Saddle-point systems have received a lot of attention in recent years since they arise in a wide
variety of applications such as constrained optimization, computational uid dynamics and
optimal control. In this study, we focus on the saddle point representation of a variational
large-scale data assimilation problem where the computation with the constraint blocks are
computationally more expensive than the (1,1) block of the underlying saddle point system.
Low-rank limited memory preconditioners exploiting the structure of this particular problem
are proposed and their performance are presented by the numerical experiments obtained
from the Object Oriented Prediction System (OOPS) developed by ECMWF.