Randomised preconditioning for the forcing formulation of weak‐constraint 4D‐Var

There is growing awareness that errors in the model equations cannot be ignored data assimilation methods such as four-dimensional variational (4D-Var). If allowed for, more information can extracted from observations, longer time windows are possible, and minimisation process easier, at least principle. Weak-constraint 4D-Var estimates error minimises a series of quadratic cost functions, which achieved using conjugate gradient (CG) method; minimising each function called an inner loop. CG needs preconditioning to improve its performance. In previous work, limited-memory preconditioners (LMPs) have been constructed approximations eigenvalues eigenvectors Hessian changes significantly consecutive loops, LMP may limited usefulness. To circumvent this, we propose randomised for low-rank eigenvalue decomposition use these construct LMPs cheaply current Three compared. Numerical experiments idealized systems show resulting perform better than existing LMPs. Using allow efficient robust implementations incremental weak-constraint 4D-Var.