Preconditioning of variational data assimilation and the use of a bi-conjugate gradient method

Presently, a preferred minimization for strong-constraint four-dimensional variational (4D-Var) assimilation uses a Lanczos-based conjugate gradient (CG) algorithm. This requires the availability of a square-root of the background-error covariance matrix (B). In the context of weak-constraint 4D-Var, this requirement might be too restrictive for the formulations of the model error term. It might therefore be desirable to avoid a square-root decomposition of the augmented background term. An appealing minimization scheme is the double CG minimization employed, for example, in the grid-point statistical interpolation (GSI) analysis. Realizing the double CG algorithm is a special case of the more general bi-conjugate gradient (BiCG) method for solving non-symmetric problems, the present work introduces a Lanczos-based preconditioning strategy when B, instead of its square-root, is used initially. Implementation of the scheme is done in the context of the GSI analysis system, and preliminary experiments are presented using its 3D-Var version. Comparison of the Lanczos-based CG and the BiCG shows that the algorithms converge at the same rate and to the same solution. Despite the additional computational cost, the importance of the re-orthogonalization step is also shown to be fundamental to any of these CG algorithms. Furthermore, when using the Hessian eigenvectors for preconditioning, the BiCG behaviour is shown to be comparable to that of the Lanczos-CG algorithm. Both schemes construct the same approximation of the Hessian with the same number of eigenvectors, and benefit in the same way from the reduction of the condition number. The efficiency, computational cost, and stability of the three algorithms are discussed. Copyright (c) 2012 Royal Meteorological Society