September  2021, 3(3): 647-675. doi: 10.3934/fods.2021025

Stability of non-linear filter for deterministic dynamics

1. 

Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India-400076

2. 

International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore, India-560089

* Corresponding author: Amit Apte

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

Fund Project: The work of ASR was partially supported by Infosys Foundation Excellence Program of ICTS. AA acknowledges support from US Office of Naval Research under grant N00014-18-1-2204. Authors acknowledge the support of the Department of Atomic Energy, Government of India, under projects no.12-R & D-TFR-5.10-1100, and no.RTI4001

This papers shows that nonlinear filter in the case of deterministic dynamics is stable with respect to the initial conditions under the conditions that observations are sufficiently rich, both in the context of continuous and discrete time filters. Earlier works on the stability of the nonlinear filters are in the context of stochastic dynamics and assume conditions like compact state space or time independent observation model, whereas we prove filter stability for deterministic dynamics with more general assumptions on the state space and observation process. We give several examples of systems that satisfy these assumptions. We also show that the asymptotic structure of the filtering distribution is related to the dynamical properties of the signal.

Citation: Anugu Sumith Reddy, Amit Apte. Stability of non-linear filter for deterministic dynamics. Foundations of Data Science, 2021, 3 (3) : 647-675. doi: 10.3934/fods.2021025
References:
[1]

B. D. O. Anderson and J. B. Moore, New results in linear system stability, SIAM J. Control, 7 (1969), 398-414.  doi: 10.1137/0307029.

[2]

A. Apte, M. Hairer, A. M. Stuart and J. Voss, Sampling the posterior: An approach to non-Gaussian data assimilation, Phys. D, 230 (2007), 50–64. doi: 10.1016/j.physd.2006.06.009.

[3]

M. Asch, M. Bocquet and M. Nodet, Data Assimilation. Methods, Algorithms, and Applications, Fundamentals of Algorithms, 11, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974546.pt1.

[4]

R. Atar, Exponential stability for nonlinear filtering of diffusion processes in a noncompact domain, Ann. Probab., 26 (1998), 1552-1574.  doi: 10.1214/aop/1022855873.

[5]

R. Atar and O. Zeitouni, Exponential stability for nonlinear filtering, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 697-725.  doi: 10.1016/S0246-0203(97)80110-0.

[6]

A. Bain and D. Crisan, Fundamentals of Stochastic Filtering, Stochastic Modelling and Applied Probability, 60, Springer, New York, 2009. doi: 10.1007/978-0-387-76896-0.

[7]

L. Barreira and Y. Pesin, Introduction to Smooth Ergodic Theory, Graduate Studies in Mathematics, 148, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/gsm/148.

[8]

P. Billingsley, Convergence of Probability Measures, 2$^{nd}$ edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons Inc., New York, 1999. doi: 10.1002/9780470316962.

[9]

A. N. Bishop and P. Del Moral, On the stability of Kalman–Bucy diffusion processes, SIAM J. Control Optim., 55 (2017), 4015-4047.  doi: 10.1137/16M1102707.

[10]

M. BocquetK. S. GurumoorthyA. ApteA. CarrassiC. Grudzien and C. K. R. T. Jones, Degenerate Kalman filter error covariances and their convergence onto the unstable subspace, SIAM/ASA J. Uncertain. Quantif., 5 (2017), 304-333.  doi: 10.1137/16M1068712.

[11]

A. Budhiraja, Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear filter, Ann. Inst. H. Poincaré Probab. Statist., 39 (2003), 919-941.  doi: 10.1016/S0246-0203(03)00022-0.

[12]

A. Budhiraja and D. Ocone, Exponential stability in discrete-time filtering for non-ergodic signals, Stochastic Process. Appl., 82 (1999), 245-257.  doi: 10.1016/S0304-4149(99)00032-0.

[13]

A. Budhiraja and D. Ocone, Exponential stability of discrete-time filters for bounded observation noise, Systems Control Lett., 30 (1997), 185-193.  doi: 10.1016/S0167-6911(97)00012-1.

[14]

A. Carrassi, M. Bocquet, L. Bertino and G. Evensen, Data assimilation in the geosciences: An overview of methods, issues, and perspectives, WIREs Clim. Change, 9 (2018). doi: 10.1002/wcc.535.

[15]

F. Cérou, Long Time Asymptotics for Some Dynamical Noise Free Non-Linear Filtering Problems, New Cases, Research Report, RR-2541, INRIA, 1995. Available from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.57.1616&rep=rep1&type=pdf.

[16]

F. Cérou, Long time behavior for some dynamical noise free nonlinear filtering problems, SIAM J. Control Optim., 38 (2000), 1086-1101.  doi: 10.1137/S0363012995290124.

[17]

P. Chigansky, Stability of nonlinear filters: A survey, Mini-Course Lecture Notes, Petropolis, Brazil, 2006. Available from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.169.1819&rep=rep1&type=pdf.

[18]

P. Chigansky, R. Liptser and R. Van Handel, Intrinsic methods in filter stability, in Oxford Handbook of Nonlinear Filtering, Oxford Univ. Press, Oxford, 2011, 319-351.

[19]

J. M. C. ClarkD. L. Ocone and C. Coumarbatch, Relative entropy and error bounds for filtering of Markov processes, Math. Control Signals Systems, 12 (1999), 346-360.  doi: 10.1007/PL00009856.

[20]

A. D'Aristotile, P. Diaconis and D. Freedman, On merging of probabilities, Sankhyā Ser. A, 50 (1988), 363–380.

[21]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075.  doi: 10.1088/0951-7715/6/6/014.

[22]

H. E. Gollwitzer, A note on a functional inequality, Proc. Amer. Math. Soc., 23 (1969), 642-647.  doi: 10.1090/S0002-9939-1969-0247016-9.

[23]

K. S. GurumoorthyC. GrudzienA. ApteA. Carrassi and C. K. R. T. Jones, Rank deficiency of Kalman error covariance matrices in linear time-varying system with deterministic evolution, SIAM J. Control Optim., 55 (2017), 741-759.  doi: 10.1137/15M1025839.

[24]

G. Kallianpur, Stochastic Filtering Theory, Applications of Mathematics, 13, Springer-Verlag, New York-Berlin, 1980. doi: 10.1007/978-1-4757-6592-2.

[25]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.

[26]

D. T. B. KellyK. J. H. Law and A. M. Stuart, Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time, Nonlinearity, 27 (2014), 2579-2604.  doi: 10.1088/0951-7715/27/10/2579.

[27]

Koro, Uniform expansivity., Available from: https://planetmath.org/UniformExpansivity.

[28]

K. Law, A. Stuart and K. Zygalakis, Data Assimilation. A Mathematical Introduction, Texts in Applied Mathematics, 62, Springer, Cham, 2015. doi: 10.1007/978-3-319-20325-6.

[29]

M. Ledoux, Isoperimetry and Gaussian analysis, in Lectures on Probability Theory and Statistics, Lecture Notes in Math., 1648, Springer, Berlin, 1996,165–294. doi: 10.1007/BFb0095676.

[30]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

[31]

E. N. Lorenz, Predictability - A problem partly solved, in Predictability of Weather and Climate, Cambridge University Press, 2006, 40-58. doi: 10.1017/CBO9780511617652.004.

[32]

C. McDonald and S. Yüksel, Converse results on filter stability criteria and stochastic non-linear observability, preprint, arXiv: 1812.01772.

[33]

B. Ni and Q. Zhang, Stability of the Kalman filter for continuous time output error systems, Systems Control Lett., 94 (2016), 172-180.  doi: 10.1016/j.sysconle.2016.06.006.

[34]

D. Ocone and E. Pardoux, Asymptotic stability of the optimal filter with respect to its initial condition, SIAM J. Control Optim., 34 (1996), 226-243.  doi: 10.1137/S0363012993256617.

[35]

D. L. Ocone, Asymptotic stability of Beneš filters, Stochastic Anal. Appl., 17 (1999), 1053-1074.  doi: 10.1080/07362999908809648.

[36]

T. N. Palmer, Stochastic weather and climate models, Nature Reviews Physics, 1 (2019), 463-471.  doi: 10.1038/s42254-019-0062-2.

[37]

A. S. Reddy, A. Apte and S. Vadlamani, Asymptotic properties of linear filter for deterministic processes, Systems Control Lett., 139 (2020), 8 pp. doi: 10.1016/j.sysconle.2020.104676.

[38] S. Reich and C. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, New York, 2015.  doi: 10.1017/CBO9781107706804.
[39]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-1947-5.

[40]

R. van Handel, Filtering, Stability, and Robustness, Ph.D thesis, California Institute of Technology, 2007.

[41]

R. van Handel, Nonlinear filtering and systems theory, in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS Semi-Plenary Paper), 2010. Available from: https://web.math.princeton.edu/rvan/mtns10.pdf.

[42]

R. van Handel, Observability and nonlinear filtering, Probab. Theory Related Fields, 145 (2009), 35-74.  doi: 10.1007/s00440-008-0161-y.

[43]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[44]

D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511813658.

[45]

J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford Graduate Texts in Mathematics, 18, Oxford University Press, Oxford, 2008.

show all references

References:
[1]

B. D. O. Anderson and J. B. Moore, New results in linear system stability, SIAM J. Control, 7 (1969), 398-414.  doi: 10.1137/0307029.

[2]

A. Apte, M. Hairer, A. M. Stuart and J. Voss, Sampling the posterior: An approach to non-Gaussian data assimilation, Phys. D, 230 (2007), 50–64. doi: 10.1016/j.physd.2006.06.009.

[3]

M. Asch, M. Bocquet and M. Nodet, Data Assimilation. Methods, Algorithms, and Applications, Fundamentals of Algorithms, 11, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974546.pt1.

[4]

R. Atar, Exponential stability for nonlinear filtering of diffusion processes in a noncompact domain, Ann. Probab., 26 (1998), 1552-1574.  doi: 10.1214/aop/1022855873.

[5]

R. Atar and O. Zeitouni, Exponential stability for nonlinear filtering, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 697-725.  doi: 10.1016/S0246-0203(97)80110-0.

[6]

A. Bain and D. Crisan, Fundamentals of Stochastic Filtering, Stochastic Modelling and Applied Probability, 60, Springer, New York, 2009. doi: 10.1007/978-0-387-76896-0.

[7]

L. Barreira and Y. Pesin, Introduction to Smooth Ergodic Theory, Graduate Studies in Mathematics, 148, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/gsm/148.

[8]

P. Billingsley, Convergence of Probability Measures, 2$^{nd}$ edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons Inc., New York, 1999. doi: 10.1002/9780470316962.

[9]

A. N. Bishop and P. Del Moral, On the stability of Kalman–Bucy diffusion processes, SIAM J. Control Optim., 55 (2017), 4015-4047.  doi: 10.1137/16M1102707.

[10]

M. BocquetK. S. GurumoorthyA. ApteA. CarrassiC. Grudzien and C. K. R. T. Jones, Degenerate Kalman filter error covariances and their convergence onto the unstable subspace, SIAM/ASA J. Uncertain. Quantif., 5 (2017), 304-333.  doi: 10.1137/16M1068712.

[11]

A. Budhiraja, Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear filter, Ann. Inst. H. Poincaré Probab. Statist., 39 (2003), 919-941.  doi: 10.1016/S0246-0203(03)00022-0.

[12]

A. Budhiraja and D. Ocone, Exponential stability in discrete-time filtering for non-ergodic signals, Stochastic Process. Appl., 82 (1999), 245-257.  doi: 10.1016/S0304-4149(99)00032-0.

[13]

A. Budhiraja and D. Ocone, Exponential stability of discrete-time filters for bounded observation noise, Systems Control Lett., 30 (1997), 185-193.  doi: 10.1016/S0167-6911(97)00012-1.

[14]

A. Carrassi, M. Bocquet, L. Bertino and G. Evensen, Data assimilation in the geosciences: An overview of methods, issues, and perspectives, WIREs Clim. Change, 9 (2018). doi: 10.1002/wcc.535.

[15]

F. Cérou, Long Time Asymptotics for Some Dynamical Noise Free Non-Linear Filtering Problems, New Cases, Research Report, RR-2541, INRIA, 1995. Available from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.57.1616&rep=rep1&type=pdf.

[16]

F. Cérou, Long time behavior for some dynamical noise free nonlinear filtering problems, SIAM J. Control Optim., 38 (2000), 1086-1101.  doi: 10.1137/S0363012995290124.

[17]

P. Chigansky, Stability of nonlinear filters: A survey, Mini-Course Lecture Notes, Petropolis, Brazil, 2006. Available from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.169.1819&rep=rep1&type=pdf.

[18]

P. Chigansky, R. Liptser and R. Van Handel, Intrinsic methods in filter stability, in Oxford Handbook of Nonlinear Filtering, Oxford Univ. Press, Oxford, 2011, 319-351.

[19]

J. M. C. ClarkD. L. Ocone and C. Coumarbatch, Relative entropy and error bounds for filtering of Markov processes, Math. Control Signals Systems, 12 (1999), 346-360.  doi: 10.1007/PL00009856.

[20]

A. D'Aristotile, P. Diaconis and D. Freedman, On merging of probabilities, Sankhyā Ser. A, 50 (1988), 363–380.

[21]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075.  doi: 10.1088/0951-7715/6/6/014.

[22]

H. E. Gollwitzer, A note on a functional inequality, Proc. Amer. Math. Soc., 23 (1969), 642-647.  doi: 10.1090/S0002-9939-1969-0247016-9.

[23]

K. S. GurumoorthyC. GrudzienA. ApteA. Carrassi and C. K. R. T. Jones, Rank deficiency of Kalman error covariance matrices in linear time-varying system with deterministic evolution, SIAM J. Control Optim., 55 (2017), 741-759.  doi: 10.1137/15M1025839.

[24]

G. Kallianpur, Stochastic Filtering Theory, Applications of Mathematics, 13, Springer-Verlag, New York-Berlin, 1980. doi: 10.1007/978-1-4757-6592-2.

[25]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.

[26]

D. T. B. KellyK. J. H. Law and A. M. Stuart, Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time, Nonlinearity, 27 (2014), 2579-2604.  doi: 10.1088/0951-7715/27/10/2579.

[27]

Koro, Uniform expansivity., Available from: https://planetmath.org/UniformExpansivity.

[28]

K. Law, A. Stuart and K. Zygalakis, Data Assimilation. A Mathematical Introduction, Texts in Applied Mathematics, 62, Springer, Cham, 2015. doi: 10.1007/978-3-319-20325-6.

[29]

M. Ledoux, Isoperimetry and Gaussian analysis, in Lectures on Probability Theory and Statistics, Lecture Notes in Math., 1648, Springer, Berlin, 1996,165–294. doi: 10.1007/BFb0095676.

[30]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

[31]

E. N. Lorenz, Predictability - A problem partly solved, in Predictability of Weather and Climate, Cambridge University Press, 2006, 40-58. doi: 10.1017/CBO9780511617652.004.

[32]

C. McDonald and S. Yüksel, Converse results on filter stability criteria and stochastic non-linear observability, preprint, arXiv: 1812.01772.

[33]

B. Ni and Q. Zhang, Stability of the Kalman filter for continuous time output error systems, Systems Control Lett., 94 (2016), 172-180.  doi: 10.1016/j.sysconle.2016.06.006.

[34]

D. Ocone and E. Pardoux, Asymptotic stability of the optimal filter with respect to its initial condition, SIAM J. Control Optim., 34 (1996), 226-243.  doi: 10.1137/S0363012993256617.

[35]

D. L. Ocone, Asymptotic stability of Beneš filters, Stochastic Anal. Appl., 17 (1999), 1053-1074.  doi: 10.1080/07362999908809648.

[36]

T. N. Palmer, Stochastic weather and climate models, Nature Reviews Physics, 1 (2019), 463-471.  doi: 10.1038/s42254-019-0062-2.

[37]

A. S. Reddy, A. Apte and S. Vadlamani, Asymptotic properties of linear filter for deterministic processes, Systems Control Lett., 139 (2020), 8 pp. doi: 10.1016/j.sysconle.2020.104676.

[38] S. Reich and C. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, New York, 2015.  doi: 10.1017/CBO9781107706804.
[39]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4757-1947-5.

[40]

R. van Handel, Filtering, Stability, and Robustness, Ph.D thesis, California Institute of Technology, 2007.

[41]

R. van Handel, Nonlinear filtering and systems theory, in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS Semi-Plenary Paper), 2010. Available from: https://web.math.princeton.edu/rvan/mtns10.pdf.

[42]

R. van Handel, Observability and nonlinear filtering, Probab. Theory Related Fields, 145 (2009), 35-74.  doi: 10.1007/s00440-008-0161-y.

[43]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[44]

D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511813658.

[45]

J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford Graduate Texts in Mathematics, 18, Oxford University Press, Oxford, 2008.

Figure 1.  Dependence of $ \frac{D_N(x, y)}{\sum_{i = 0}^N \rho_{i\tau}} $ vs $ t = N\tau $ with $ \tau = 0.01 $ for $ 100 $ samples. We have $ t = N\tau $ with $ \tau = 0.01 $. We chose $ 100 $ different pairs of $ (x, y) $ for five different choices of $ \rho_t = 1000, \; t+1000, \; \log(t+1000); t^2+1000; t^3 + 1000 $. The initial conditions for the samples are randomly chosen from uniform distribution on $ [-10, 10]^{p} $ where the dimension $ p = 3 $ for Lorenz 63 model (left panel) and $ p = 36 $ for the Lorenz 96 model (right panel). The insets show the plots for large $ t $. (Note that the Lyapunov time scale for both these models is $ O(1) $.)
[1]

Xin Li, Feng Bao, Kyle Gallivan. A drift homotopy implicit particle filter method for nonlinear filtering problems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 727-746. doi: 10.3934/dcdss.2021097

[2]

Marc Bocquet, Julien Brajard, Alberto Carrassi, Laurent Bertino. Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization. Foundations of Data Science, 2020, 2 (1) : 55-80. doi: 10.3934/fods.2020004

[3]

Alexandre J. Chorin, Fei Lu, Robert N. Miller, Matthias Morzfeld, Xuemin Tu. Sampling, feasibility, and priors in data assimilation. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4227-4246. doi: 10.3934/dcds.2016.36.4227

[4]

John Maclean, Elaine T. Spiller. A surrogate-based approach to nonlinear, non-Gaussian joint state-parameter data assimilation. Foundations of Data Science, 2021, 3 (3) : 589-614. doi: 10.3934/fods.2021019

[5]

Andrey V. Kremnev, Alexander S. Kuleshov. Nonlinear dynamics and stability of the skateboard. Discrete and Continuous Dynamical Systems - S, 2010, 3 (1) : 85-103. doi: 10.3934/dcdss.2010.3.85

[6]

Débora A. F. Albanez, Maicon J. Benvenutti. Continuous data assimilation algorithm for simplified Bardina model. Evolution Equations and Control Theory, 2018, 7 (1) : 33-52. doi: 10.3934/eect.2018002

[7]

Jochen Bröcker. Existence and uniqueness for variational data assimilation in continuous time. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021050

[8]

Jules Guillot, Emmanuel Frénod, Pierre Ailliot. Physics informed model error for data assimilation. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022059

[9]

Gerasimos G. Rigatos, Efthymia G. Rigatou, Jean Daniel Djida. Change detection in the dynamics of an intracellular protein synthesis model using nonlinear Kalman filtering. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1017-1035. doi: 10.3934/mbe.2015.12.1017

[10]

Z. G. Feng, Kok Lay Teo, N. U. Ahmed, Yulin Zhao, W. Y. Yan. Optimal fusion of sensor data for Kalman filtering. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 483-503. doi: 10.3934/dcds.2006.14.483

[11]

Issam S. Strub, Julie Percelay, Olli-Pekka Tossavainen, Alexandre M. Bayen. Comparison of two data assimilation algorithms for shallow water flows. Networks and Heterogeneous Media, 2009, 4 (2) : 409-430. doi: 10.3934/nhm.2009.4.409

[12]

Joshua Hudson, Michael Jolly. Numerical efficacy study of data assimilation for the 2D magnetohydrodynamic equations. Journal of Computational Dynamics, 2019, 6 (1) : 131-145. doi: 10.3934/jcd.2019006

[13]

Yuan Pei. Continuous data assimilation for the 3D primitive equations of the ocean. Communications on Pure and Applied Analysis, 2019, 18 (2) : 643-661. doi: 10.3934/cpaa.2019032

[14]

Juan Carlos De los Reyes, Estefanía Loayza-Romero. Total generalized variation regularization in data assimilation for Burgers' equation. Inverse Problems and Imaging, 2019, 13 (4) : 755-786. doi: 10.3934/ipi.2019035

[15]

Valerii Maltsev, Michael Pokojovy. On a parabolic-hyperbolic filter for multicolor image noise reduction. Evolution Equations and Control Theory, 2016, 5 (2) : 251-272. doi: 10.3934/eect.2016004

[16]

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

[17]

Fanze Kong, Qi Wang. Stability, free energy and dynamics of multi-spikes in the minimal Keller-Segel model. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2499-2523. doi: 10.3934/dcds.2021200

[18]

Renhai Wang, Bixiang Wang. Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2461-2493. doi: 10.3934/dcdsb.2020019

[19]

Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725

[20]

Matthew Gardner, Adam Larios, Leo G. Rebholz, Duygu Vargun, Camille Zerfas. Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations. Electronic Research Archive, 2021, 29 (3) : 2223-2247. doi: 10.3934/era.2020113

 Impact Factor: 

Metrics

  • PDF downloads (92)
  • HTML views (288)
  • Cited by (0)

Other articles
by authors

[Back to Top]