Skip to main content
Springer Nature Link
Log in

Splitting-up Spectral Method for Nonlinear Filtering Problems with Correlation Noises

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we study nonlinear filtering problems via solving their corresponding Zakai equations. Using the splitting-up technique, we approximate the Zakai equation with two equations consisting of a first-order stochastic partial differential equation and a deterministic second-order partial differential equation. For the splitting-up equations, we use a spectral Galerkin method for the spatial discretization and a finite difference scheme for the temporal discretization. The main results are an error estimate for the semi-discretized scheme with respect to the spatial variable, and an error estimate for the full discretized scheme. To improve the numerical performance, we apply an adaptive technique to accurately locate the support domain of the solution in each time iteration. Finally, we present numerical experiments to demonstrate our theoretical analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Data Availability

Enquiries about data availability should be directed to the authors.

References

  1. Ahmed, N.U., Radaideh, S.M.: A powerful numerical technique solving Zakai equation for nonlinear filtering. Dynam. Control 7(3), 293–308 (1997). https://doi.org/10.1023/A:1008206719489

    Article  MathSciNet  MATH  Google Scholar 

  2. Bain, A., Crisan, D.: Fundamentals of stochastic filtering, Stochastic Modelling and Applied Probability, vol. 60. Springer, New York (2009). https://doi.org/10.1007/978-0-387-76896-010.1007/978-0-387-76896-0

    Book  MATH  Google Scholar 

  3. Baker, S., Poskar, H., Schreiber, F., Junker, B.: An improved constraint filtering technique for inferring hidden states and parameters of a biological model. Bioinformatics (Oxford, England) 29, 1052–1059 (2013). https://doi.org/10.1093/bioinformatics/btt097

    Article  Google Scholar 

  4. Bao, F., Cao, Y., Webster, C., Zhang, G.: A hybrid sparse-grid approach for nonlinear filtering problems based on adaptive-domain of the Zakai equation approximations. SIAM/ASA J. Uncertain. Quantif. 2(1), 784–804 (2014). https://doi.org/10.1137/140952910

    Article  MathSciNet  MATH  Google Scholar 

  5. Beneš, V.E.: Exact finite-dimensional filters for certain diffusions with nonlinear drift. Stochastics 5(1–2), 65–92 (1981). https://doi.org/10.1080/17442508108833174

    Article  MathSciNet  MATH  Google Scholar 

  6. Bensoussan, A.: On a general class of stochastic partial differential equations. Stochastic Hydrology & Hydraulics 1(4), 297–302 (1987)

    Article  MATH  Google Scholar 

  7. Bensoussan, A., Glowinski, R., Răşcanu, A.: Approximation of the Zakai equation by the splitting up method. SIAM J. Control. Optim. 28(6), 1420–1431 (1990). https://doi.org/10.1137/0328074

    Article  MathSciNet  MATH  Google Scholar 

  8. Bensoussan, A., Keppo, J., Sethi, S.: Optimal consumption and portfolio decisions with partially observed real prices. Math. Financ. 19, 215–236 (2009). https://doi.org/10.1111/j.1467-9965.2009.00362.x

    Article  MathSciNet  MATH  Google Scholar 

  9. Blömker, D., Schillings, C., Wacker, P., Weissmann, S.: Well posedness and convergence analysis of the ensemble Kalman inversion. Inverse Prob. 35(8), 085007, 32 (2019). https://doi.org/10.1088/1361-6420/ab149c

    Article  MathSciNet  MATH  Google Scholar 

  10. Bolic, M., Djuric, P.M.: Sangjin Hong: Resampling algorithms and architectures for distributed particle filters. IEEE Trans. Signal Process. 53(7), 2442–2450 (2005). https://doi.org/10.1109/TSP.2005.849185

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, D., Yu, Y., Xu, L., Liu, X.: Kalman filtering for discrete stochastic systems with multiplicative noises and random two-step sensor delays. Discrete Dyn. Nat. Soc. 2015, 1–11 (2015). https://doi.org/10.1155/2015/809734

  12. Chen, J., Zhu, Q., Liu, Y.: Modified Kalman filtering based multi-step-length gradient iterative algorithm for ARX models with random missing outputs. Automatica J. IFAC 118, 109034, 8 (2020). https://doi.org/10.1016/j.automatica.2020.109034

    Article  MathSciNet  MATH  Google Scholar 

  13. Crisan, D.: Exact rates of convergence for a branching particle approximation to the solution of the Zakai equation. Ann. Probab. 31(2), 693–718 (2003). https://doi.org/10.1214/aop/1048516533

    Article  MathSciNet  MATH  Google Scholar 

  14. Crisan, D., Gaines, J., Lyons, T.: Convergence of a branching particle method to the solution of the Zakai equation. SIAM J. Appl. Math. 58(5), 1568–1590 (1998). https://doi.org/10.1137/S0036139996307371

    Article  MathSciNet  MATH  Google Scholar 

  15. Crisan, D., Obanubi, O.: Particle filters with random resampling times. Stochastic Process. Appl. 122(4), 1332–1368 (2012). https://doi.org/10.1016/j.spa.2011.12.012

    Article  MathSciNet  MATH  Google Scholar 

  16. Da Prato, G., Kunstmann, P.C., Lasiecka, I., Lunardi, A., Schnaubelt, R., Weis, L.: Functional analytic methods for evolution equations, Lecture Notes in Mathematics, vol. 1855. Springer-Verlag, Berlin (2004). https://doi.org/10.1007/b100449. Edited by M. Iannelli, R. Nagel and S. Piazzera

  17. Djogatović, M.S., Stanojević, M.J., Mladenović, N.: A variable neighborhood search particle filter for bearings-only target tracking. Computers & Operations Research 52, 192 – 202 (2014). https://doi.org/10.1016/j.cor.2013.11.013. http://www.sciencedirect.com/science/article/pii/S030505481300333X. Recent advances in Variable neighborhood search

  18. Dong, W., Luo, X., Yau, S.S.T.: Solving nonlinear filtering problems in real time by Legendre Galerkin spectral method. IEEE Trans. Automat. Control 66(4), 1559–1572 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. Weinan, E., Han, J., Jentzen, A.: Algorithms for solving high dimensional PDEs: from nonlinear monte carlo to machine learning. Nonlinearity 35(1), 278–310 (2021). https://doi.org/10.1088/1361-6544/ac337f

    Article  MathSciNet  MATH  Google Scholar 

  20. Elliott, R.J., Glowinski, R.: Approximations to solutions of the Zakai filtering equation. Stochastic Anal. Appl. 7(2), 145–168 (1989). https://doi.org/10.1080/07362998908809174

    Article  MathSciNet  MATH  Google Scholar 

  21. Evensen, G.: Sequential data assimilation with a nonlinear quasi-geostrophic model using monte carlo methods to forecast error statistics. J. Geophys. Res. 99, 10143–10162 (1994)

    Article  Google Scholar 

  22. Evensen, G., et al.: Data assimilation: the ensemble Kalman filter, vol. 2. Springer (2009)

  23. Florchinger, P., Le Gland, F.: Time-discretization of the Zakai equation for diffusion processes observed in correlated noise. Stochastics Stochastics Rep. 35(4), 233–256 (1991). https://doi.org/10.1080/17442509108833704

    Article  MathSciNet  MATH  Google Scholar 

  24. Frey, R., Schmidt, T., Xu, L.: On Galerkin approximations for the Zakai equation with diffusive and point process observations. SIAM J. Numer. Anal. 51(4), 2036–2062 (2013). https://doi.org/10.1137/110837395

    Article  MathSciNet  MATH  Google Scholar 

  25. Gobet, E., Pagès, G., Pham, H., Printems, J.: Discretization and simulation of the Zakai equation. SIAM J. Numer. Anal. 44(6), 2505–2538 (2006). https://doi.org/10.1137/050623140

    Article  MathSciNet  MATH  Google Scholar 

  26. Gottlieb, D.: Numerical analysis of spectral methods : theory and applications. Society For Industrial And Applied Mathematics (1977)

  27. Gyöngy, I., Krylov, N.: On the splitting-up method and stochastic partial differential equations. Ann. Probab. 31(2), 564–591 (2003). https://doi.org/10.1214/aop/1048516528

    Article  MathSciNet  MATH  Google Scholar 

  28. Hairer, M., Stuart, A., Voss, J.: Signal processing problems on function space: Bayesian formulation, stochastic PDEs and effective MCMC methods. In: The Oxford handbook of nonlinear filtering, pp. 833–873. Oxford Univ. Press, Oxford (2011)

  29. Han, X., Li, J., Xiu, D.: Error analysis for numerical formulation of particle filter. Discrete Contin. Dyn. Syst. Ser. B 20(5), 1337–1354 (2015). https://doi.org/10.3934/dcdsb.2015.20.1337

    Article  MathSciNet  MATH  Google Scholar 

  30. Hu, J., Hu, X.: Nonlinear filtering in target tracking using cooperative mobile sensors. Automatica 46(12), 2041–2046 (2010). https://doi.org/10.1016/j.automatica.2010.08.016

    Article  MathSciNet  MATH  Google Scholar 

  31. Inoue, A., Nakano, Y., Anh, V.: Linear filtering of systems with memory and application to finance. J. Appl. Math. Stoch. Anal. 2006, 1–26 (2006). https://doi.org/10.1155/JAMSA/2006/53104

  32. Ito, K.: Approximation of the zakai equation for nonlinear filtering. Siam Journal on Control and Optimization - SIAM J CONTR OPTIMIZAT 34, 620–634 (1996). https://doi.org/10.1137/S0363012993254783

    Article  MathSciNet  MATH  Google Scholar 

  33. Jazwinski, A.H.: Stochastic Processes and Filtering Theory, Mathematics in Science and Engineering, vol. 64. Elsevier (1970). https://doi.org/10.1016/S0076-5392(09)60378-7

  34. Julier, S., Jr., L., : On kalman filter with nonlinear equality constraints. Signal Processing, IEEE Transactions on 55, 2774–2784 (2007). https://doi.org/10.1109/TSP.2007.893949

  35. Kai, X., Liangdong, L., Yiwu, L.: Robust extended Kalman filtering for nonlinear systems with multiplicative noises. Optimal Control Appl. Methods 32(1), 47–63 (2011). https://doi.org/10.1002/oca.928

    Article  MathSciNet  MATH  Google Scholar 

  36. Kalman, R.E., Bucy, R.S.: New results in linear filtering and prediction theory. Trans. ASME Ser. D. J. Basic Engrg. 83, 95–108 (1961)

    Article  MathSciNet  Google Scholar 

  37. Kim, S., Deshpande, V.M., Bhattacharya, R.: Robust Kalman filtering with probabilistic uncertainty in system parameters. IEEE Control Syst. Lett. 5(1), 295–300 (2021)

    Article  MathSciNet  Google Scholar 

  38. Le Gland, F.: Splitting-up approximation for SPDEs and SDEs with application to nonlinear filtering. In: Stochastic partial differential equations and their applications (Charlotte, NC, 1991), Lect. Notes Control Inf. Sci., vol. 176, pp. 177–187. Springer, Berlin (1992). https://doi.org/10.1007/BFb0007332

  39. Li, J., Hu, J., Chen, D., Wu, Z.: Distributed extended Kalman filtering for state-saturated nonlinear systems subject to randomly occurring cyberattacks with uncertain probabilities. Adv. Difference Equ. 2020, 437 (2020). https://doi.org/10.1186/s13662-020-02896-3

  40. Llopis, F.P., Kantas, N., Beskos, A., Jasra, A.: Particle filtering for stochastic Navier-Stokes signal observed with linear additive noise. SIAM J. Sci. Comput. 40(3), A1544–A1565 (2018). https://doi.org/10.1137/17M1151900

    Article  MathSciNet  MATH  Google Scholar 

  41. Luo, X., Yau, S.S.T.: Complete real time solution of the general nonlinear filtering problem without memory. IEEE Trans. Automat. Control 58(10), 2563–2578 (2013). https://doi.org/10.1109/TAC.2013.2264552

    Article  MathSciNet  MATH  Google Scholar 

  42. Luo, X., Yau, S.S.T.: Hermite spectral method to 1-D forward Kolmogorov equation and its application to nonlinear filtering problems. IEEE Trans. Automat. Control 58(10), 2495–2507 (2013). https://doi.org/10.1109/TAC.2013.2259975

    Article  MathSciNet  MATH  Google Scholar 

  43. Majda, A.J., Tong, X.T.: Performance of ensemble Kalman filters in large dimensions. Comm. Pure Appl. Math. 71(5), 892–937 (2018). https://doi.org/10.1002/cpa.21722

    Article  MathSciNet  MATH  Google Scholar 

  44. Masreliez, C., Martin, R.: Robust bayesian estimation for the linear model and robustifying the kalman filter. IEEE Trans. Autom. Control 22(3), 361–371 (1977). https://doi.org/10.1109/TAC.1977.1101538

    Article  MathSciNet  MATH  Google Scholar 

  45. Nagata, M., Sawaragi, Y.: Error analysis of the Schmidt-Kalman filter. Internat. J. Systems Sci. 7(7), 769–778 (1976). https://doi.org/10.1080/00207727608941963

    Article  MathSciNet  MATH  Google Scholar 

  46. Nakano, Y.: Kernel-based collocation methods for Zakai equations. Stoch. Partial Differ. Equ. Anal. Comput. 7(3), 476–494 (2019). https://doi.org/10.1007/s40072-019-00132-y

    Article  MathSciNet  MATH  Google Scholar 

  47. Özbek, L.: An adaptive extended Kalman filtering approach to nonlinear dynamic gene regulatory networks via short gene expression time series. Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat. 69(2), 1205–1214 (2020). https://doi.org/10.31801/cfsuasmas.749624

    Article  MathSciNet  MATH  Google Scholar 

  48. Pardoux, E.: Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3(2), 127–167 (1979). https://doi.org/10.1080/17442507908833142

    Article  MathSciNet  MATH  Google Scholar 

  49. Pardoux, E.: Équations du filtrage non linéaire de la prédiction et du lissage. Stochastics An International Journal of Probability and Stochastic Processes 6, 193–231 (1982). https://doi.org/10.1080/17442508208833204

    Article  MATH  Google Scholar 

  50. Ruzayqat, H.M., Jasra, A.: Unbiased estimation of the solution to Zakai’s equation. Monte Carlo Methods Appl. 26(2), 113–129 (2020). https://doi.org/10.1515/mcma-2020-2061

    Article  MathSciNet  MATH  Google Scholar 

  51. Shen, J., Tang, T., Wang, L.L.: Spectral methods: algorithms, analysis and applications, vol. 41. Springer Science & Business Media (2011)

  52. Song, W., Wang, Z., Wang, J., Alsaadi, F.E., Shan, J.: Particle filtering for nonlinear/non-Gaussian systems with energy harvesting sensors subject to randomly occurring sensor saturations. IEEE Trans. Signal Process. 69, 15–27 (2021). https://doi.org/10.1109/TSP.2020.3042951

    Article  MathSciNet  MATH  Google Scholar 

  53. Song, W., Wang, Z., Wang, J., Shan, J.: Particle filtering for a class of cyber-physical systems under Round-Robin protocol subject to randomly occurring deception attacks. Inform. Sci. 544, 298–307 (2021). https://doi.org/10.1016/j.ins.2020.07.047

    Article  MathSciNet  MATH  Google Scholar 

  54. Thomée, V.: Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, vol. 25, second edn. Springer-Verlag, Berlin (2006)

  55. Zakai, M.: On the optimal filtering of diffusion processes. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 11(3), 230–243 (1969). https://doi.org/10.1007/BF00536382

    Article  MathSciNet  MATH  Google Scholar 

  56. Zhang, G., Lan, J., Zhang, L., He, F., Li, S.: Filtering in pairwise markov model with student’s t non-stationary noise with application to target tracking. IEEE Trans. Signal Process. 69, 1627–1641 (2021). https://doi.org/10.1109/TSP.2021.3062170

    Article  MathSciNet  MATH  Google Scholar 

  57. Zou, Y., Ghanem, R.: A multiscale data assimilation with the ensemble Kalman filter. Multiscale Model. Simul. 3(1), 131–150 (2004/05). https://doi.org/10.1137/030601168

Download references

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130012, China

    Fengshan Zhang, Yongkui Zou, Shimin Chai & Ran Zhang

  2. Department of Mathematics and Statistics, Auburn University, AL 36840, Auburn, USA

    Yanzhao Cao

Authors
  1. Fengshan Zhang
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Yongkui Zou
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Shimin Chai
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Ran Zhang
    View author publications

    You can also search for this author inPubMed Google Scholar

  5. Yanzhao Cao
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Yanzhao Cao.

Ethics declarations

Competing Interests

The authors have not disclosed any competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research of Yongkui Zou and Ran Zhang is partly supported by the National Key R &D Program (2020YFA0714100, 2020YFA0713601), NSFC (12171199, 11971198), Jilin Provincial Department of science and technology (20210201015GX), and the Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education at Jilin University. Research of Yanzhao Cao is partially supported by U.S. Department of Energy under grant number DE-SC0022253.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, F., Zou, Y., Chai, S. et al. Splitting-up Spectral Method for Nonlinear Filtering Problems with Correlation Noises. J Sci Comput 93, 25 (2022). https://doi.org/10.1007/s10915-022-01994-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-022-01994-6

Keywords