Publication Date:
2018-03-15
Description:
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 that compensates for the condition number of the weighting matrix. A low-dimension analytical example is given illustrating the resulting potential inefficiency of such preconditioners. Consequences of these results in the context of the state formulation of the weakly-constrained 4D-Var data assimilation problem are finally discussed. It is shown that the common approximations of the tangent linear model that maintain parallelization-in-time properties (identity or null matrix) can result in large bounds on the eigenvalues of the preconditioned matrix system.
Print ISSN:
0035-9009
Electronic ISSN:
1477-870X
Topics:
Geography
,
Physics
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