Computation of the analysis error covariance in variational data assimilation problems with nonlinear dynamics

I.Yu Gejadze, G.J.M Copeland, F.X. Le Dimet, V. Shutyaev

Research output: Contribution to journalArticle

26 Citations (Scopus)

Abstract

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The data contain errors (observation and background errors), hence there will be errors in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can often be approximated by the inverse Hessian of the cost functional. Here we focus on highly nonlinear dynamics, in which case this approximation may not be valid. The equation relating the optimal solution error and the errors of the input data is used to construct an approximation of the optimal solution error covariance. Two new methods for computing this covariance are presented: the fully nonlinear ensemble method with sampling error compensation and the ‘effective inverse Hessian’ method. The second method relies on the efficient computation of the inverse Hessian by the quasi-Newton BFGS method with preconditioning. Numerical examples are presented for the model governed by Burgers equation with a nonlinear viscous term.
LanguageEnglish
Pages7923-7943
Number of pages21
JournalJournal of Computational Physics
Volume230
Issue number22
Early online date30 Mar 2011
DOIs
Publication statusPublished - 10 Sep 2011

Fingerprint

error analysis
assimilation
Error analysis
Error compensation
preconditioning
Newton methods
Newton-Raphson method
Burger equation
Covariance matrix
optimal control
approximation
Sampling
sampling
costs

Keywords

  • posterior covariance
  • computational physics
  • nonlinear dynamics

Cite this

Gejadze, I.Yu ; Copeland, G.J.M ; Le Dimet, F.X. ; Shutyaev, V. / Computation of the analysis error covariance in variational data assimilation problems with nonlinear dynamics. In: Journal of Computational Physics. 2011 ; Vol. 230, No. 22. pp. 7923-7943.
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Computation of the analysis error covariance in variational data assimilation problems with nonlinear dynamics. / Gejadze, I.Yu; Copeland, G.J.M; Le Dimet, F.X.; Shutyaev, V.

In: Journal of Computational Physics, Vol. 230, No. 22, 10.09.2011, p. 7923-7943.

Research output: Contribution to journalArticle

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AB - The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The data contain errors (observation and background errors), hence there will be errors in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can often be approximated by the inverse Hessian of the cost functional. Here we focus on highly nonlinear dynamics, in which case this approximation may not be valid. The equation relating the optimal solution error and the errors of the input data is used to construct an approximation of the optimal solution error covariance. Two new methods for computing this covariance are presented: the fully nonlinear ensemble method with sampling error compensation and the ‘effective inverse Hessian’ method. The second method relies on the efficient computation of the inverse Hessian by the quasi-Newton BFGS method with preconditioning. Numerical examples are presented for the model governed by Burgers equation with a nonlinear viscous term.

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