Multigrid solvers and multigrid preconditioners for the solution of variational data assimilation problems

In order to lower the computational cost of the variational data assimilation process, we investigate the use of multigrid methods to solve the associated optimal control system. In a linear advection equation, we study the impact of the regularization term on the optimal control and the impact of discretization errors on the efficiency of the coarse-grid correction step. We show that, even if the optimal control problem leads to the solution of an elliptic system, numerical errors introduced by the discretization can alter the success of the multigrid method. The view of multigrid iteration as a preconditioner for a Krylov optimization method leads to a more robust algorithm. A scale-dependent weighting of the multigrid preconditioner and the usual background-error covariance-matrix based preconditioner is proposed and brings significant improvements.