I³MS - Vexler Seminar
Prof. Dr. Boris Vexler - Adaptive Finite Element Methods for PDE-Constrained Optimization Problems
Centre for Mathematical Sciences, TU München
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 numerical treatment these infinite dimensional problems have to be discretized. We consider the discretization of the state equation by finite element methods, which leads to the discretization error between the optimal solutions of the continuous and the discretized optimization problems.
For a given quantity of interest, which describes the goal of the computation, we derive a posteriori error estimates accessing the discretization error with respect to this quantity. The resulting error estimators are used in an adaptive algorithm for successive improvement of the accuracy by construction of locally refined meshes.
We discuss several aspects including separation of the temporal and the spatial discretization errors and the treatment of pointwise inequality constraints on the control and the state variable within the adaptive algorithm.
The efficiency of our approach is demonstrated on numerical examples.