A domain decomposition method for optimization problems for partial differential equations

A nonoverlapping domain decomposition method for optimization problems for partial differential equations is presented. The domain decomposition is effected through an auxiliary optimization problem. This results in an multiobjective optimization problem involving the given functional and the auxiliary functional. The existence of an optimal solution to the multiobjective optimization problem is proved as are convergence estimates as the parameters used to regularize the problem (penalty parameters) and to combine the two objective functionals tend to zero. An optimality system for the optimal solution is derived and used to define a gradient method. Convergence results are obtained for the gradient method and the results of some numerical experiments are obtained. Then, unregularized problems having vanishing penalty parameters are discussed.