A low-rank approach to the solution of weak constraint variational data assimilation problems

Highlights: • The weak constraint four dimensional variational data assimilation problem is written in saddle point formulation. • Using low-rank methods from matrix equation theory, a new low-rank GMRES solver is proposed. • Several preconditioning approaches are investigated with low-rank GMRES. • Computations show that this new approach is successful and we achieve close approximations to the full-rank solutions. • Storage requirements of the new low-rank approach are only up to 10% of those needed by the full-rank approach. 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. In this paper, 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 Krylov subspace solver when compared to a traditional solver.