L. Méthodes-multigrilles, Sommaire 3.1 Les méthodes multigrilles pour la résolution de systèmes linéaires, p.54

L. Etude-de, 59 3.2.1 L'ellipticité, p.61

L. Non-linéarité-sommaire and .. , 1 Les algorithmes étudiés

A. Variational-data, 136 6.2.1 A complex control optimal system, 136 6.2.2 Incremental methods to solve the optimal system . . . . . . . . . . . . . 136

.. Multigrid-methods, 138 6.3.1 A way to accelerate iterative methods, p.139

.. The-multigrid-configuration, 144 6.5.3 Experimental set-up, p.150

L. Amodei, Solution approchée pour un problème d'assimilation de données météorologiques avec prise en compte de l'erreur de modèle, C.R. Acad. Sci. Paris, vol.321, issue.12, pp.1087-1094, 1995.

A. Bennett, Inverse methods in Physical Oceanography, 1992.
DOI : 10.1017/CBO9780511600807

A. Bennett, Inverse Modeling of the Ocean and Atmosphere, 2002.
DOI : 10.1017/CBO9780511535895

A. F. Bennett, L. M. Leslie, C. F. Hagelberg, and P. Powers, Tropical Cyclone Prediction Using a Barotropic Model Initialized by a Generalized Inverse Method, Monthly Weather Review, vol.121, issue.6, pp.1714-1729, 1993.
DOI : 10.1175/1520-0493(1993)121<1714:TCPUAB>2.0.CO;2

A. Brandt, Multi-level adaptive solutions to boundary-value problems, Mathematics of Computation, vol.31, issue.138, pp.333-390, 1977.
DOI : 10.1090/S0025-5718-1977-0431719-X

A. Brandt, Multi-level adaptive computations in fluid dynamics, 4th Computational Fluid Dynamics Conference, pp.1165-1172, 1980.
DOI : 10.2514/6.1979-1455

A. Brandt, Multigrid solutions to steady-state compressible navier-stokes equations i, Proc. Fifth International Symposium, pp.407-423, 1981.

A. Brandt, Multigrid techniques 1984 Guide with applications to fluid dynamics, Gesellschaft fuer Mathematik und Datenverarbeitung m.b.H. Bonn (GMD), St. Augustin (Germany ), 1984.

P. Courtier, Dual formulation of four-dimensional variational assimilation, Quarterly Journal of the Royal Meteorological Society, vol.121, issue.544, pp.2449-2461, 1997.
DOI : 10.1002/qj.49712354414

P. Courtier, J. N. Thépaut, and A. Hollingsworth, A strategy for operational implementation of 4D-Var, using an incremental approach, Quarterly Journal of the Royal Meteorological Society, vol.45, issue.519, pp.1367-1387, 1994.
DOI : 10.1002/qj.49712051912

D. Silva, A. Pfaendtner, J. Guo, J. Sienkiewicz, M. Cohn et al., Assessing the effects of data selection with dao's physicalspace statistical analysis system, Proceedings of the second international WMO symposium on assimilation of observations in meteorology and oceanography, pp.273-278, 1995.

D. Leon and D. , Wavelets Operators Applied to Multigrid Methods, 2000.

L. Debreu, E. Simon, and E. Blayo, 4d variational data assimilation for locally nested models : optimality system and preliminary numerical experiments, Int. J. Numer. Meth. Fluids, vol.00, pp.1-6, 2000.

G. Evensen, Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model, Journal of Geophysical Research: Oceans, vol.93, issue.C2, pp.11-17905, 1992.
DOI : 10.1029/92JC01972

G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, Journal of Geophysical Research, vol.109, issue.Part 4, pp.5-10143, 1994.
DOI : 10.1029/94JC00572

G. Evensen, Data assimilation. The Ensemble Kalman filter, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00229825

D. Ferreira, J. Marshall, and P. Heimbach, Estimating Eddy Stresses by Fitting Dynamics to Observations Using a Residual-Mean Ocean Circulation Model and Its Adjoint, Journal of Physical Oceanography, vol.35, issue.10, pp.1891-1910, 2005.
DOI : 10.1175/JPO2785.1

L. Gandin, Objective analysis of meteorological fields, Gidrometeorol. Izd, 1963.

M. Ghil, S. Cohn, J. Tavantzis, K. Bube, and E. Isaacson, Applications of Estimation Theory to Numerical Weather Prediction, Dynamic Meteorology : Data assimilation Methods, pp.139-224, 1981.
DOI : 10.1007/978-1-4612-5970-1_5

J. Gilbert and C. Lemaréchal, Some numerical experiments with variable-storage quasi-Newton algorithms, Mathematical Programming, vol.11, issue.2, pp.407-435, 1989.
DOI : 10.1007/BF01589113

S. Goedecker and C. Chauvin, COMBINING MULTIGRID AND WAVELET IDEAS TO CONSTRUCT MORE EFFICIENT MULTISCALE ALGORITHMS, Journal of Theoretical and Computational Chemistry, vol.02, issue.04, p.483, 2003.
DOI : 10.1142/S021963360300063X
URL : https://hal.archives-ouvertes.fr/hal-00450034

S. Gratton, A. S. Lawless, N. , and N. K. , Approximate Gauss???Newton Methods for Nonlinear Least Squares Problems, SIAM Journal on Optimization, vol.18, issue.1, pp.106-132, 2007.
DOI : 10.1137/050624935

S. Gratton, A. Sartenaer, and P. Toint, Recursive Trust-Region Methods for Multiscale Nonlinear Optimization, SIAM Journal on Optimization, vol.19, issue.1, pp.414-444, 2008.
DOI : 10.1137/050623012
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.329.5701

S. Gratton and P. Toint, Approximate invariant subspaces and quasi-newton optimization methods, Optimization Methods and Software, vol.25, issue.4, pp.507-529, 2010.
DOI : 10.1080/10556780902992746
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.414.9566

W. Hackbusch, Multi-grid Methods and Applications, 1985.
DOI : 10.1007/978-3-662-02427-0

L. Hascoët and V. Pascual, Tapenade 2.1 user's guide, 2004.

P. Hemker, A note on defect correction processes with an approximate inverse of deficient rank, Journal of Computational and Applied Mathematics, vol.8, issue.2, pp.137-139, 1982.
DOI : 10.1016/0771-050X(82)90070-5

P. Hemker, On the order of prolongations and restrictions in multigrid procedures, Journal of Computational and Applied Mathematics, vol.32, issue.3, pp.423-429, 1990.
DOI : 10.1016/0377-0427(90)90047-4

P. Hemker, Lecture notes on defect correction, 2001.

W. R. Holland and P. Malanotte-rizzoli, Assimilation of Altimeter Data into an Ocean Circulation Model: Space versus Time Resolution Studies, Journal of Physical Oceanography, vol.19, issue.10, pp.1507-1534, 1989.
DOI : 10.1175/1520-0485(1989)019<1507:AOADIA>2.0.CO;2

K. Ide, P. Courtier, M. Ghil, and A. C. Lorenc, Unified notation for data assimilation : Operational, and variational, J. Met. Soc. of Japan, vol.75, pp.1-181, 1997.

. Ifremer and . Www, ifremer.fr. Institut français de recherche pour l'exploitation de la mer

R. H. Jones, Optimal Estimation of Initial Conditions for Numerical Prediction, Journal of the Atmospheric Sciences, vol.22, issue.6, pp.658-1965, 1965.
DOI : 10.1175/1520-0469(1965)022<0658:OEOICF>2.0.CO;2

R. E. Kalman, A new approach to linear filtering and prediction problems. Transactions of the ASME, Journal of Basic Engineering Series D, vol.82, pp.33-45, 1960.

R. Kettler, Analysis and comparison of relaxation schemes in robust multigrid and preconditioned conjugate gradient methods, Multigrid methods Proc. conference held at Köln-Porz, 1981.
DOI : 10.1137/0705044

J. Krüger, Simulated Annealing: A Tool for Data Assimilation into an Almost Steady Model State, Journal of Physical Oceanography, vol.23, issue.4, pp.679-688, 1993.
DOI : 10.1175/1520-0485(1993)023<0679:SAATFD>2.0.CO;2

L. Dimet and F. , A general formalism of variational analysis, 1982.

L. Dimet, F. , T. , and O. , Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects, Tellus A: Dynamic Meteorology and Oceanography, vol.109, issue.2, pp.97-110, 1986.
DOI : 10.3402/tellusa.v38i2.11706

R. M. Lewis and S. G. Nash, Model Problems for the Multigrid Optimization of Systems Governed by Differential Equations, SIAM Journal on Scientific Computing, vol.26, issue.6, pp.1811-1837, 2005.
DOI : 10.1137/S1064827502407792

J. L. Lions, Contrôle optimal des systèmes gouvernès par des équations aux dérivées partielles, 1968.

E. N. Lorenz, Deterministic Nonperiodic Flow, Journal of the Atmospheric Sciences, vol.20, issue.2, pp.130-141, 1963.
DOI : 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2

S. F. Mccormick, An Algebraic Interpretation of Multigrid Methods, SIAM Journal on Numerical Analysis, vol.19, issue.3, pp.548-560, 1982.
DOI : 10.1137/0719036

A. J. Miller, C. , and B. D. , Forecasts from fits of frontal fluctuations, Dynamics of Atmospheres and Oceans, vol.29, issue.2-4, pp.305-333, 1999.
DOI : 10.1016/S0377-0265(99)00009-3

R. N. Miller, Toward the Application of the Kalman Filter to Regional Open Ocean Modeling, Journal of Physical Oceanography, vol.16, issue.1, pp.72-86, 1986.
DOI : 10.1175/1520-0485(1986)016<0072:TTAOTK>2.0.CO;2

R. N. Miller, E. F. Carter, J. Blue, and S. T. , Data assimilation into nonlinear stochastic models, pp.167-194, 1999.

A. M. Moore, Data assimilation in a quasi-geostrophic open-ocean model of the gulf stream region using the adjoint method data assimilation in a quasi-geostrophic open-ocean model of the gulf stream region using the adjoint method, J. Phys. Oceanogr, vol.21, pp.382-427, 1991.

S. G. Nash, A multigrid approach to discretized optimization problems, Optimization Methods and Software, vol.14, issue.1, pp.99-116, 2000.
DOI : 10.1080/10556780008805795

C. Pflaum, A multigrid conjugate gradient method, Applied Numerical Mathematics, vol.58, issue.12, pp.1803-1817, 2008.
DOI : 10.1016/j.apnum.2007.11.020

D. T. Pham, J. Verron, and M. Roubaud, A singular evolutive extended kalman filter for data assimilation in oceanography, Journal of Marine Systems, vol.16, pp.3-4, 1998.

W. H. Press, B. P. Flannery, and S. A. Teukolsky, Numerical Recipes, 1992.

R. Robert and C. Rosier, Long range predictability of atmospheric flows, Nonlinear Processes in Geophysics, vol.8, issue.1/2, pp.55-67, 2001.
DOI : 10.5194/npg-8-55-2001
URL : https://hal.archives-ouvertes.fr/hal-00331055

Y. Sasaki, A fundamental study of the numerical prediction based on the variational principle, J. Met. Soc. of Japan, vol.33, pp.262-275, 1955.

A. F. Shchepetkin and J. C. Mcwilliams, The regional oceanic modeling system (ROMS): a split-explicit, free-surface, topography-following-coordinate oceanic model, Ocean Modelling, vol.9, issue.4, pp.347-404, 2005.
DOI : 10.1016/j.ocemod.2004.08.002

E. Simon, Assimilation variationnelle de données pour des modèles emboîtés, 2007.

R. V. Southwell, Stress-Calculation in Frameworks by the Method of "Systematic Relaxation of Constraints". I and II, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.151, issue.872, pp.56-95, 1935.
DOI : 10.1098/rspa.1935.0134

S. Ta-'asan, One shot methods for optimal control of distributed parameters systems i : Finite dimensional control, 1991.

S. Ta-'asan, Multigrid one-shot methods and Design Strategy, 1997.

J. N. Thépaut and P. Courtier, Four-dimensional variational data assimilation using the adjoint of a multilevel primitive-equation model, Quarterly Journal of the Royal Meteorological Society, vol.110, issue.502, pp.1225-1254, 1991.
DOI : 10.1002/qj.49711750206

Y. Trémolet, Incremental 4D-Var convergence study, Tellus A: Dynamic Meteorology and Oceanography, vol.124, issue.1, pp.706-718, 2007.
DOI : 10.1111/j.1600-0870.2007.00271.x

F. Veersé and J. N. Thépaut, Multiple-truncation incremental approach for four-dimensional variational data assimilation, Quarterly Journal of the Royal Meteorological Society, vol.124, issue.1, pp.1889-1908, 1998.
DOI : 10.1002/qj.49712455006

M. Verlaan and A. W. Heemink, Tidal flow forecasting using reduced rank square root filters, Stochastic Hydrology and Hydraulics, vol.19, issue.4, pp.349-368, 1997.
DOI : 10.1007/BF02427924
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.6723

J. Verron, Nudging satellite altimeter data into quasi-geostrophic ocean models, Journal of Geophysical Research, vol.78, issue.C3, pp.7479-7491, 1992.
DOI : 10.1029/92JC00200

P. A. Vidard, Vers une prise en compte des erreurs modèle en assimilation de données 4D-variationnelle. Application à un modèle réaliste d'océan, 2001.

Z. Wang, I. M. Navon, L. Dimet, F. Zhou, and X. , The second order adjoint analysis: Theory and applications, Meteorology and Atmospheric Physics, vol.44, issue.3, pp.3-20, 1992.
DOI : 10.1007/BF01025501

A. T. Weaver and P. Courtier, Correlation modelling on the sphere using a generalized diffusion equation, Quarterly Journal of the Royal Meteorological Society, vol.108, issue.575, pp.1815-1846, 2001.
DOI : 10.1002/qj.49712757518

A. T. Weaver and S. Ricci, Constructing a background-error correlation model using generalized diffusion operators Recent developments in data assimilation for atmosphere and ocean, ECWMF procreedings on, 2003.

I. Yavneh and G. Dardyk, A Multilevel Nonlinear Method, SIAM Journal on Scientific Computing, vol.28, issue.1, pp.24-46, 2006.
DOI : 10.1137/040613809