Erratum: Limited-memory preconditioners, with application to incremental four-dimensional variational data assimilation (Quarterly Journal of the Royal Meteorological Society (2008) vol. 134 (632) (751-769) 10.1002/qj.228);Limited-memory preconditioners, with application to incremental four-dimensional variational data assimilation

Quarterly Journal of the Royal Meteorological Society

Volumn 134, Issue 632, 2008, Pages 751-769

  Tshimanga, J ^a,b   Gratton, S ^b,c   Weaver, Anthony T ^b   Sartenaer, A ^a  

^a UNIVERSITY OF NAMUR   (Belgium)

^b CERFACS   (France)

^c CNES   (France)

Author keywords

4D Var; Conjugate gradient; Minimization; Preconditioning

Indexed keywords

CONJUGATE GRADIENT METHOD; COST FUNCTIONS; OPTIMIZATION; PROBLEM SOLVING; VALUE ENGINEERING; VECTORS;

4D-VAR; APPROXIMATE EIGENVECTORS; FOUR DIMENSIONAL VARIATIONAL; FOUR-DIMENSIONAL VARIATIONAL DATA ASSIMILATION; NONLINEAR MINIMIZATION; PRECONDITIONING; QUADRATIC MINIMIZATION; SYMMETRIC POSITIVE DEFINITE MATRICES;

MATRIX ALGEBRA;

ALGORITHM; DATA ASSIMILATION; FLOW FIELD; GLOBAL OCEAN;

EID: 46849085278     PISSN: 00359009     EISSN: None     Source Type: Journal    
DOI: 10.1002/qj.292     Document Type: Erratum

References (42)

1. Benzi M. 2002. Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182: 418-477.
2. Björck Å. 1996. Numerical Methods for Least Squares Problems. SIAM: Philadelphia, USA.
3. Björck Å, Elfving T, Strakos Z. 1998. Stability of conjugate gradient and Lanczos methods for linear least squares problems. SIAM J. Matrix Anal. A. 19: 720-736.
4. Chen K. 2005. Matrix Preconditioning Techniques and Applications. SIAM: Philadelphia, USA.
5. Courtier P. 1997. Dual formulations of four-dimensional variational assimilation. Q. J. R. Meteorol. Soc. 123: 2449-2461.
6. Courtier P, Thépaut JN, Hollingsworth A. 1994. A strategy for operational implementation of 4D-Var, using an incremental approach. Q. J. R. Meteorol. Soc. 120: 1367-1388.
7. Cullum JK, Willoughby RA. 2002. Lanczos Algorithms for Large Symmetric Eigenvalue Computations. SIAM: Philadelphia, USA.
8. Dennis JE, Schnabel RB. 1983. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall: Englewood Cliffs, New Jersey, USA. Reprinted as Classics in Applied Mathematics 16. SIAM: Philadelphia, USA, 1996.
9. Derber J, Bouttier F. 1999. A reformulation of the background error covariance in the ECMWF global data assimilation system. Tellus 51A: 195-221.
10. Di Lorenzo E, Moore AM, Arango HG, Cornuelle BD, Miller AJ, Powell B, Chua BS, Bennett AF. 2006. Weak and strong constraint data assimilation in the inverse Regional Ocean Modeling System (ROMS): Development and application for a baroclinic coastal upwelling system. Ocean Model. 16: 160-187.
11. Fisher M. 1998. 'Minimization algorithms for variational data assimilation'. Pp 364-385 in Recent Developments in Numerical Methods for Atmospheric Modelling. ECMWF.
12. Fisher M, Nocedal J, Trémolet Y, Wright S. 2006. 'Data assimilation in weather forecasting: a case study in PDE-constrained optimization'. Technical Report, Optimization Technology Center, Northwestern University.
13. Gilbert JC, Lemaréchal C. 1989. Some numerical experiments with variable-storage quasi-Newton algorithms Math. Program. 45: 407-435.
14. Golub GH, van Loan CF. 1996. Matrix Computations, 3rd edn. Johns Hopkins University Press: Baltimore.
15. Gratton S, Lawless A, Nichols N. 2007. Approximate Gauss-Newton methods for nonlinear least squares problems. SIAM J. Optimiz. 18: 106-132.
16. Hestenes MR, Stiefel E. 1952. Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49: 409-436.
17. Horn RA, Johnson CR. 1999. Matrix Analysis. Cambridge University Press.
18. Ingleby B, Huddleston M. 2007. Quality control of ocean temperature and salinity profiles - historical and real-time data. J. Marine Syst. 65: 158-175.
19. Kelley CT. 1999. Iterative Methods for Optimization. SIAM: Philadelphia, USA.
20. Lawless AS, Nichols NK. 2006. Inner-loop stopping criteria for incremental four-dimensional variational assimilation. Mon. Weather Rev. 134: 3425-3435.
21. Lawless AS, Gratton S, Nichols NK. 2005. Approximate iterative methods for variational data assimilation. Int. J. Numer. Meth. Fl. 1: 1-6.
22. Le Dimet FX, Talagrand O. 1986. Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects. Tellus 38A: 97-110.
23. Lorenc AC. 2003. Modelling of error covariances by 4D-Var data assimilation. Q. J. R. Meteorol. Soc. 129: 3167-3182.
24. Madec G, Delecluse P, Imbard M, Levy C. 1998. 'OPA8.1 Ocan General Circulation Model Reference Manual'. Technical Note 11, LODYC/IPSL, Paris, France.
25. Meurant G. 2006. The Lanczos and Conjugate Gradient Algorithms: From Theory to Finite Precision Computations SIAM: Philadelphia, USA.
26. Morales J, Nocedal J. 2000. Automatic preconditioning by limited memory quasi-Newton updating. SIAM J. Optimiz. 10 (4): 1079-1096.
27. Nocedal J. 1980. Updating quasi-Newton matrices with limited storage. Math. Comput. 35: 773-782.
28. Nocedal J, Wright SJ. 1999. Numerical Optimization. Series in Operations Research, Springer: Heidelberg, Berlin, New York.
29. Paige CC, Saunders MA. 1982. LSQR: an algorithm for sparse linear equations and sparse least squares. ACM T. Math. Software 8: 43-71.
30. Rabier F. 2005. Overview of global data assimilation developments in numerical weather-prediction centres. Q. J. R. Meteorol. Soc. 131: 3215-3233.
31. Rabier F, Järvinen H, Klinker E, Mahfouf JF, Simmons A. 2000. The ECMWF operational implementation of four-dimensional variational assimilation. I: Experimental results with simplified physics. Q. J. R. Meteorol. Soc. 117: 1225-1254.
32. Rigal RL, Gaches J. 1967. On the compatibility of a given solution with the data of a linear system. J. ACM 14: 543-548.
33. Toint PL. 1987. On large scale nonlinear least squares calculations. SIAM J. Sci. Stat. Comp. 8: 416-435.
34. Trémolet Y. 2007. Incremental 4D-Var convergence study. Tellus 59A: 706-718.
35. Tshimanga J. 2007. On a Class of Limited Memory Preconditioners for Large-Scale Nonlinear Least-Squares Problems (with Application to Variational Ocean Data Assimilation). PhD thesis, University of Namur, Belgium. 167 pp.
36. Uppala SM, Kållberg PW, Simmons AJ, Andrae U, Bechtold da Costa V, Fiorino M, Gibson JK, Haseler J, Hernandez A, Kelly GA, Li X, Onogi K, Saarinen S, Sokka N, Allan RP, AnderssonE, Arpe K, Balmaseda MA, Beljaars ACM, van de Berg L, Bidlot J, Bormann N, Caires S, Chevallier F, Dethof A, Dragosavac M, Fisher M, Fuentes M, Hagemann S, Hólm E, Hoskins BJ, Isaksen L, Janssen PAEM, Jenne R, McNally AP, Mahfouf I-F, Morcette J-J, Rayner NA, Saunders RW, Simon P, Sterl A, Trenberth KE, Untch A, Vasiljevic D, Viterbo P, Woollen J. 2005. The ERA-40 re-analysis. Q. J. R. Meteorol. Soc. 131: 2961-3012.
37. Van der Vorst HA. 2003. Iterative Krylov Methods for Large Linear Systems. Cambridge University Press: Cambridge, UK.
38. Weaver AT, Vialard J, Anderson DLT. 2003. Three- and four-dimensional variational assimilation with a general circulation model of the tropical Pacific Ocean. Part I: Formulation, internal diagnostics, and consistency checks. Mon. Weather Rev. 131: 1360-1378.
39. Weaver AT, Deltel C, Machu E, Ricci S, Daget N. 2005. A multivariate balance operator for variational ocean data assimilation. Q. J. R. Meteorol. Soc. 131: 3605-3625.
40. Yang W, Navon IM, Courtier P. 1996. A new Hessian preconditioning method applied to variational data assimilation experiments using NASA general circulation models. Mon. Weather Rev. 124: 1000-1017.
41. Zupanski M. 1993. A preconditioning algorithm for large-scale minimization problems. Tellus 45A: 478-492.
42. Zupanski M. 1996. A preconditioning algorithm for four-dimensional variational data assimilation. Mon. Weather Rev. 124: 2562-2573.