Weak constraint four-dimensional variational data assimilation is an important method for incorporating data (typically observations) into a model. The linearised system arising within the minimisation process can be formulated as a saddle point problem. A disadvantage of this formulation is the large storage requirements involved in the linear system.
Here we present a low-rank approach which exploits the structure of the saddle point system using techniques and theory from solving large scale matrix equations.
Numerical experiments with the linear advection-diffusion equation, and the non-linear Lorenz-95 model demonstrate the effectiveness of a low-rank approximate Krylov subspace solver compared to a standard one.