B ‐preconditioned minimization algorithms for variational data assimilation with the dual formulation

Variational data assimilation problems in meteorology and oceanography require the solution of a regularized nonlinear least‐squares problem. Practical solution algorithms are based on the incremental (truncated Gauss–Newton) approach, which involves the iterative solution of a sequence of linear least‐squares (quadratic minimization) sub‐problems. Each sub‐problem can be solved using a primal approach, where the minimization is performed in a space spanned by vectors of the size of the model control vector, or a dual approach, where the minimization is performed in a space spanned by vectors of the size of the observation vector. The dual formulation can be advantageous for two reasons. First, the dimension of the minimization problem with the dual formulation does not increase when additional control variables are considered, such as those accounting for model error in a weak‐constraint formulation. Second, whenever the dimension of observation space is significantly smaller than that of the model control space, the dual formulation can reduce both memory usage and computational cost. In this article, a new dual‐based algorithm called Restricted B ‐preconditioned Lanczos (RBLanczos) is introduced, where B denotes the background‐error covariance matrix. RBLanczos is the Lanczos formulation of the Restricted B ‐preconditioned Conjugate Gradient (RBCG) method. RBLanczos generates mathematically equivalent iterates to those of RBCG and the corresponding B ‐preconditioned Conjugate Gradient and Lanczos algorithms used in the primal approach. All these algorithms can be implemented without the need for a square‐root factorization of B . RBCG and RBLanczos, as well as the corresponding primal algorithms, are implemented in two operational ocean data assimilation systems and numerical results are presented. Practical diagnostic formulae for monitoring the convergence properties of the minimization are also presented.