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Optimization and Engineering

, Volume 10, Issue 3, pp 409–426 | Cite as

Data assimilation in weather forecasting: a case study in PDE-constrained optimization

  • Mike Fisher
  • Jorge Nocedal
  • Yannick Trémolet
  • Stephen J. Wright
Article

Abstract

Variational data assimilation is used at major weather prediction centers to produce the initial conditions for 7- to 10-day weather forecasts. This technique requires the solution of a very large data-fitting problem in which the major element is a set of partial differential equations that models the evolution of the atmosphere over a time window for which observational data has been gathered. Real-time solution of this difficult computational problem requires sophisticated models of atmospheric physics and dynamics, effective use of supercomputers, and specialized algorithms for optimization and linear algebra. The optimization algorithm can be accelerated by using a spectral preconditioner based on the Lanczos method. This paper shows how practical demands of the application dictate the various algorithmic choices that are made in the nonlinear optimization solver, with particular reference to the system in operation at the European Centre for Medium-Range Weather Forecasts.

Keywords

Weather forecasting Variational assimilation PDE-constrained optimization Large-scale optimization 

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References

  1. Biegler L, Waecther A (2003) DAE-constrained optimization. SIAG/OPT Views-and-News 14:10–15 Google Scholar
  2. Biegler LT, Ghattas O, Heinkenschloss M, van Bloemen Waanders B (2003) Large-scale PDE-constrained optimization. Lecture notes in computational science and engineering, vol 30. Springer, Berlin MATHGoogle Scholar
  3. Biros G, Ghattas O (2005) Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. Part I: The Krylov-Schur solver. SIAM J Sci Comput 27:687–713 MATHCrossRefMathSciNetGoogle Scholar
  4. Bock H-G, Plitt KJ (1984) A multiple shooting algorithm for direct solution of optimal control problems. In: Proceedings of the 9th IFAC world conference, Budapest, 1984. Pergamon, Elmsford, pp 242–247 Google Scholar
  5. Courtier P, Thépaut J-N, Hollingsworth A (1994) A strategy for operational implementation of 4D-Var, using an incremental approach. Q J R Meteorol Soc 120:1367–1387 CrossRefGoogle Scholar
  6. Fisher M, Leutbecher M, Kelly G (2005) On the equivalence between Kalman smoothing and weak-constraint four-dimensional variational data assimilation. Q J R Meteorol Soc 131:3235–3246 CrossRefGoogle Scholar
  7. Giraud L, Gratton S (2006) On the sensitivity of some spectral preconditioners. SIAM J Matrix Anal Appl 27:1089–1105 MATHCrossRefMathSciNetGoogle Scholar
  8. Golub GH, Van Loan CF (1989) Matrix computations, 2nd edn. Johns Hopkins Press, Baltimore MATHGoogle Scholar
  9. Haltiner GJ, Williams R (1980) Numerical prediction and dynamic meteorology, 2nd edn. Wiley, New York Google Scholar
  10. Morales JL, Nocedal J (2000) Automatic preconditioning by limited memory quasi-Newton updating. SIAM J Optim 10:1079–1096 MATHCrossRefMathSciNetGoogle Scholar
  11. Nabben R, Vuik C (2004) A comparison of deflation and coarse grid correction applied to porous media flow. SIAM J Numer Anal 42:1631–1647 MATHCrossRefMathSciNetGoogle Scholar
  12. Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edn. Springer, Berlin MATHGoogle Scholar
  13. Notay Y (1993) On the convergence rate of the conjugate gradients in presence of rounding errors. Numer Math 65:301–317 MATHCrossRefMathSciNetGoogle Scholar
  14. Simmons A, Hollingsworth A (2002) Some aspects of the improvement in skill of numerical weather prediction. Q J R Meteorol Soc 128:647–677 CrossRefGoogle Scholar
  15. Trémolet Y (2005) Incremental 4D-Var convergence study. Technical Memorandum 469, European Center for Medium-Range Weather Forecasts, July 2005 Google Scholar
  16. Trémolet Y (2006) Accounting for an imperfect model in 4D-Var. Q J R Meteorol Soc 132:2483–2504 CrossRefGoogle Scholar
  17. Young DP, Huffman WP, Melvin RG, Hilmes CL, Johnson FT (2002) Nonlinear elimination in aerodynamic analysis and design optimization. In: Proceedings of the first Sandia workshop on large-scale PDE constrained optimization. Lecture notes in computational science and engineering. Springer, Berlin Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Mike Fisher
    • 1
  • Jorge Nocedal
    • 2
  • Yannick Trémolet
    • 1
  • Stephen J. Wright
    • 3
  1. 1.European Centre for Medium-Range Weather ForecastsReadingUK
  2. 2.Department of Electrical Engineering and Computer ScienceNorthwestern UniversityEvanstonUSA
  3. 3.Department of Computer SciencesUniversity of WisconsinMadisonUSA

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