Analysis of the 3DVAR filter for thepartially observed Lorenz'63 model

Kody Law; Abhishek Shukla; Andrew Stuart

doi:10.3934/dcds.2014.34.1061

Article Contents

2014, Volume 34, Issue 3: 1061-1078. Doi: 10.3934/dcds.2014.34.1061

This issue Previous Article Bernstein-type approximation of set-valued functions in the symmetric difference metric Next Article Variable step size multiscale methods for stiff and highly oscillatory dynamical systems

Analysis of the 3DVAR filter for the partially observed Lorenz'63 model

1.
Warwick Mathematics Institute, University of Warwick, CV4 7AL

Received: December 2012

Revised: April 2013

Published: March 2014

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Abstract

The problem of effectively combining data with a mathematical model constitutes a major challenge in applied mathematics. It is particular challenging for high-dimensional dynamical systems where data is received sequentially in time and the objective is to estimate the system state in an on-line fashion; this situation arises, for example, in weather forecasting. The sequential particle filter is then impractical and ad hoc filters, which employ some form of Gaussian approximation, are widely used. Prototypical of these ad hoc filters is the 3DVAR method. The goal of this paper is to analyze the 3DVAR method, using the Lorenz '63 model to exemplify the key ideas. The situation where the data is partial and noisy is studied, and both discrete time and continuous time data streams are considered. The theory demonstrates how the widely used technique of variance inflation acts to stabilize the filter, and hence leads to asymptotic accuracy.

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Mathematics Subject Classification: Primary: 34F05, 37H10; Secondary: 60H10.

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References

[1]	A. Apte, C. K. R. T. Jones, A. M. Stuart and J. Voss, Data assimilation: Mathematical and statistical perspectives, International Journal for Numerical Methods in Fluids, 56 (2008), 1033-1046.doi: 10.1002/fld.1698.
[2]	A. Bennett, "Inverse Modeling of the ocean and Atmosphere," Cambridge, 2002.doi: 10.1017/CBO9780511535895.
[3]	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.
[4]	D. Blömker, K. Law, A. M. Stuart and K. C. Zygalakis, Accuracy and stability of the continuous-time 3dvar filter for the navier-stokes equation, Preprint, arXiv:1210.1594, 2012.
[5]	C. E. A. Brett, K. F. Lam, K. J. H. Law, D. S. McCormick, M. R. Scott and A. M. Stuart, Accuracy and stability of filters for dissipative pdes, Physica D: Nonlinear Phenomena, 245 (2013), 34-45.doi: 10.1016/j.physd.2012.11.005.
[6]	A. Doucet, N. De Freitas and N. Gordon, "Sequential Monte Carlo Methods in Practice," Springer Verlag, 2001.
[7]	G. Evensen, The ensemble Kalman filter: Theoretical formulation and practical implementation, Ocean Dynamics, 53 (2003), 343-367.doi: 10.1007/s10236-003-0036-9.
[8]	G. Evensen, "Data Assimilation: The Ensemble Kalman Filter," Springer Verlag, 2009.doi: 10.1007/978-3-642-03711-5.
[9]	C. Foias, M. S. Jolly, I. Kukavica and E. S. Titi, The lorenz equationas a metaphor for the navier-stokes equation, discrete and continous dynamical systems, Discrete and Continous Dynamical Systems, 7 (2001), 403-429.doi: 10.3934/dcds.2001.7.403.
[10]	C. Foias and G. Prodi, Sur le comportement global des solutions nonstationnaires des équations de navier-stokes en dimension 2, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.
[11]	A. C. Harvey, "Forecasting, Structural Time Series Models and the Kalman Filter," Cambridge Univ Pr, 1991.
[12]	K. Hayden, E. Olson and E. S. Titi, Discrete data assimilation in the lorenz and 2d navier-stokes equations, Physica D: Nonlinear Phenomena, 240 (2011), 1416-1425.doi: 10.1016/j.physd.2011.04.021.
[13]	A. H. Jazwinski, "Stochastic Processes and Filtering Theory," Academic Pr, 1970.
[14]	D. A. Jones and E. S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the navier-stokes equations, Indiana University Mathematics Journal, 42 (1993), 875-888.doi: 10.1512/iumj.1993.42.42039.
[15]	E. Kalnay, "Atmospheric Modeling, Data Assimilation, and Predictability," Cambridge Univ. Pr., 2002.doi: 10.1017/CBO9780511802270.
[16]	E. Kalnay, H. Li, T. Miyoshi, S.-C. Yang and J. Ballabrera-Poy, 4DVAR or ensemble Kalman filter?, Tellus A, 59 (2008), 758-773.
[17]	K. J. H. Law and A. M. Stuart, Evaluating data assimilation algorithms, Monthly Weather Review, 140 (2012), 3757-3782.doi: 10.1175/MWR-D-11-00257.1.
[18]	A. C. Lorenc, Analysis methods for numerical weather prediction, Quart. J. R. Met. Soc., 112 (2000), 1177-1194.doi: 10.1002/qj.49711247414.
[19]	E. N. Lorenz, Deterministic nonperiodic flow, Atmos. J. Sci., 20 (1963), 130-141.
[20]	E. N. Lorenz, Predictability: A problem partly solved, in "Proc. Seminar on Predictability," 1 (1996), 1-18.doi: 10.1017/CBO9780511617652.004.
[21]	A. J. Majda and J. Harlim, "Filtering Complex Turbulent Systems," Cambridge University Press, 2012.doi: 10.1017/CBO9781139061308.
[22]	A. J. Majda, J. Harlim and B. Gershgorin, Mathematical strategies for filtering turbulent dynamical systems, Discrete and Continuous Dynamical Systems - Series A, 27 (2010), 441-486.doi: 10.3934/dcds.2010.27.441.
[23]	X. Mao, "Stochastic Differential Equations And Applications," Horwood, 1997.doi: 10.1533/9780857099402.
[24]	D. S. Oliver, A. C. Reynolds and N. Liu, "Inverse Theory for Petroleum Reservoir Characterization and History Matching," Cambridge University Press, 2008.doi: 10.1017/CBO9780511535642.
[25]	E. Olson and E. S. Titi, Determining modes for continuous data assimilation in 2D turbulence, Journal of Statistical Physics, 113 (2003), 799-840.doi: 10.1023/A:1027312703252.
[26]	D. F. Parrish and J. C. Derber, The National Meteorological Center's spectral statistical-interpolation analysis system, Monthly Weather Review, 120 (1992), 1747-1763.doi: 10.1175/1520-0493(1992)120<1747:TNMCSS>2.0.CO;2.
[27]	J. C. Robinson, "Infinite-Dimensional Dynamical Systems," Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001.doi: 10.1007/978-94-010-0732-0.
[28]	C. Sparrow, "The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors," Springer, 1982.
[29]	A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559.doi: 10.1017/S0962492910000061.
[30]	R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 68 of Applied Mathematical Sciences. Springer-Verlag, New York, second edition, 1997.
[31]	W. Tucker, A rigorous ode solver and smale's 14th problem, Journal of Foundations of Computational Mathematics, 2 (2002), 53-117.
[32]	P. J. Van Leeuwen, Particle filtering in geophysical systems, Monthly Weather Review, 137 (2009), 4089-4114.