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

Abstract

Variational data assimilation problems arising 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, 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.In this paper, 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.

A multiple length scale correlation operator for ocean data assimilation

“…(9), using such a preconditioning increases the size of the minimisation vector. It is therefore preferable in this case to precondition the cost function using the full background error covariance matrix, as suggested in earlier studies by Derber and Rosati (1989) and more recently by Gratton and Tshimanga (2009) and Gu¨rol et al (2014). Both methods of preconditioning are equivalent and will lead to the same solution.…”

Observing System Impacts on Estimates of California Current Transport

“…The minimization proceeds via a Lanczos formulation of the restricted B-preconditioned conjugate gradient (CG) method. 24 Each sequence of linear minimizations of Eqn (1) proceeds via so-called inner-loops. When the dz k that minimizes J k has been identified, the estimate z k about which H is linearized is updated (a so-called outer-loop), and minimization of Eqn (1) proceeds again.…”

Improving spatial localization in 4DEnVar

“…Still another formulation of 4DEnVar was suggested by Desroziers et al (), based on the minimization of the following cost function in observation space: and In this case, a reduced B conjugate gradient (Gürol et al ) can be used, which is similar to a double preconditioned conjugate gradient but with a size of control variable now equal to the number of observations P and a transformation of successive gradients, with the same size P , by matrix : …”