A null controllability data assimilation methodology applied to a large scale ocean circulation model

García, Galina C.; Osses, Axel; Puel, Jean Pierre

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 2, pp. 361-386.

Data assimilation refers to any methodology that uses partial observational data and the dynamics of a system for estimating the model state or its parameters. We consider here a non classical approach to data assimilation based in null controllability introduced in [Puel, C. R. Math. Acad. Sci. Paris 335 (2002) 161-166] and [Puel, SIAM J. Control Optim. 48 (2009) 1089-1111] and we apply it to oceanography. More precisely, we are interested in developing this methodology to recover the unknown final state value (state value at the end of the measurement period) in a quasi-geostrophic ocean model from satellite altimeter data, which allows in fact to make better predictions of the ocean circulation. The main idea of the method is to solve several null controllability problems for the adjoint system in order to obtain projections of the final state on a reduced basis. Theoretically, we have to prove the well posedness of the involved systems associated to the method and we also need an observability property to show the existence of null controls for the adjoint system. To this aim, we use a global Carleman inequality for the associated velocity-pressure formulation of the problem which was previously proved in [Fernández-Cara et al., J. Math. Pures Appl. 83 (2004) 1501-1542]. We present numerical simulations using a regularized version of this data assimilation methodology based on null controllability for elements of a reduced spectral basis. After proving the convergence of the regularized solutions, we analyze the incidence of the observatory size and noisy data in the recovery of the initial value for a quality prediction.

[1] A. Belmiloudi and F. Brossier, A control method for assimilation of surface data in a linearized Navier-Stokes-type problem related to oceanography. SIAM J. Control Optim. 35 (1997) 2183-2197. | MR | Zbl

[2] A.F. Bennett, Inverse Methods in Physical Oceanography. Cambridge University Press, Cambridge (1992). | MR | Zbl

[3] R. Bermejo and P. Galán Del Sastre, Numerical studies of the long-term dynamics of the 2D Navier-Stokes equations applied to ocean circulation, in XVII CEDYA: Congress on Differential Equations and Applications, L. Ferragut and A. Santos Eds., Universidad de Salamanca, Salamanca (2001) 15-34. | MR | Zbl

[4] C. Bernardi, E. Godlewski and G. Raugel, A mixed method for time-dependent Navier-Stokes problem. IMA J. Numer. Anal. 7 (1987) 165-189. | MR | Zbl

[5] E. Blayo, J. Blum and J. Verron, Assimilation variationnelle de données en océanographie et réduction de la dimension de l'espace de contrôle, in Équations aux dérivées partielles et applications, Articles dédiés à Jacques-Louis Lions, Gauthier-Villars, éd. Sci. Méd. Elsevier, Paris (1998) 199-219. | MR | Zbl

[6] J. Blum, B. Luong and J. Verron, Variational assimilation of altimeter data into a non-linear ocean model: Temporal strategies. ESAIM: Proc. 4 (1998) 21-57. | MR | Zbl

[7] C. Carthel, R. Glowinski and J.L. Lions, On exact and approximate boundary controllabilities for heat equation: a numerical approach. J. Optim. Theory Appl. 82 (1994) 429-484. | MR | Zbl

[8] P. Courtier, O. Talagrand, Variational assimilation of meteorological observations with the adjoint vorticity equation. I: Theory. Quart. J. Roy. Meteorol. Soc. 113 (1987) 1311-1328.

[9] C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 31-61. | MR | Zbl

[10] E. Fernández-Cara, S. Guerrero, O.Y. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501-1542. | MR | Zbl

[11] E. Fernández-Cara, G.C. García and A. Osses, Controls insensitizing the observation of a quasi-geostrophic ocean model. SIAM J. Control Optim. 43 (2005) 1616-1639. | MR | Zbl

[12] A.V. Fursikov and O.Y. Imanuilov, Local exact controllability of the two-dimensional Navier-Stokes equations. Matematicheskiĭ Sbornik 187 (1996) 103-138. | MR | Zbl

[13] A. Fursikov and O.Y. Imanuvilov, Controllability of evolution equations. Lecture Notes, Research Institute of Mathematics, Seoul National University, Korea (1996). | MR | Zbl

[14] M. Ghil and P. Malanotte-Rizzoli, Data assimilation in meteorology and oceanography. Adv. Geophys. 33 (1991) 141-266.

[15] V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, New York (1986). | MR | Zbl

[16] C. Hansen, Analysis of ill-posed problems by means of the L-curve. SIAM Rev. 34 (1992) 561-580. | MR | Zbl

[17] F.-X. Le Dimet and O. Talagrand, Variational algorithms for analysis and assimilation of meteorological observations. Tellus 38A (1986) 97 -110.

[18] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin (1971). | MR | Zbl

[19] J.-L. Lions, Remarks on approximate controllability, Festschrift on the occasion of the 70th birthday of Samuel Agmon. J. Anal. Math. 59 (1992) 103-116. | MR | Zbl

[20] J.-L. Lions, Exact and approximate controllability for distributed parameter system, in VI Escuela de Otoño Hispano-Francesa sobre simulación numérica en física e ingeniería, Universidad de Sevilla, España (1994) 1-238. | MR | Zbl

[21] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications 1. Dunod (1968). | MR | Zbl

[22] B. Luong, J. Blum and J. Verron, A variational method for the resolution of a data assimilation problem in oceanography. Inv. Probl. 14 (1998) 979-997. | MR | Zbl

[23] G.I. Marchuk, Formulation of theory of perturbations for complicated models. Appl. Math. Optim. 2 (1975) 1-33. | MR | Zbl

[24] P.G. Myers and A.J. Weaver, A diagnostic barotropic finite-element ocean circulation model. J. Atmos. Ocean Tech. 12 (1995) 511-526.

[25] A. Osses and J.-P. Puel, Boundary controllability of a stationary Stokes system with linear convection observed on an interior curve. J. Optim. Theory Appl. 99 (1998) 201-234. | MR | Zbl

[26] A. Osses and J.-P. Puel, On the controllability of the Laplace equation observed on an interior curve. Rev. Mat. Complut. 11 (1998) 403-441. | MR | Zbl

[27] J.-P. Puel, Une approche non classique d'un problème d'assimilation de données. C. R. Math. Acad. Sci. Paris 335 (2002) 161-166. | MR | Zbl

[28] J.-P. Puel, A nonstandard approach to a data assimilation problem and Tychonov regularization revisited. SIAM J. Control Optim. 48 (2009) 1089-1111. | MR | Zbl

[29] L. Quartapelle, Numerical Solution of the Incompressible Navier-Stokes Equations. Birkhauser Verlag (1993). | MR | Zbl

[30] J. Verron, Altimeter data assimilation into ocean model: sensitivity to orbital parameters. J. Geophys. Res. 95 (1990) 11443-11459.

[31] J. Verron, Nudging satellite altimeter data into quasi-geostrophic ocean models. J. Geophys. Res. 97 (1992) 7479-7492.