## Contribution de S. GUROL, A. WEAVER, S. GRATTON, A. MOORE, A. PIACENTINI & H. ARANGO:

*B-Preconditioned Minimization Algorithms for Variational Data Assimilation with the Dual Formulation*

*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 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, such as those accounting for model error in a weak-constraint formulation, are considered. 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. This presentation describes recent work in developing the dual approach for two operational ocean data assimilation systems: NEMOVAR and ROMS 4D-Var. NEMOVAR employs the Restricted B-preconditioned Conjugate Gradient (RBCG) method, while ROMS 4D-Var employs the B-preconditioned Lanczos (RBLanczos) method. RBCG and RBLanczos, and the corresponding B-preconditioned Conjugate Gradient and Lanczos algorithms used in the primal approach, generate mathematically equivalent iterates. All these algorithms can be implemented without the need for a square-root factorization of the background-error covariance matrix (B). Numerical results comparing the dual and primal algorithms in NEMOVAR and ROMS 4D-Var will be presented.*