### 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.

Language | English |
---|---|

Pages | 934-940 |

Number of pages | 7 |

Journal | Quarterly Journal of the Royal Meteorological Society |

Volume | 144 |

Issue number | 712 |

DOIs | |

Publication status | Published - Apr 2018 |

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### Keywords

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

### Cite this

*Quarterly Journal of the Royal Meteorological Society*,

*144*(712), 934-940. https://doi.org/10.1002/qj.3262

}

*Quarterly Journal of the Royal Meteorological Society*, vol. 144, no. 712, pp. 934-940. https://doi.org/10.1002/qj.3262

**A note on preconditioning weighted linear least-squares with consequences for weakly constrained variational data assimilation.** / Gratton, Serge; Selime, Gürol; Simon, Ehouarn; Toint, Philippe.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A note on preconditioning weighted linear least-squares with consequences for weakly constrained variational data assimilation

AU - Gratton, Serge

AU - Selime, Gürol

AU - Simon, Ehouarn

AU - Toint, Philippe

PY - 2018/4

Y1 - 2018/4

N2 - 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.

AB - 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.

KW - linear least-squares

KW - preconditioning

KW - data assimilation

KW - weakly-constrained 4D-Var

KW - earth sciences

UR - http://www.scopus.com/inward/record.url?scp=85052490834&partnerID=8YFLogxK

U2 - 10.1002/qj.3262

DO - 10.1002/qj.3262

M3 - Article

VL - 144

SP - 934

EP - 940

JO - Quarterly Journal of the Royal Meteorological Society

T2 - Quarterly Journal of the Royal Meteorological Society

JF - Quarterly Journal of the Royal Meteorological Society

SN - 0035-9009

IS - 712

ER -