A note on preconditioning weighted linear least squares, with consequences for weakly-constrained variational data assimilation
This work provides theoretical insights for practitioners using preconditioners in weighted least-squares problems, particularly in data assimilation, though the findings are incremental.
The paper analyzes the effect of preconditioning weighted linear least-squares with an approximate model matrix, revealing potential inefficiencies due to eigenstructure interplay, and discusses implications for weakly-constrained 4D-Var data assimilation.
The effect of preconditioning linear weighted least-squares using an approximation of the model matrix is analyzed, showing the interplay of the eigenstructures of both the model and weighting matrices. A small example is given illustrating the resulting potential inefficiency of such preconditioners. Consequences of these results in the context of the weakly-constrained 4D-Var data assimilation problem are finally discussed.