nach oben

Journal of Scientific Computing

Erschienen in:

01.10.2022

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

verfasst von: Fengshan Zhang, Yongkui Zou, Shimin Chai, Ran Zhang, Yanzhao Cao

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2022

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

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.

Vorheriger Artikel Inertial Version of Generalized Projected Reflected Gradient Method

Nächster Artikel Anisotropic a Posteriori Error Analysis for the Two-Step Backward Differentiation Formula for Parabolic Integro-Differential Equation

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

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:1008206719489MathSciNetCrossRefMATH

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

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/btt097CrossRef

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/10.1137/140952910MathSciNetCrossRefMATH

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

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

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/10.1137/0328074MathSciNetCrossRefMATH

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.xMathSciNetCrossRefMATH

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/ab149cMathSciNetCrossRefMATH

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.2005.849185MathSciNetCrossRefMATH

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.2020.109034MathSciNetCrossRefMATH

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/1048516533MathSciNetCrossRefMATH

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/S0036139996307371MathSciNetCrossRefMATH

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.2011.12.012MathSciNetCrossRefMATH

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)MathSciNetCrossRefMATH

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/ac337fMathSciNetCrossRefMATH

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/10.1080/07362998908809174MathSciNetCrossRefMATH

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)CrossRef

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/10.1080/17442509108833704MathSciNetCrossRefMATH

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/10.1137/110837395MathSciNetCrossRefMATH

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/10.1137/050623140MathSciNetCrossRefMATH

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/1048516528MathSciNetCrossRefMATH

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.2015.20.1337MathSciNetCrossRefMATH

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.2010.08.016MathSciNetCrossRefMATH

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/S0363012993254783MathSciNetCrossRefMATH

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.928MathSciNetCrossRefMATH

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)MathSciNetCrossRef

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)MathSciNetCrossRef

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/10.1137/17M1151900MathSciNetCrossRefMATH

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.2013.2264552MathSciNetCrossRefMATH

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.2013.2259975MathSciNetCrossRefMATH

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.21722MathSciNetCrossRefMATH

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.1977.1101538MathSciNetCrossRefMATH

45.

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

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-yMathSciNetCrossRefMATH

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.749624MathSciNetCrossRefMATH

48.

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

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/10.1080/17442508208833204CrossRefMATH

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-2020-2061MathSciNetCrossRefMATH

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.2020.3042951MathSciNetCrossRefMATH

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.2020.07.047MathSciNetCrossRefMATH

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/BF00536382MathSciNetCrossRefMATH

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.2021.3062170MathSciNetCrossRefMATH

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

Titel: Splitting-up Spectral Method for Nonlinear Filtering Problems with Correlation Noises
verfasst von: Fengshan Zhang
Yongkui Zou
Shimin Chai
Ran Zhang
Yanzhao Cao
Publikationsdatum: 01.10.2022
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2022
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-022-01994-6

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 1/2022

Troubled-Cell Indication Using K-means Clustering with Unified Parameters

Numerical Factorization of Propagation Operator for Hyperbolic Equations and Application to One-way, True Amplitude One-way Equations and Bremmer Series

High Order Asymptotic Preserving Hermite WENO Fast Sweeping Method for the Steady-State Transport Equations

A Lowest-Degree Conservative Finite Element Scheme for Incompressible Stokes Problems on General Triangulations

The Abstract Laplacian Tensor of a Hypergraph with Applications in Clustering

A Second-order Modified Ghost Fluid Method (2nd-MGFM) with Discontinuous Galerkin Method for 1-D compressible Multi-medium Problem with Cylindrical and Spherical Symmetry

Premium Partner