Adaptive Multiple Shooting for Boundary Value Problems and Constrained Parabolic Optimization Problems

by Michael Ernst Geiger

Institution: Universität Heidelberg
Department: The Faculty of Mathematics and Computer Science
Degree: PhD
Year: 2015
Record ID: 1111415
Full text PDF: http://www.ub.uni-heidelberg.de/archiv/18774


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. Gegenstand dieser Arbeit ist die Entwicklung adaptiver Techniken für Mehrfachschießmethoden. Im Fokus liegt hierbei die Anwendung auf Optimalsteuerungsprobleme, welche durch parabolische partielle Differentialgleichungen beschränkt sind. Um möglichst viel Freiheit bei der späteren Wahl von Diskretisierungsschemata zu bewahren, werden die Details von direkten wie indirekten Verfahrensvarianten im abstrakten Funktionenraum ausgearbeitet. Schießverfahren stellen daher keine Diskretisierungsmethode dar. Eine eingehende Untersuchung der Zusammenhänge zwischen den Ansätzen liefert eine Übersicht der verschiedenen Verfahrensformulierungen und ermöglicht ihren Vergleich im Rahmen von linearen wie nichtlinearen Problemstellungen. Wir erweitern den aktuellen Forschungsstand, indem wir zusätzliche Beschränkungen an die Steuervariable im Kontext von Mehrfachschießverfahren betrachten. Unter Einbezug sogenannter Box-Schranken wird zunächst ein Optimierungsproblem im Rahmen von Mehrfachschießmethoden entwickelt. Mehrere moderne…