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

TL;DR

This paper analyzes how preconditioning affects weighted linear least squares problems, revealing potential inefficiencies and discussing implications for weakly-constrained 4D-Var data assimilation.

Contribution

It provides a theoretical analysis of preconditioning effects on weighted least squares and explores implications for data assimilation methods.

Findings

Preconditioning can lead to inefficiencies depending on eigenstructure interactions.

Eigenstructure interplay impacts the effectiveness of preconditioners.

Implications for weakly-constrained 4D-Var are discussed.

Abstract

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.