Augmented Lagrangian Methods for the Solution of Variational Problems.

Abstract : This report describes the use of augmented Langrangian methods and of decomposition principles for the numerical solution of variational problems. After a brief introduction, augmented Langrangian methods are introduced on a simple finite-dimensional model problem taking its origin in quadratic programming. These methods mainly consist in: 1) introducing an augmented Lagrangian and an associated dual formulation of the model problem; and 2) proposing different algorithms for its numerical solution, which all can be interpreted as descent methods operating on this dual formulation. Then, these initial ideas are generalized to an infinite dimensional framework in the particular case of the Stokes problem.