Adaptive Finite Element Methods for Optimization Problems with PDEs

We present a general approach for error estimation and adaptivity for optimization problems governed by partial differential equations.
We treat this topic for a large class of optimization problems involving optimal control and parameter identification problems governed by either elliptic or parabolic partial differential equations.

For treatment of time-dependent problems we use space-time finite element methods and derive a posteriori error estimates which assess the discretization error with respect to a given quantity of interest and separate the influences of different parts of the discretization (time, space, and control discretization).
This allows to set up an efficient adaptive algorithm which successively improves the accuracy of the computed solution by construction of locally refined meshes for time and space discretizations.

Numerical examples illustrate the capability of the method.