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

Keywords

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

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