, A survey of convergence results on particle filtering methods for practitioners, IEEE Trans. Sig. Proc., 50 (2002), 736-746.  doi: 10.1109/78.984773.  Google Scholar

[17]

A. Doucet and A. M. Johansen, A tutorial on particle filtering and smoothing: Fifteen years later, The Oxford Handbook of Nonlinear Filtering, 2011,656–704.  Google Scholar

[18]

O. Dyck, M. Ziatdinov, S. Jesse, F. Bao, A. Yousefzadi Nobakht, A. Maksov, B. G. Sumpter, R. Archibald, K. J. H. Law and S. V. Kalinin, Probing potential energy landscapes via electron-beam-induced single atom dynamics, Acta Materialia, 203 (2021), 116508. Google Scholar

[19]

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

[20]

G. Evensen, The ensemble Kalman filter for combined state and parameter estimation: Monte Carlo techniques for data assimilation in large systems, IEEE Control Syst. Mag., 29 (2009), 83-104.  doi: 10.1109/MCS.2009.932223.  Google Scholar

[21]

E. Gobet, G. Pagès, H. Pham and J. Printems, Discretization and simulation of the Zakai equation, SIAM J. Numer. Anal., 44 (2006), 2505–2538 (electronic). doi: 10.1137/050623140.  Google Scholar

[22]

N. J GordonD. J Salmond and A. F. M. Smith, Novel approach to nonlinear/non-gaussian bayesian state estimation, IEE Proceeding-F, 140 (1993), 107-113.  doi: 10.1049/ip-f-2.1993.0015.  Google Scholar

[23]

S. J. Julier and J. K. Uhlmann, Unscented filtering and nonlinear estimation, Proceedings of the IEEE, 92 (2004), 401-422.  doi: 10.1109/JPROC.2003.823141.  Google Scholar

[24]

K. KangV. MaroulasI. Schizas and F. Bao, Improved distributed particle filters for tracking in a wireless sensor network, Comput. Statist. Data Anal., 117 (2018), 90-108.  doi: 10.1016/j.csda.2017.07.009.  Google Scholar

[25]

H. R. Kunsch, Particle filters, Bernoulli, 19 (2013), 1391-1403.  doi: 10.3150/12-BEJSP07.  Google Scholar

[26]

F. Le Gland, Time discretization of nonlinear filtering equations, In Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 1–3 (Tampa, FL, 1989), 2601–2606, New York, 1989. IEEE.  Google Scholar

[27]

V. Maroulas and P. Stinis, Improved particle filters for multi-target tracking, Journal of Computational Physics, 231 (2012), 602-611.  doi: 10.1016/j.jcp.2011.09.023.  Google Scholar

[28]

M. MorzfeldX. TuE. Atkins and A. J. Chorin, A random map implementation of implicit filters, J. Comput. Phys., 231 (2012), 2049-2066.  doi: 10.1016/j.jcp.2011.11.022.  Google Scholar

[29]

M. K. Pitt and N. Shephard, Filtering via simulation: Auxiliary particle filters, J. Amer. Statist. Assoc., 94 (1999), 590-599.  doi: 10.1080/01621459.1999.10474153.  Google Scholar

[30]

C. SnyderT. BengtssonP. Bickel and J. Anderson, Obstacles to high-dimensional particle filtering, Mon. Wea. Rev., 136 (2008), 4629-4640.  doi: 10.1175/2008MWR2529.1.  Google Scholar

[31]

T. Song and J. Speyer, A stochastic analysis of a modified gain extended kalman filter with applications to estimation with bearings only measurements, IEEE Transactions on Automatic Control, 30 (1985), 940-949.   Google Scholar

[32]

X. T. 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

[33]

P. J. van Leeuwen, Nonlinear data assimilation in geosciences: An extremely efficient particle filter, Q. J. Roy. Meteor. Soc., 136 (2010), 1991-1999.  doi: 10.1002/qj.699.  Google Scholar

[34]

B. WangX. Zou and J. Zhu, Data assimilation and its applications, Proceedings of the National Academy of Sciences, 97 (2000), 11143-11144.  doi: 10.1073/pnas.97.21.11143.  Google Scholar

[35]

M. Zakai, On the optimal filtering of diffusion processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 11 (1969), 230-243.  doi: 10.1007/BF00536382.  Google Scholar

Figure 1.  Double well potential case 1: $ \alpha = 1 $, $ \sigma = 1.5 $, $ R = 1.5 $
Figure 2.  Double well potential case 2: $ \alpha = 1 $, $ \sigma = 1 $, $ R = 1 $ with state switch
Figure 3.  Double well potential case 3: $ \alpha = 10 $, $ \sigma = 1 $, $ R = 2 $ with state switch
Figure 4.  Tracking performance for $ 4000 $ steps
Figure 5.  Tracking errors for $ 4000 $ steps
Figure 6.  Tracking performance with rapid change in the state
Figure 7.  Tracking errors with rapid change in the state
Figure 8.  Mean square errors with respect to observation gaps
Table 1.   
Algorithm: Drift homotopy implicit particle filter (DHIPF)
Initialize the particle cloud $ \{x_0^{(i)}\}_{i=1}^{N_p} $, the number of drift homotopy levels $ L $ with the intermediate drift function $ b $ and the constant sequence $ \{\beta_l\}_{l=0}^{L} $, and the reference random variable $ \xi $ for the implicit particle filter procedure.
while $ n =0, 1, 2, \cdots $, do
    for: particles $ i = 1, 2, \cdots, N_p $,
        for: drift homotopy levels $ l = 0, 1, 2, \cdots, L-1 $,
            -: Construct the drift homotopy dynamics (10);
            -: Solve for $ \psi_{l}^{n+1, i} $ in the equation (17) with the initial guess $ \hat{x}_{n+1, l}^{(i)} $;
            -: Generate the sample $ \hat{x}_{n+1, l+1}^{(i)} $ through $ \exp\big(- \frac{\xi^{T} \xi}{2}\big) J_{\psi_l^{n+1, i}}^{-1} $;
        end for
    end for
  The particles $ \{x_{n+1}^{(i)}\}_{i = 1}^{N_p} : = \{\hat{x}_{n+1, L}^{(i)}\}_{i = 1}^{N_p} $ provide an empirical distribution for the filtering density $ p(X_{n+1} | Y_{1:n+1}) $
end while
Algorithm: Drift homotopy implicit particle filter (DHIPF)
Initialize the particle cloud $ \{x_0^{(i)}\}_{i=1}^{N_p} $, the number of drift homotopy levels $ L $ with the intermediate drift function $ b $ and the constant sequence $ \{\beta_l\}_{l=0}^{L} $, and the reference random variable $ \xi $ for the implicit particle filter procedure.
while $ n =0, 1, 2, \cdots $, do
    for: particles $ i = 1, 2, \cdots, N_p $,
        for: drift homotopy levels $ l = 0, 1, 2, \cdots, L-1 $,
            -: Construct the drift homotopy dynamics (10);
            -: Solve for $ \psi_{l}^{n+1, i} $ in the equation (17) with the initial guess $ \hat{x}_{n+1, l}^{(i)} $;
            -: Generate the sample $ \hat{x}_{n+1, l+1}^{(i)} $ through $ \exp\big(- \frac{\xi^{T} \xi}{2}\big) J_{\psi_l^{n+1, i}}^{-1} $;
        end for
    end for
  The particles $ \{x_{n+1}^{(i)}\}_{i = 1}^{N_p} : = \{\hat{x}_{n+1, L}^{(i)}\}_{i = 1}^{N_p} $ provide an empirical distribution for the filtering density $ p(X_{n+1} | Y_{1:n+1}) $
end while
Table 2.  Example 1. Performance comparison for Case 1
APF EnKF IPF DHPF DHIPF
CPU Time $ 9.703 $ $ 0.365625 $ $ 0.0938 $ $ 48.563 $ $ \bf{0.312} $
MSE $ 2.03 E-3 $ $ 5.22 E-3 $ $ 1.10 E-3 $ $ 5.92 E-4 $ $ \bf{4.60 E-4} $
APF EnKF IPF DHPF DHIPF
CPU Time $ 9.703 $ $ 0.365625 $ $ 0.0938 $ $ 48.563 $ $ \bf{0.312} $
MSE $ 2.03 E-3 $ $ 5.22 E-3 $ $ 1.10 E-3 $ $ 5.92 E-4 $ $ \bf{4.60 E-4} $
Table 3.  Example 1. Performance comparison for Case 2
APF EnKF IPF DHPF DHIPF
CPU Time $ 9.391 $ $ 0.578 $ $ 0.109 $ $ 43.344 $ $ \bf{0.297} $
MSE $ 6.29 E-1 $ $ 1.36 $ $ 5.28 E-3 $ $ 1.78 E-3 $ $ \bf{1.08 E-3} $
APF EnKF IPF DHPF DHIPF
CPU Time $ 9.391 $ $ 0.578 $ $ 0.109 $ $ 43.344 $ $ \bf{0.297} $
MSE $ 6.29 E-1 $ $ 1.36 $ $ 5.28 E-3 $ $ 1.78 E-3 $ $ \bf{1.08 E-3} $
Table 4.  Example 1. Performance comparison for Case 3
APF EnKF IPF DHPF DHIPF
CPU Time $ 9.563 $ $ 0.453 $ $ 0.156 $ $ 50.188 $ $ \bf{0.422} $
MSE $ 1.80 $ $ 1.99 $ $ 5.02 E-3 $ $ 1.35 E-2 $ $ \bf{1.59 E-3} $
APF EnKF IPF DHPF DHIPF
CPU Time $ 9.563 $ $ 0.453 $ $ 0.156 $ $ 50.188 $ $ \bf{0.422} $
MSE $ 1.80 $ $ 1.99 $ $ 5.02 E-3 $ $ 1.35 E-2 $ $ \bf{1.59 E-3} $
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