We consider a combined state and drift estimation problem for the linear stochastic heat equation. The infinite-dimensional Bayesian inference problem is formulated in terms of the Kalman–Bucy filter over an extended state space, and its long-time asymptotic properties are studied. Asymptotic posterior contraction rates in the unknown drift function are the main contribution of this paper. Such rates have been studied before for stationary non-parametric Bayesian inverse problems, and here we demonstrate the consistency of our time-dependent formulation with these previous results building upon scale separation and a slow manifold approximation.
Citation: |
Figure 1.
We display the time evolution of the bias
[1] |
R. Altmeyer and M. Reiß, Nonparametric estimation for linear SPDEs from local measurements, Annals of Appl. Probability, preprint, arXiv: 1903.06984.
![]() |
[2] |
K. Bergemann and S. Reich, An ensemble Kalman–Bucy filter for continuous data assimilation, Meteorolog. Zeitschrift, 21 (2012), 213-219.
doi: 10.1127/0941-2948/2012/0307.![]() ![]() |
[3] |
I. Cialenco, Statistical inference for SPDEs: An overview, Stat. Inference Stoch. Process., 21 (2018), 309-329.
doi: 10.1007/s11203-018-9177-9.![]() ![]() ![]() |
[4] |
R. Curtain, A survey of infinite-dimensional filtering, SIAM Review, 17 (1975), 395-411.
doi: 10.1137/1017041.![]() ![]() ![]() |
[5] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 1st edition, Encyclopedia of Mathematics and its Applications, 45, Cambridge University Press, Cambridge,
1992.
doi: 10.1017/CBO9780511666223.![]() ![]() ![]() |
[6] |
J. de Wiljes, S. 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. Appl. Dyn. Syst., 17 (2018), 1152-1181.
doi: 10.1137/17M1119056.![]() ![]() ![]() |
[7] |
S. Ghosal and A. van der Vaart, Fundamentals of Nonparametric Bayesian Inference, Cambridge Series in Statistical and Probabilistic Mathematics, 44, Cambridge University Press, Cambridge, 2017.
doi: 10.1017/9781139029834.![]() ![]() ![]() |
[8] |
E. Giné and R. Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, Cambridge Series in Statistical and Probabilistic Mathematics, 40, Cambridge University Press, Cambridge, 2015.
doi: 10.1017/CBO9781107337862.![]() ![]() ![]() |
[9] |
F. Götze, A. Naumov, V. Spokoiny and V. Ulyanov, Large ball probabilities, Gaussian comparison and anti-concentration, Bernoulli, 25 (2019), 2538-2563.
doi: 10.3150/18-BEJ1062.![]() ![]() ![]() |
[10] |
M. Hairer, An introduction to stochastic PDEs, unpublished lecture notes, arXiv: 0907.4178.
![]() |
[11] |
A. Jazwinski, Stochastic Processes and Filtering Theory, Mathematics in Science and Engineering, 64, Academic Press, New York, 1970.
![]() |
[12] |
B. T. Knapik, B. T. Szabó, A. W. van der Vaart and J. H. van Zanten, Bayes procedures for adaptive inference in inverse problems for the white noise model, Probab. Theory Relat. Fields, 164 (2016), 771-813.
doi: 10.1007/s00440-015-0619-7.![]() ![]() ![]() |
[13] |
B. T. Knapik, A. W. van der Vaart and J. H. van Zanten, Bayesian inverse problems with Gaussian priors, Annals of Statistics, 39 (2011), 2626-2657.
doi: 10.1214/11-AOS920.![]() ![]() ![]() |
[14] |
G. J. Lord, C. E. Powell and T. Shardlow, An Introduction to Computational Stochastic PDEs, 1st edition, Cambridge Texts in Applied Mathematics, 50, Cambridge University Press, Cambridge, 2014.
doi: 10.1017/CBO9781139017329.![]() ![]() ![]() |
[15] |
N. Nüsken, S. Reich and P. J. Rozdeba, State and parameter estimation from observed signal increments, Entropy, 21 (2019), 505.
doi: 10.3390/e21050505.![]() ![]() ![]() |
[16] |
E. Shchepakina, V. Sobolev and M. P. Mortell, Singular Perturbations: Introduction to System Order Reduction Methods with Applications, Lecture Notes in Mathematics, 2114, Springer International Publishing, 2014.
doi: 10.1007/978-3-319-09570-7.![]() ![]() ![]() |
[17] |
D. Simon, Optimal State Estimation, Wiley, Hoboken, 2006.
doi: 10.1002/0470045345.![]() ![]() |
[18] |
A. Taghvaei, J. de Wiljes, P. Mehta and S. Reich, Kalman filter and its modern extensions for the continuous-time nonlinear filtering problem, J. Dyn. Sys., Meas., Control., 140 (2017), 030904.
doi: 10.1115/1.4037780.![]() ![]() |
[19] |
F. Verhulst, Singular perturbation methods for slow-fast dynamics, Nonlinear Dynamics, 50 (2007), 747-753.
doi: 10.1007/s11071-007-9236-z.![]() ![]() ![]() |
[20] |
D. Yan, Bayesian inference for Gaussian models: Inverse problems and evolution equations, PhD thesis, Universiteit Leiden, Leiden, 2020.
doi: 1887/86070.![]() ![]() |
[21] |
T. Yang, P. Mehta and S. Meyn, Feedback particle filter, IEEE Trans. Automatic Control, 58 (2013), 2465-2480.
doi: 10.1109/TAC.2013.2258825.![]() ![]() ![]() |