The technique of automatic differentiation provides directional derivatives and discrete adjoints with working accuracy. A complete complexity analysis of the basic modes of automatic differentiation is available. Therefore, the research activities are focused now on different aspects of the derivative calculation, as for example the efficient implementation by exploitation of structural information, studies of the theoretical properties of the provided derivatives in the context of optimization problems, and the development and analysis of new mathematical algorithms based on discrete adjoint information. According to this motivation, this habilitation presents an analysis of different checkpointing strategies to reduce the memory requirement of the discrete adjoint computation. Additionally, a new algorithm for computing sparse Hessian matrices is presented including a complexity analysis and a report on practical experiments. Hence, the first two contributions of this thesis are dedicated to an efficient computation of discrete adjoints. The analysis of discrete adjoints with respect to their theoretical properties is another important research topic. The third and fourth contribution of this thesis focus on the relation of discrete adjoint information and continuous adjoint information for optimal control problems. Here, differences resulting from different discretization strategies as well as convergence properties of the discrete adjoints are analyzed comprehensively. In the fifth contribution, checkpointing approaches that are successfully applied for the computation of discrete adjoints, are adapted such that they can be used also for the computation of continuous adjoints. Additionally, the fifth contributions presents a new proof of optimality for the binomial checkpointing that is based on new theoretical results. Discrete adjoint information can be applied for example for the approximation of dense Jacobian matrices. The development and analysis of new mathematical algorithms based on these approximate Jacobians is the topic of the sixth contribution. Is was possible to show global convergence to first-order critical points for a whole class of trust-region methods. Here, the usage of inexact Jacobian matrices allows a considerable reduction of the computational complexity.

The technique of automatic differentiation provides directional derivatives and discrete adjoints with working accuracy. A complete complexity analysis of the basic modes of automatic differentiation is available. Therefore, the research activities are focused now on different aspects of the derivative calculation, as for example the efficient implementation by exploitation of structural information, studies of the theoretical properties of the provided derivatives in the context of optimization problems, and the development and analysis of new mathematical algorithms based on discrete adjoint information. According to this motivation, this habilitation presents an analysis of different checkpointing strategies to reduce the memory requirement of the discrete adjoint computation. Additionally, a new algorithm for computing sparse Hessian matrices is presented including a complexity analysis and a report on practical experiments. Hence, the first two contributions of this thesis are dedicated to an efficient computation of discrete adjoints. The analysis of discrete adjoints with respect to their theoretical properties is another important research topic. The third and fourth contribution of this thesis focus on the relation of discrete adjoint information and continuous adjoint information for optimal control problems. Here, differences resulting from different discretization strategies as well as convergence properties of the discrete adjoints are analyzed comprehensively. In the fifth contribution, checkpointing approaches that are successfully applied for the computation of discrete adjoints, are adapted such that they can be used also for the computation of continuous adjoints. Additionally, the fifth contributions presents a new proof of optimality for the binomial checkpointing that is based on new theoretical results. Discrete adjoint information can be applied for example for the approximation of dense Jacobian matrices. The development and analysis of new mathematical algorithms based on these approximate Jacobians is the topic of the sixth contribution. Is was possible to show global convergence to first-order critical points for a whole class of trust-region methods. Here, the usage of inexact Jacobian matrices allows a considerable reduction of the computational complexity.

Formoptimierung bezeichnet die Bestimmung der Geometrischen Gestalt eines Gebietes auf dem eine partielle Differentialgleichung (PDE) wirkt, sodass bestimmte gegebene Zielgrößen, welche von der Lösung der PDE abhängen, Extrema annehmen. Bei der Diskret Adjungierten Methode wird der Gradient einer Zielgröße bezüglich einer beliebigen Anzahl von Formparametern mit Hilfe der Lösung einer adjungierten Gleichung der diskretisierten PDE effizient ermittelt. Dieser Gradient wird dann in Verfahren der numerischen Optimierung verwendet um eine optimale Lösung zu suchen. Da sowohl die Zielgröße als auch der Gradient für die diskretisierte PDE ermittelt werden, sind beide zunächst vom verwendeten Netz abhängig. Bei groben Netzen sind sogar Unstetigkeiten der diskreten Zielfunktion zu erwarten, wenn bei Änderungen der Formparameter sich das Netz unstetig ändert (z.B. Änderung Anzahl Knoten, Umschalten der Konnektivität). Mit zunehmender Feinheit der Netze verschwinden jedoch diese Unstetigkeiten aufgrund der Konvergenz der Diskretisierung. Da im Zuge der Formoptimierung Zielgröße und Gradient für eine Vielzahl von Iterierten der Lösung bestimmt werden müssen, ist man bestrebt die Kosten einer einzelnen Auswertung möglichst gering zu halten, z.B. indem man mit nur moderat feinen oder adaptiv verfeinerten Netzen arbeitet. Aufgabe dieser Diplomarbeit ist es zu untersuchen, ob mit gängigen Methoden adaptiv verfeinerte Netze hinreichende Genauigkeit der Auswertung von Zielgröße und Gradient erlauben und ob eventuell Anpassungen der Optimierungsstrategie an die adaptive Vernetzung notwendig sind. Für die Untersuchungen sind geeignete Modellprobleme aus der Festigkeitslehre zu wählen und zu untersuchen. / Shape optimization describes the determination of the geometric shape of a domain with a partial differential equation (PDE) with the purpose that a specific given performance function is minimized, its values depending on the solution of the PDE. The Discrete Adjoint Method can be used to evaluate the gradient of a performance function with respect to an arbitrary number of shape parameters by solving an adjoint equation of the discretized PDE. This gradient is used in the numerical optimization algorithm to search for the optimal solution. As both function value and gradient are computed for the discretized PDE, they both fundamentally depend on the discretization. In using the coarse meshes, discontinuities in the discretized objective function can be expected if the changes in the shape parameters cause discontinuous changes in the mesh (e.g. change in the number of nodes, switching of connectivity). Due to the convergence of the discretization these discontinuities vanish with increasing fineness of the mesh. In the course of shape optimization, function value and gradient require evaluation for a large number of iterations of the solution, therefore minimizing the costs of a single computation is desirable (e.g. using moderately or adaptively refined meshes). Overall, the task of the diploma thesis is to investigate if adaptively refined meshes with established methods offer sufficient accuracy of the objective value and gradient, and if the optimization strategy requires readjustment to the adaptive mesh design. For the investigation, applicable model problems from the science of the strength of materials will be chosen and studied.

Formoptimierung bezeichnet die Bestimmung der Geometrischen Gestalt eines Gebietes auf dem eine partielle Differentialgleichung (PDE) wirkt, sodass bestimmte gegebene Zielgrößen, welche von der Lösung der PDE abhängen, Extrema annehmen. Bei der Diskret Adjungierten Methode wird der Gradient einer Zielgröße bezüglich einer beliebigen Anzahl von Formparametern mit Hilfe der Lösung einer adjungierten Gleichung der diskretisierten PDE effizient ermittelt. Dieser Gradient wird dann in Verfahren der numerischen Optimierung verwendet um eine optimale Lösung zu suchen. Da sowohl die Zielgröße als auch der Gradient für die diskretisierte PDE ermittelt werden, sind beide zunächst vom verwendeten Netz abhängig. Bei groben Netzen sind sogar Unstetigkeiten der diskreten Zielfunktion zu erwarten, wenn bei Änderungen der Formparameter sich das Netz unstetig ändert (z.B. Änderung Anzahl Knoten, Umschalten der Konnektivität). Mit zunehmender Feinheit der Netze verschwinden jedoch diese Unstetigkeiten aufgrund der Konvergenz der Diskretisierung. Da im Zuge der Formoptimierung Zielgröße und Gradient für eine Vielzahl von Iterierten der Lösung bestimmt werden müssen, ist man bestrebt die Kosten einer einzelnen Auswertung möglichst gering zu halten, z.B. indem man mit nur moderat feinen oder adaptiv verfeinerten Netzen arbeitet. Aufgabe dieser Diplomarbeit ist es zu untersuchen, ob mit gängigen Methoden adaptiv verfeinerte Netze hinreichende Genauigkeit der Auswertung von Zielgröße und Gradient erlauben und ob eventuell Anpassungen der Optimierungsstrategie an die adaptive Vernetzung notwendig sind. Für die Untersuchungen sind geeignete Modellprobleme aus der Festigkeitslehre zu wählen und zu untersuchen. / Shape optimization describes the determination of the geometric shape of a domain with a partial differential equation (PDE) with the purpose that a specific given performance function is minimized, its values depending on the solution of the PDE. The Discrete Adjoint Method can be used to evaluate the gradient of a performance function with respect to an arbitrary number of shape parameters by solving an adjoint equation of the discretized PDE. This gradient is used in the numerical optimization algorithm to search for the optimal solution. As both function value and gradient are computed for the discretized PDE, they both fundamentally depend on the discretization. In using the coarse meshes, discontinuities in the discretized objective function can be expected if the changes in the shape parameters cause discontinuous changes in the mesh (e.g. change in the number of nodes, switching of connectivity). Due to the convergence of the discretization these discontinuities vanish with increasing fineness of the mesh. In the course of shape optimization, function value and gradient require evaluation for a large number of iterations of the solution, therefore minimizing the costs of a single computation is desirable (e.g. using moderately or adaptively refined meshes). Overall, the task of the diploma thesis is to investigate if adaptively refined meshes with established methods offer sufficient accuracy of the objective value and gradient, and if the optimization strategy requires readjustment to the adaptive mesh design. For the investigation, applicable model problems from the science of the strength of materials will be chosen and studied.