Résumé
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.
| langue originale | Anglais |
|---|---|
| Pages (de - à) | 934-940 |
| Nombre de pages | 7 |
| journal | Quarterly Journal of the Royal Meteorological Society |
| Volume | 144 |
| Numéro de publication | 712 |
| Les DOIs | |
| état | Publié - avr. 2018 |
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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.
Dans: Quarterly Journal of the Royal Meteorological Society, Vol 144, Numéro 712, 04.2018, p. 934-940.Résultats de recherche: Contribution à un journal/une revue › 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
JF - Quarterly Journal of the Royal Meteorological Society
SN - 0035-9009
IS - 712
ER -