Finite Element Methods for PDE-Constrained Optimal Control Problems

Prof. Dr. Boris Vexler , Dr. Dominik Meidner

Abstract

In this course, we present theoretical and practical aspects of finite element methods applied to discretizations of PDE-constrained optimization problems.

In the theoretical part of the course, we discuss different discretization concepts carefully taking into account the difference between discretization of a single equation and the discretization of an optimization problem. We derive a priori error estimates for different situations including problems with inequality constraints as well as problems governed by semilinear PDEs. Moreover we discuss a posteriori error estimates and adaptive mesh refinement algorithms for PDE-constrained optimization. Last but not least, we give an overview of current research topics in this area.

In the practical part of the course the students get to know the software packages Gascoigne and RoDoBo (both written in C++). The finite element library Gascoigne is a flexible solver for elliptic and parabolic PDEs and RoDoBo is a package for solving PDE-constrained optimization problems with an interface to Gascoigne. The students learn to use these packages for simulation and optimization of PDEs and to carry out numerical experiments. In particular, the following aspects of numerics for PDEs will be investigated: Solving linear and nonlinear systems of stationary and nonstationary PDEs with different types of boundary conditions, evaluation of discretization errors and functionals, mesh generation and adaptation, and evaluation of a posteriori error estimators. Finally, the mentioned aspects will be extended to the treatment of PDE-constrained optimization and parameter estimation problems.

Date and Place

 

TUM Mathematik Rutschen Universit√§t der Bundeswehr M√ľnchen Technische Universit√§t Graz Karl-Franzens-Universit√§t Graz Technische Universit√§t M√ľnchen
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