TY  - GEN
UR  - https://archiv.ub.uni-heidelberg.de/volltextserver/18774/
KW  - Adaptivität
TI  - Adaptive Multiple Shooting for Boundary Value Problems and Constrained Parabolic Optimization Problems
Y1  - 2015///
A1  - Geiger, Michael Ernst
AV  - public
ID  - heidok18774
N2  - Subject of this thesis is the development of adaptive techniques for multiple shooting methods. The focus is on the application to optimal control problems governed by parabolic partial differential equations. In order to retain as much freedom as possible in the later choice of discretization schemes, the details of both direct and indirect multiple shooting variants are worked out on an abstract function space level. Therefore, shooting techniques do not constitute a way of discretizing a problem. A thorough examination of the connections between the approaches provides an overview of different shooting formulations and enables their comparison for both linear and nonlinear problems.

We extend current research by considering additional constraints on the control variable in the multiple shooting context. An optimization problem is developed which includes so-called box constraints in the multiple shooting context. Several modern algorithms treating control constraints are adapted to the requirements of shooting methods. The modified algorithms permit an extended comparison of the different shooting approaches.

The efficiency of numerical methods can often be increased by developing grid adaptation techniques. While adaptive discretization schemes can be readily transferred to the multiple shooting context, questions of conditioning and stability make it difficult to develop adaptive features for shooting point distribution in multiple shooting processes. We concentrate on the design and comparison of two different approaches to shooting grid adaptation in the framework of ordinary differential equations. A residual-based adaptive algorithm is transferred to parabolic optimization problems with control constraints.  

The presented concepts and methods are verified by means of several examples, whereby theoretical results are numerically confirmed. We choose the test problems so that the simple shooting method becomes unstable and therefore a genuine multiple shooting technique is required.
ER  -