<b>B</b>‐preconditioned minimization algorithms for variational data assimilation with the dual formulation

Quarterly Journal of the Royal Meteorological Society - Tập 140 Số 679 - Trang 539-556 - 2014
Selime Gürol1,2, Anthony Weaver1, Andrew M. Moore3, Andrea Piacentini4, Hernan G. Arango5, Serge Gratton6,1,7
1Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique
2IRT Saint Exupéry - Institut de Recherche Technologique
3University of California, Santa Cruz
4Cerfacs, Toulouse, France
5Rutgers, The State University of New Jersey, New Brunswick;
6Algorithmes Parallèles et Optimisation
7Institut National Polytechnique (Toulouse)

Tóm tắt

AbstractVariational data assimilation problems in meteorology and oceanography require the solution of a regularized nonlinear least‐squares problem. Practical solution algorithms are based on the incremental (truncated Gauss–Newton) approach, which involves the iterative solution of a sequence of linear least‐squares (quadratic minimization) sub‐problems. Each sub‐problem can be solved using a primal approach, where the minimization is performed in a space spanned by vectors of the size of the model control vector, or a dual approach, where the minimization is performed in a space spanned by vectors of the size of the observation vector. The dual formulation can be advantageous for two reasons. First, the dimension of the minimization problem with the dual formulation does not increase when additional control variables are considered, such as those accounting for model error in a weak‐constraint formulation. Second, whenever the dimension of observation space is significantly smaller than that of the model control space, the dual formulation can reduce both memory usage and computational cost.In this article, a new dual‐based algorithm called RestrictedB‐preconditioned Lanczos (RBLanczos) is introduced, whereBdenotes the background‐error covariance matrix. RBLanczos is the Lanczos formulation of the RestrictedB‐preconditioned Conjugate Gradient (RBCG) method. RBLanczos generates mathematically equivalent iterates to those of RBCG and the correspondingB‐preconditioned Conjugate Gradient and Lanczos algorithms used in the primal approach. All these algorithms can be implemented without the need for a square‐root factorization ofB. RBCG and RBLanczos, as well as the corresponding primal algorithms, are implemented in two operational ocean data assimilation systems and numerical results are presented. Practical diagnostic formulae for monitoring the convergence properties of the minimization are also presented.

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Tài liệu tham khảo

Axelsson O., 1996, Iterative Solution Methods

10.1002/qj.2063

10.1017/CBO9780511535895

10.1016/j.dynatmoce.2009.03.001

Broquet G, 2009, Ocean state and surface forcing correction using the ROMS‐IS 4DVAR data assimilation system, Mercator Ocean Quarterly Newsletter, 34, 5

10.1256/qj.04.15

10.1256/qj.03.205

10.1137/S1064827596311554

10.1175/1520-0485(1985)015<1060:NTOCSO>2.0.CO;2

10.1016/j.pocean.2009.07.028

10.1002/nla.306

10.1016/S1463-5003(01)00006-3

10.1175/1520-0493(1998)126<2913:ATEODS>2.0.CO;2

10.1002/qj.49712354414

10.1002/qj.49712051912

Da Silva A, 1995, Proceedings of the 2nd WMO Symposium on assimilation of observations in meteorology and oceanography, 273

10.1175/1520-0493(2001)129<0869:NFAD>2.0.CO;2

10.1175/1520-0485(1989)019<1333:AGODAS>2.0.CO;2

10.1002/qj.1886

10.1016/j.dsr2.2008.08.009

10.1029/94JC01894

10.1002/qj.545

10.1002/qj.257

El Akkraoui A, 2012, Preconditioning of variational data assimilation and the use of a bi‐conjugate gradient method, Q. J. R. Meteorol. Soc.

10.1007/978-3-642-03711-5

10.1029/95JC03190

10.1029/95JC03205

Fisher M., 2003, Proceedings of Seminar on Recent Developments in Data Assimilation for Atmosphere and Ocean, 87

10.1007/s11081-008-9051-5

Flather RA., 1976, A tidal model of the northwest European continental shelf, Mem. Soc. R. Sci. Liege, 10, 141

10.1175/1520-0493(2001)129<2089:IOTDFA>2.0.CO;2

10.1111/j.1600-0870.2008.00388.x

Golub GH, 1996, Matrix Computations

10.1002/qj.477

10.1137/050624935

GrattonS TointPL TshimangaJ.2009.‘Inexact range‐space Krylov solvers for linear systems arising from inverse problems’. Technical Report 09/20 Department of Mathematics FUNDP University of Namur: Belgium.

10.1002/qj.743

10.1137/08074008

10.1007/s10589-012-9478-7

10.1016/j.jcp.2007.06.016

Hickey BM., 1998, Coastal oceanography of western North America from the tip of Baja California to Vancouver Island, The Sea, 11, 345

Horn RA, 1999, Matrix Analysis

10.1002/fld.851

10.1175/1520-0469(1979)036<1722:BPOASE>2.0.CO;2

10.1002/qj.49711447911

10.2151/jmsj1965.75.1B_339

MadecG.2008. ‘NEMO ocean engine’. Technical Report 27 Note du Pôle de modélisation Institut Pierre‐Simon Laplace (IPSL) France.http://www.nemo‐ocean.eu.

10.1016/S1463-5003(00)00013-5

Mogensen K, 2009, ‘NEMOVAR: A variational data assimilation system for the NEMO model’, ECMWF Newsletter, 120, 17

MogensenK BalmasedaMA WeaverAT.2012. ‘The NEMOVAR ocean data assimilation system as implemented in the ECMWF ocean analysis for System 4’. Tech. Report 668 ECMWF: Reading UK.

10.1016/j.pocean.2011.05.004

10.1016/j.pocean.2011.05.003

10.1016/j.pocean.2011.05.005

Nocedal J, 2006, Numerical Optimization

Nour‐Omid B, 1988, Proceedings of First International Symposium on Domain Decomposition Methods for Partial Differential Equations, 250

10.1137/0712047

10.1002/nme.1620290110

Parlett BN., 1980, The symmetric eigenvalue problem

10.1175//1520-0493(2003)131<1524:NAOTAO>2.0.CO;2

Saad Y., 1996, Iterative Methods for Sparse Linear Systems

10.1175/1520-0493(1970)098<0875:SBFINV>2.3.CO;2

Shchepetkin AF, 2003, A method for computing horizontal pressure‐gradient force in an oceanic model with a nonaligned vertical grid, J. Geophys. Res., 108, 10.1029/2001JC001047

10.1016/j.ocemod.2004.08.002

10.1137/1.9780898717921

10.1256/qj.05.224

10.1002/qj.228

Veneziani M, 2009, A central California coastal ocean modeling study: 1. Forward model and the influence of realistic versus climatological forcing., J. Geophys. Res., 114

Veneziani M, 2009, A central California coastal ocean modeling study: 2. Adjoint sensitivities to local and remote forcing mechanisms., J. Geophys. Res., 114

10.1175/MWR3282.1

10.1175/2008MWR2444.1

10.1002/qj.1955

10.1256/qj.05.119

10.1175/1520-0493(2003)131<1360:TAFVAW>2.0.CO;2

10.1016/j.physd.2006.09.040

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Quarterly Journal of the Royal Meteorological Society

Tập 140 Số 679

539-556

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