Theoretical and experimental study of preconditioners in numerical optimization in the framework of data assimilation in oceanography and meteorology

We are interested in comparing theoretically and experimentally some preconditioners in numerical optimization in the field of data assimilation in oceanographic prevision. The information to be computed which is called "initial state" is used to calculate forcast of the ocean state as a function of time. The goal is to approximate the minimizer of a non linear large scale fonctional using the Gauss-Newton method. This method approximate the solution by incremental resolving linear. These systems are produced by constructing a new minimizing
quadratic problem which appoximates the original one in the neighborhood of the current solution approximation. Conjugate gradient method is used to solve each linear system. The main idea is to use some informations collected during the solution of a system (search direction, residuals,...) to precondition the next one. We discuss the choice of the information to be kept from a CG iteration and compare the efficieny of various preconditionners including spectral or BFGS preconditionners.