Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-18T15:45:50.619Z Has data issue: false hasContentIssue false

The Back and Forth Nudging algorithm for data assimilation problems : theoretical results on transport equations

Published online by Cambridge University Press:  19 January 2011

Didier Auroux
Laboratoire Dieudonné, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 2, France. INRIA, Grenoble, France
Maëlle Nodet
INRIA, Grenoble, France Université de Grenoble, Laboratoire Jean Kuntzmann, UMR 5224, Grenoble, France;


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Burgers’ equation). Our aim is to prove some theoretical results on the convergence, and convergence properties, of this algorithm. We show that for non viscous equations (both linear transport and Burgers), the convergence of the algorithm holds under observability conditions. Convergence can also be proven for viscous linear transport equations under some strong hypothesis, but not for viscous Burgers’ equation. Moreover, the convergence rate is always exponential in time. We also notice that the forward and backward system of equations is well posed when no nudging term is considered.

Research Article
© EDP Sciences, SMAI, 2011


Auroux, D. and Blum, J., Back and forth nudging algorithm for data assimilation problems. C. R. Acad. Sci. Paris Sér. I 340 (2005) 873878. Google Scholar
Auroux, D. and Blum, J., A nudging-based data assimilation method for oceanographic problems : the back and forth nudging (BFN) algorithm. Nonlin. Proc. Geophys. 15 (2008) 305319. Google Scholar
D. Auroux and S. Bonnabel, Symmetry-based observers for some water-tank problems. IEEE Trans. Automat. Contr. (2010) DOI : 10.1109/TAC.2010.2067291.
H. Brezis, Analyse fonctionnelle : théorie et applications. Dunod, Paris (1999).
R. Courant and D. Hilbert, Methods of Mathematical Physics II. Wiley-Interscience (1962).
L.C. Evans, Partial Differential Equations. American Mathematical Society, Providence (1998).
Evensen, G. and van Leeuwen, P.J., An ensemble Kalman smoother for nonlinear dynamics. Mon. Weather Rev. 128 (1999) 18521867. Google Scholar
Guo, B.-Z. and Guo, W., The strong stabilization of a one-dimensional wave equation by non-collocated dynamic boundary feedback control. Automatica 45 (2009) 790797. Google Scholar
Guo, B.-Z. and Shao, Z.-C., Stabilization of an abstract second order system with application to wave equations under non-collocated control and observations. Syst. Control Lett. 58 (2009) 334341. Google Scholar
Hoke, J. and Anthes, R.A., The initialization of numerical models by a dynamic initialization technique. Mon. Weather Rev. 104 (1976) 15511556. Google Scholar
Kalman, R.E., A new approach to linear filtering and prediction problems. Trans. ASME – J. Basic Eng. 82 (1960) 3545. Google Scholar
E. Kalnay, Atmospheric modeling, data assimilation and predictability. Cambridge University Press (2003).
Krstic, M., Magnis, L. and Vazquez, R., Nonlinear control of the viscous burgers equation : Trajectory generation, tracking, and observer design. J. Dyn. Sys. Meas. Control 131 (2009) 18. Google Scholar
Le Dimet, F.-X., and Talagrand, O., Variational algorithms for analysis and assimilation of meteorological observations : theoretical aspects. Tellus 38A (1986) 97110. Google Scholar
Luenberger, D., Observers for multivariable systems. IEEE Trans. Automat. Contr. 11 (1966) 190197. Google Scholar
Moireau, Ph., Chapelle, D. and Tallec, P. Le, Filtering for distributed mechanical systems using position measurements : perspectives in medical imaging. Inver. Probl. 25 (2009) 035010. Google Scholar
Ramdani, K., Tucsnak, M. and Weiss, G., Recovering the initial state of an infinite-dimensional system using observers. Automatica 46 (2010) 16161625. Google Scholar
Russell, D.L., Controllability and stabilizability theory for linear partial differential equations : recent progress and open questions. SIAM Rev. 20 (1978) 639739. Google Scholar
Smyshlyaev, A. and Krstic, M., Backstepping observers for a class of parabolic PDEs. Syst. Control Lett. 54 (2005) 613625. Google Scholar