A note on preconditioning weighted linear least-squares with consequences for weakly constrained variational data assimilation

Serge Gratton, Gürol Selime, Ehouarn Simon, Philippe Toint

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Abstract

The effect of preconditioning linear weighted least-squares using an approximation of the model matrix is analyzed. The aim is to investigate from a theoretical point of view the inefficiencies of this approach as observed in the application of the weakly constrained 4D-Var algorithm in geosciences. Bounds on the eigenvalues of the preconditioned system matrix are provided. It highlights the interplay of the eigenstructures of both the model and weighting matrices: maintaining a low bound on the eigenvalues of the preconditioned system matrix requires an approximation error of the model matrix which compensates for the condition number of the weighting matrix. A low-dimension analytical example is given illustrating the resulting potential inefficiency of such preconditioners. The consequences of these results in the context of the state formulation of the weakly constrained 4D-Var data assimilation problem are discussed. It is shown that the common approximations of the tangent linear model which maintain parallelization-in-time properties (identity or null matrix) can result in large bounds on the eigenvalues of the preconditioned matrix system.

Original languageEnglish
Pages (from-to)934-940
Number of pages7
JournalQuarterly Journal of the Royal Meteorological Society
Volume144
Issue number712
DOIs
Publication statusPublished - Apr 2018

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Data Assimilation
Linear Least Squares
Weighted Least Squares
Preconditioning
data assimilation
matrix
eigenvalue
Matrix Models
Eigenvalue
Weighting
Approximation Error
Approximation
Condition number
Parallelization
Preconditioner
Tangent line
Null
Linear Model
Lower bound
Formulation

Keywords

  • linear least-squares
  • preconditioning
  • data assimilation
  • weakly-constrained 4D-Var
  • earth sciences

Cite this

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abstract = "The effect of preconditioning linear weighted least-squares using an approximation of the model matrix is analyzed. The aim is to investigate from a theoretical point of view the inefficiencies of this approach as observed in the application of the weakly constrained 4D-Var algorithm in geosciences. Bounds on the eigenvalues of the preconditioned system matrix are provided. It highlights the interplay of the eigenstructures of both the model and weighting matrices: maintaining a low bound on the eigenvalues of the preconditioned system matrix requires an approximation error of the model matrix which compensates for the condition number of the weighting matrix. A low-dimension analytical example is given illustrating the resulting potential inefficiency of such preconditioners. The consequences of these results in the context of the state formulation of the weakly constrained 4D-Var data assimilation problem are discussed. It is shown that the common approximations of the tangent linear model which maintain parallelization-in-time properties (identity or null matrix) can result in large bounds on the eigenvalues of the preconditioned matrix system.",
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A note on preconditioning weighted linear least-squares with consequences for weakly constrained variational data assimilation. / Gratton, Serge; Selime, Gürol; Simon, Ehouarn; Toint, Philippe.

In: Quarterly Journal of the Royal Meteorological Society, Vol. 144, No. 712, 04.2018, p. 934-940.

Research output: Contribution to journalArticle

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