Data assimilation is a methodology for estimating the initial state of a dynamical system by combining the information from observational data and from a numerical prediction model that describes the evolution of the system. The most important fields of application of data assimilation are the ocean and weather forecasts. In this talk, we briefly survey the two main approaches used in data assimilation: the sequential one, based on the statistical estimation theory (Kalman filter) and the variational one, based on the optimal control theory. This last approach amounts to solve a very large weighted nonlinear least-squares problem called 4D-Var (four-dimensional variational problem).
Focussing on the solution of the 4D-Var problem, we discuss two challenging issues in the context of large-scale operational data assimilation: preconditioning techniques, for accelerating the convergence, and derivative-free techniques, to avoid the computation of derivatives.
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