Saddle point preconditioners for weak-constraint 4d-var

Data assimilation algorithms combine information from observations and prior model information to obtain the most likely state of a dynamical system. The linearised weak-constraint four-dimensional variational assimilation problem can be reformulated as a saddle point problem, which admits more scope for preconditioners than the primal form. In this paper we design new terms that can be used within existing preconditioners, such as block diagonal and constraint-type preconditioners. Our novel preconditioning approaches (i) incorporate model information and (ii) are designed to target correlated observation error covariance matrices. To our knowledge, (i) has not been considered previously for data assimilation problems. We develop a theory demonstrating the effectiveness of the new preconditioners within Krylov subspace methods. Linear and non-linear numerical experiments reveal that our new approach leads to faster convergence than existing state-of-the-art preconditioners for a broader range of problems than indicated by the theory alone. We present a range of numerical experiments performed in serial.