Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects

Two general algorithms for solving constrained minimization problems are presented and discussed in the context of analysis and assimilation of meteorological observations. In both algorithms, the original constrained problem is transformed by appropriate modifications into one unconstrained problem, or into a sequence of unconstrained problems. The main advantage of proceeding in this way is that the new unconstrained problems can be solved by classical descent algorithms, thus avoiding the need of directly solving the Euler-Lagrange equations of the original constrained problem. The first algorithm presented in the augmented lagrangian algorithm. It generalizes the more classical penalty and duality algorithms. The second algorithm, inspired from optimal control techniques, is based on an appropriate use of an adjoint dynamical equation, and seems to be particularly well adapted to the assimilation of observations distributed in time. Simple numerical examples show the ability of these algorithms to solve non-linear minimization problems of the type encountered in meteorology. Their possible use in more complex situations is discussed, in particular in terms of their computational cost. DOI: 10.1111/j.1600-0870.1986.tb00459.x