The multigrid strategy for solving elliptic optimal control problems

Recent efforts in the development and analysis of multigrid methods for solving elliptic PDE-based control problems are illustrated.
These problems consist of an elliptic model including an optimization mechanism and an objective of the control. The solution to these problems is characterized by an optimality system that also provides gradient information for the related minimization procedure.

It is shown that fast and robust solution schemes for elliptic optimal control problems are obtained based on the multigrid strategy.
On the one hand, direct one-shot multigrid methods can be implemented that solve the optimality system within the hierarchy of grid levels. The essential component of this approach is the formulation of appropriate smoothers.

On the other hand, it is possible to formulate optimization schemes where the multigrid method defines the outer solver and the inner solver is given by a classical optimization procedure. This is an emerging research field leading to reinterpreting multigrid algorithms from an optimization point of view.