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The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The equation for the analysis error is derived through the errors of the input data (background and observation errors). This equation is used to show that in a nonlinear case the analysis error covariance operator can be approximated by the inverse Hessian of an auxiliary data assimilation problem which involves the tangent linear model constraints. The inverse Hessian is constructed by the quasi-Newton BFGS algorithm when solving the auxiliary data assimilation problem. A fully nonlinear ensemble procedure is developed to verify the accuracy of the proposed algorithm. Numerical examples are presented.

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This paper presents a reduced-order approach for four-dimensional variational data assimilation, based on a prior EOF analysis of a model trajectory. This method implies two main advantages: a natural model-based definition of a multivariate background error covariance matrix B r , and an important decrease of the computational burden of the method, due to the drastic reduction of the dimension of the control space. An illustration of the feasibility and the effectiveness of this method is given in the academic framework of twin experiments for a model of the equatorial Pacific ocean. It is shown that the multivariate aspect of B r brings additional information which substantially improves the identification procedure. Moreover the computational cost can be decreased by one order of magnitude with regard to the full-space 4D-Var method.

Computational experience with several limited-memory quasi-Newton and truncated Newton methods for unconstrained nonlinear optimization is described. Comparative tests were conducted on a well-known test library [J. J. Mor, B. S. Garbow, and K. E. Hillstrom, ACM Trans. Math. Software, 7 (1981), pp. 17-41], on several synthetic problems allowing control of the clustering of eigenvalues in the Hessian spectrum, and on some large-scale problems in oceanography and meteorology. The results indicate that among the tested limited-memory quasi-Newton methods, the L-BFGS method [D. C. Liu and J. Nocedal, Math. Programming, 45 (1989), pp. 503-528] has the best overall performance for the problems examined. The numerical performance of two truncated Newton methods, differing in the inner-loop solution for the search vector, is competitive with that of L-BFGS. Key words, limited-memory quasi-Newton methods, truncated Newton methods, synthetic cluster functions, large-scale unconstrained minimization AMS subject classifications. 90C30, 93C20, 93C75, 65K10, 76C20 1. Introduction. Limited-memory quasi-Newton (LMQN) and truncated Newton (TN) methods represent two classes of algorithms that are attractive for large-scale problems because of their modest storage requirements. They use a low and adjustable amount of storage and require the function and gradient values at each iteration. Preconditioning of the Newton equations may be used for both algorithms. In this case, additional function information (e.g., a sparse approximation to the Hessian) may also be required at each iteration. LMQN methods can be viewed as extensions of conjugate-gradient (CG) methods in which the addition of some modest storage serves to accelerate the convergence rate. TN methods attempt to retain the rapid convergence rate of classical Newton methods while economizing storage and computational requirements so as to become feasible for large-scale applications. They can be particularly powerful when structure information of the objective function is exploited LMQN originated from the works of Nazareth [21] and Perry [25], [26] and were further extended by Shanno [31], [32] resulting in the CONMIN code of Shanno and *