Variational Data Assimilation : from optimal control to large scale data assimilation

Optimal control of systems governed by PDEs is a wide, rich and di cult topic. The present course is mainly limited to the numerical resolution of the first order optimality system using adjoint equations.

In a first part, fundamental results of functional analysis, optimization and descent algorithms are recalled. Many exercises from the literature are presented. The students must be comfortable with this basic part.

This chapter focuses on the derivation of the first order optimality system (necessary condi- tions characterizing an optimal control), and its numerical resolution, in the case of a system governed by an non-linear elliptic PDE (the system is distributed in space).

A result of existence and uniqueness of the optimal control is presented in the case linear model - quadratic cost function (LQ PDE case).

A key point of the present resolution is the introduction of the adjoint equations. The optimality system, characterizing the solutions, is derived rigorously in the case of a non-linear elliptic PDE direct model. Also, a presentation based on the introduction of a Lagrangian is done. Finally, the resulting computational control algorithm is elaborated. Concrete examples are presented.

In an engineering point of view, once the optimality system is solved, we obtain either the identification of the input parameters which was known with uncertainties (or unknown), and/or a better calibration of the model.