Data assimilation of time under-sampled measurements using observers, the wave-like equation example

Cîndea, Nicolae; Imperiale, Alexandre; Moireau, Philippe

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 3, pp. 635-669.

We propose a sequential data assimilation scheme using Luenberger type observers when only some space restricted time under-sampled measurements are available. More precisely, we consider a wave-like equation for which we assume known the restriction of the solution to an open non-empty subset of the spatial domain and for some time samples (typically the sampling step in time is much larger than the time discretization step). To assimilate the available data, two strategies are proposed and analyzed. The first strategy consists in assimilating data only if they are available and the second one in assimilating interpolation of the available data at all the discretization times. In order to tackle the spurious high frequencies which appear when we discretize the wave equation, for both strategies, we introduce a numerical viscous term. In this case, we prove some error estimates between the exact solution and our observers. Numerical simulations illustrate the theoretical results in the case of the one dimensional wave equation.

Classification : 35L05, 35L20, 65D05, 65M12, 93B07, 93B70, 93D15
Mots clés : Data assimilation, time under-sampled measurements, Luenberger observers, numerical analysis, interpolation

R.A. Anthes, Data assimilation and initialization of hurricane prediction model. J. Atmospheric Sci. 31 (1974) 702–719. | DOI

H T Banks, K. Ito and C. Wang, Exponentially stable approximations of weakly damped wave equations, in Estimation and control of distributed parameter systems (Vorau, 1990). Birkhäuser, Basel (1991) 1–33. | MR | Zbl

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024–1065. | DOI | MR | Zbl

A. Bensoussan, Filtrage optimal des systèmes linéaires. Dunod (1971). | Zbl

J. Blum, F.X. LeDimet and I.N. Navon, Data assimilation for geophysical fluids. In vol. 14 of Handbook of Numerical Analysis: Computational Methods for the Atmosphere and the Oceans. Elsevier, Amsterdam (2008) 377–434. | MR

R. Chabiniok, P. Moireau, P.-F. Lesault, A. Rahmouni, J.-F. Deux and D. Chapelle, Trials on tissue contractility estimation from cardiac cine-MRI using a biomechanical heart model. In vol. 6666, Proc. of FIMH’11. Lect. Notes Compt. Sci. (2011) 304–313.

D. Chapelle, N. Cîndea, M. De Buhan and P. Moireau, Exponential convergence of an observer based on partial field measurements for the wave equation. Math. Probl. Eng. 2012 (2012) 12. | DOI | MR | Zbl

D. Chapelle, N. Cîndea and P. Moireau, Improving convergence in numerical analysis using observers. The wave-like equation case. Math. Models Methods Appl. Sci. (2012). | MR | Zbl

D. Chapelle, M. Fragu, V. Mallet and P. Moireau, Fundamental principles of data assimilation underlying the Verdandi library: applications to biophysical model personalization within euHeart. Med. Biol. Eng. Comput. 5 (2013) 1221–1233. | DOI

S. Cox and E. Zuazua, The rate at which energy decays in a damped string. Commun. Part. Differ. Eqs. 19 (1994) 213–243. | DOI | MR | Zbl

M. Daoulatli, B. Dehman and M. Khenissi, Local energy decay for the elastic system with nonlinear damping in an exterior domain. SIAM J. Control Optim. 48 (2010) 5254–5275 | DOI | MR | Zbl

G. Evensen, Data Assimilation – The Ensemble Kalman Filter. Springer Verlag (2007). | MR | Zbl

S. Ervedoza, Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes. Numer. Math. 113 (2009) 377–415. | DOI | MR | Zbl

S. Ervedoza and E. Zuazua, Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. 91 (2009) 20–48. | DOI | MR | Zbl

S. Ervedoza, C. Zheng and E. Zuazua, On the observability of time-discrete conservative linear systems. J. functional Anal. 254 (2008) 3037–3078. | DOI | MR | Zbl

G. Haine and K. Ramdani, Reconstructing initial data using observers: error analysis of the semi-discrete and fully discrete approximations. Numer. Math. 120 (2012) 307–343. | DOI | MR | Zbl

F.M. Hante, M. Sigalotti and M. Tucsnak, On conditions for asymptotic stability of dissipative infinite-dimensional systems with intermittent damping. J. Differ. Eqs. 252 (2012) 5569–5593. | DOI | MR | Zbl

A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems. Arch. Rational Mech. Anal. 100 (1988) 191–206. | DOI | MR | Zbl

J.E. Hoke and R.A. Anthes, The initialization of numerical models by a dynamic-initialization technique (fluid flow models for wind forecasting). Monthly Weather Rev. 104 (1976) 1551–1556. | DOI

R.E. Kalman and R.S. Bucy, New results in linear filtering and prediction theory. J. Basic Eng. 83 (1961) 95–108. | DOI | MR

S. Lakshmivarahan and J.M. Lewis, Nudging methods: A critical overview. In vol. XVIII of Data Assimilation for Atmospheric, Oceanic, and Hydrologic Applications. Edited by S.K. Park and L. Xu. Springer (2008).

F.-X. Le Dimet and O. Talagrand, Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus A 38 (2010) 97–110.

X.-D. Li and C.-Z. Xu, Infinite-dimensional Luenberger-like observers for a rotating body-beam system. Systems Control Lett. 60 (2011) 138–145. | DOI | MR | Zbl

K. Liu, Locally distributed control and damping for the conservative systems. SIAM J. Control Optim. 35 (1997) 1574–1590. | DOI | MR | Zbl

D.G. Luenberger, An introduction to observers. IEEE T. Automat. Contr. 16 (1971) 596–602. | DOI

P. Moireau, D. Chapelle and P. Le Tallec. Joint state and parameter estimation for distributed mechanical systems. Comput. Methods Appl. Mech. Engrg. 197 (2007) 659–677. | DOI | MR | Zbl

P. Moireau, D. Chapelle and P. Le Tallec, Filtering for distributed mechanical systems using position measurements: Perspectives in medical imaging. Inverse Probl. 25 (2009) 035010. | DOI | MR | Zbl

I.M. Navon, Data assimilation for numerical weather prediction: a review. In vol. XVIII of Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications. Edited by S.K. Park and L. Xu. Springer (2009).

N.K. Nichols, Mathematical concepts of data assimilation, in Data Assimilation. Edited by W. Lahoz, B. Khattatov and R. Menard. Springer Berlin Heidelberg (2010) 13–39. | Zbl

K. Ramdani, T. Takahashi and M. Tucsnak, Uniformly exponentially stable approximations for a class of second order evolution equations-application to LQR problems. ESAIM: COCV 13 (2007) 503–527. | Numdam | MR | Zbl

K. Ramdani, M. Tucsnak and G. Weiss, Recovering the initial state of an infinite-dimensional system using observers. Automatica (2012) 1616–1625. | MR | Zbl

D. Simon, Optimal state estimation: Kalman, $H_{\infty }$ and nonlinear approaches. Wiley-Interscience (2006).

L.T. Tebou and E. Zuazua, Uniform boundary stabilization of the finite difference space discretization of the 1-d wave equation. Adv. Comput. Math. 26 (2007) 337–365. | DOI | MR | Zbl

D.T. Pham, J. Verron and L. Gourdeau, Singular evolutive kalman filters for data assimilation in oceanography. C. R. Acad. Sci. Paris (1997) 255–260.

M. Tucsnak and G. Weiss, Observation and control for operator semigroups. Birkhäuser Basel (2009). | MR | Zbl

X. Zhang, C. Zheng and E. Zuazua, Exact controllability of the time discrete wave equation: a multiplier approach. Discret. Contin. Dyn. Syst. (2007) 229–245. | MR | Zbl

E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197–243. | DOI | MR | Zbl