Abstract
In this chapter, we introduce the mathematical framework for discrete-time data assimilation. Section 2.1 describes the mathematical models we use for the underlying signal , which we wish to recover, and for the data , which we use for the recovery.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Here the use of \(v =\{ v(t)\}_{t\geq 0}\) for the solution of this equation should be distinguished from our use of \(v =\{ v_{j}\}_{j=0}^{\infty }\) for the solution of (2.1).
- 2.
Here the index denotes components of the solution, not discrete time.
- 3.
Again, here the index denotes components of the solution, not discrete time.
References
H. Abarbanel. Predicting the Future: Completing Models of Observed Complex Systems. Springer, 2013.
J. Anderson. A method for producing and evaluating probabilistic forecasts from ensemble model integrations. Journal of Climate, 9(7):1518–1530, 1996.
J. Anderson and S. Anderson. A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Wea. Rev., 127(12):2741–2758, 1999.
A. Apte, C. K. R. T. Jones, A. M. Stuart, and J. Voss. Data assimilation: mathematical and statistical perspectives. Int. J. Num. Meth. Fluids, 56:1033–1046, 2008.
T. Bengtsson, P. Bickel, and B. Li. Curse of dimensionality revisited: the collapse of importance sampling in very large scale systems. IMS Collections: Probability and Statistics: Essays in Honor of David Freedman, 2:316–334, 2008.
A. Bennett. Inverse Modeling of the Ocean and Atmosphere. Cambridge, 2002.
J. O. Berger. Statistical Decision Theory and Bayesian Analysis. Springer Verlag, 1985.
J. Bernardo and A. Smith. Bayesian Theory. Wiley, 1994.
A. Beskos, D. Crisan, A. Jasra, et al. On the stability of sequential Monte Carlo methods in high dimensions. The Annals of Applied Probability, 24(4):1396–1445, 2014.
A. Beskos, O. Papaspiliopoulos, G. O. Roberts, and P. Fearnhead. Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion). Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(3):333–382, 2006.
P. Bickel, B. Li, and T. Bengtsson. Sharp failure rates for the bootstrap particle filter in high dimensions. IMS Collections: Pushing the Limits of Contemporary Statistics, 3:318–329, 2008.
M. Branicki and A. Majda. Quantifying Bayesian filter performance for turbulent dynamical systems through information theory. Comm. Math. Sci, 2014.
A. Chorin, M. Morzfeld, and X. Tu. Implicit particle filters for data assimilation. Comm. Appl. Math. Comp. Sc., 5:221–240, 2010.
S. L. Cotter, M. Dashti, J. C. Robinson, and A. M. Stuart. Bayesian inverse problems for functions and applications to fluid mechanics. Inverse Problems, 25(11):115008, 2009.
S. L. Cotter, M. Dashti, and A. M. Stuart. Approximation of Bayesian inverse problems for PDEs. SIAM J. Num. Anal., 48(1):322–345, 2010.
M. Dashti, S. Harris, and A. M. Stuart. Besov priors for Bayesian inverse problems. Inverse Problems and Imaging, 6:183–200, 2012.
M. Dashti and A. M. Stuart. Uncertainty quantification and weak approximation of an elliptic inverse problem. SIAM J. Num. Anal., 49:2524–2542, 2011.
N. Doucet, A. de Frietas and N. Gordon. Sequential Monte Carlo in Practice. Springer-Verlag, 2001.
G. Evensen. Data Assimilation: The Ensemble Kalman Filter. Springer, 2006.
K. J. Falconer. The Geometry of Fractal Sets, volume 85. Cambridge University Press, 1986.
S. Ghosal, J. K. Ghosh, and R. V. Ramamoorthi. Consistency issues in Bayesian nonparametrics. In Asymptotics, Nonparametrics and Time Series: A Tribute, pages 639–667. Marcel Dekker, 1998.
A. Gibbs and F. Su. On choosing and bounding probability metrics. International Statistical Review, 70:419–435, 2002.
K. Ide, M. Courier, M. Ghil, and A. Lorenc. Unified notation for assimilation: Operational, sequential and variational. J. Met. Soc. Japan, 75:181–189, 1997.
K. Ide and C. Jones. Special issue on the mathematics of data assimilation. Physica D, 230:vii–viii, 2007.
M. A. Iglesias, K. J. H. Law, and A. M. Stuart. Evaluation of Gaussian approximations for data assimilation in reservoir models. Computational Geosciences, 17:851–885, 2013.
A. Jazwinski. Stochastic Processes and Filtering Theory, volume 63. Academic Pr, 1970.
E. Kalnay. Atmospheric Modeling, Data Assimilation and Predictability. Cambridge, 2003.
K. J. H. Law and A. M. Stuart. Evaluating data assimilation algorithms. Mon. Wea. Rev., 140:3757–3782, 2012.
E. Lorenz. Deterministic nonperiodic flow. Atmos J Sci, 20:130–141, 1963.
E. Lorenz. The problem of deducing the climate from the governing equations. Tellus, 16(1):1–11, 1964.
E. Lorenz. Predictability: A problem partly solved. In Proc. Seminar on Predictability, volume 1, pages 1–18, 1996.
A. Majda, D. Giannakis, and I. Horenko. Information theory, model error and predictive skill of stochastic models for complex nonlinear systems. Physica D, 241:1735–1752, 2012.
A. Majda and X. Wang. Nonlinear Dynamics and Statistical Theories for Geophysical Flows. Cambridge, Cambridge, 2006.
S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Communications and Control Engineering Series. Springer-Verlag London Ltd., London, 1993.
N. Nichols. Data assimilation: aims and basic concepts. In Data Assimilation for the Earth System, Editors R. Swinbank, V. Shutyaev, W.A. Lahoz, pages 9–20. Kluwer, 2003.
D. Oliver, A. Reynolds, and N. Liu. Inverse Theory for Petroleum Reservoir Characterization and History Matching. Cambridge Univ Pr, 2008.
E. Schröder. Ueber iterirte Functionen. Mathematische Annalen, 3(2):296–322, 1870.
T. Snyder, T. Bengtsson, P. Bickel, and J. Anderson. Obstacles to high-dimensional particle filtering. Mon. Wea. Rev.., 136:4629–4640, 2008.
C. Sparrow. The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, volume 41 of Applied Mathematical Sciences. Springer-Verlag, New York, 1982.
A. M. Stuart. Inverse problems: a Bayesian perspective. Acta Numer., 19:451–559, 2010.
A. M. Stuart and A. R. Humphries. Dynamical Systems and Numerical Analysis, volume 2 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 1996.
R. Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences. Springer-Verlag, New York, second edition, 1997.
W. Tucker. The Lorenz attractor exists. C. R. Acad. Sci. Paris Sér. I Math., 328(12):1197–1202, 1999.
W. Tucker. A rigorous ODE solver and Smale’s 14th problem. Found. Comput. Math., 2(1):53–117, 2002.
A. W. Van der Vaart. Asymptotic Statistics, volume 3. Cambridge University Press, 2000.
P. van Leeuwen. Nonlinear data assimilation in geosciences: an extremely efficient particle filter. Q.J.Roy.Met.Soc., 136(653):1991–1999, 2010.
S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos, volume 2 of Texts in Applied Mathematics. Springer-Verlag, New York, 2003.
Author information
Authors and Affiliations
2.1 Electronic Supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Law, K., Stuart, A., Zygalakis, K. (2015). Discrete Time: Formulation. In: Data Assimilation. Texts in Applied Mathematics, vol 62. Springer, Cham. https://doi.org/10.1007/978-3-319-20325-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-20325-6_2
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-20324-9
Online ISBN: 978-3-319-20325-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)