Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the “intersection” of these two categories is isomorphic to Frölicher spaces, another generalisation of smooth structures. We then give examples of such spaces, as well as examples of diffeological and differential spaces that do not fall into this category. We apply the theory of diffeological spaces to differential forms on a geometric quotient of a compact Lie group. We show that the subcomplex of basic forms is isomorphic to the complex of diffeological forms on the geometric quotient. We apply this to symplectic quotients coming from a regular value of the momentum map, and show that diffeological forms on this quotient are isomorphic as a complex to Sjamaar differential forms. We also compare diffeological forms to those on orbifolds, and show that they are isomorphic complexes as well. We apply the theory of differential spaces to subcartesian spaces equipped with families of vector fields. We use this theory to show that smooth stratified spaces form a full subcategory of subcartesian spaces equipped with families of vector fields. We give families of vector fields that induce the orbit-type stratifications induced by a Lie group action, as well as the orbit-type stratifications induced by a Hamiltonian group action.

Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the “intersection” of these two categories is isomorphic to Frölicher spaces, another generalisation of smooth structures. We then give examples of such spaces, as well as examples of diffeological and differential spaces that do not fall into this category. We apply the theory of diffeological spaces to differential forms on a geometric quotient of a compact Lie group. We show that the subcomplex of basic forms is isomorphic to the complex of diffeological forms on the geometric quotient. We apply this to symplectic quotients coming from a regular value of the momentum map, and show that diffeological forms on this quotient are isomorphic as a complex to Sjamaar differential forms. We also compare diffeological forms to those on orbifolds, and show that they are isomorphic complexes as well. We apply the theory of differential spaces to subcartesian spaces equipped with families of vector fields. We use this theory to show that smooth stratified spaces form a full subcategory of subcartesian spaces equipped with families of vector fields. We give families of vector fields that induce the orbit-type stratifications induced by a Lie group action, as well as the orbit-type stratifications induced by a Hamiltonian group action.

Diese Arbeit untersucht Nicht-Glattheiten im Bereich der Formoptimierung aus zwei verschiedenen Perspektiven. Einerseits werden Formoptimierungsprobleme betrachtet, die durch Variationsungleichungen eingeschränkt sind. Häufig fallen diese unter die sogenannten Hindernis-Probleme. Diese Probleme haben zahlreiche Anwendungen, beispielsweise bei der Konstruktion von Formen, die Einschränkungen durch die Lösung von partiellen Differentialgleichungen unterliegen, welche wiederum von der optimierten Geometrie abhängen. Oft werden verschiedene Regularisierungsmethoden genutzt, um der Nicht-Glattheit und Nicht-Konvexität Herr zu werden. Über die Nicht-Glattheiten hinaus ergeben sich weitere Herausforderungen, welche durch die Variationsungleichheiten sowie durch Nicht-Konvexität und Unendlich-Dimensionalität auftreten. Diese Faktoren erschweren die Formulierung von Optimalitätsbedingungen und die Entwicklung effizienter Lösungsalgorithmen. In dieser Arbeit wird ein Ansatz vorgestellt, welcher es ermöglicht, Nicht-Glattheiten ohne Regularisierung zu behandeln. Dazu wird die Hadamard-Semiableitung verwendet. Auf der anderen Seite steht die Frage nach geeigneten Formräumen im Fokus und motiviert den zweiten Teil dieser Arbeit. Herkömmliche Formräume umfassen typischerweise glatte Verformungen der Kugel und sind mit einer glatten Struktur versehen. Dadurch kommt es zu einer Vernachlässigung von Formen mit Ecken und Kanten. Die Konstruktion eines Formraums, der nicht-glatte Formen beinhaltet, ist keine Herausforderung. Das Arbeiten mit einem solchen Raum hingegen schon. Typische glatte Strukturen, wie die der riemannschen Mannigfaltigkeit, gehen unter Umständen verloren. Diese Arbeit führt daher diffeologische Räume als natürliche Verallgemeinerung glatter Mannigfaltigkeiten ein. Die Erweiterung von Optimierungstechniken von glatten Mannigfaltigkeiten auf diffeologische Räume stellt eine Herausforderung da. Besondere Aspekte sind die Existenz nicht-äquivalenter Definitionen des Tangentialraums sowie die Notwendigkeit einer erweiterten riemannschen Struktur zur Herleitung von Gradienten. Diese Arbeit präsentiert eine Erweiterung der bereits bekannten riemannschen Optimierung und ihrer Objekte. Dazu gehören unter anderem Definitionen für einen geeigneten Tangentialraum, ein diffeologisches riemannsches Setting, einen diffeologischen Gradienten, eine diffeologische Retraktion und einen diffeologischen Levi-Civita-Zusammenhang. Dies resultiert in der Formulierung eines diffeologischen Gradientenverfahrens, das auf ein Optimierungsproblem angewendet wird.:Abstract Zusammenfassung Acknowledgments Preface 1 Introduction 1.1 Motivation,Aim,andScopeoftheThesis 1.2 StructureoftheThesis 2 Background Knowledge 2.1 DifferentialGeometry 2.2 ShapeOptimization 2.2.1 ABasicIntroduction 2.2.2 A Brief Overview of Variational Inequality Constraints 2.2.3 ShapeSpaces 3 A Hadamard Approach for Variational Inequality Constrains 3.1 HadamardSemiderivative 3.1.1 A Brief Introduction into Hadamard Semiderivatives 3.1.2 HadamardShapeDerivativeCalculus 3.2 HadamardOptimalitySystem 3.3 Application 4 Optimization on Diffeological Spaces 4.1 ABriefIntroductiontoDiffeologicalSpaces 4.2 Towards Optimization Algorithms on Diffeological Spaces 4.2.1 TangentSpace 4.2.2 Examples of Diffeological Tangent Spaces 4.2.3 DiffeologicalRiemannianSpace 4.2.4 Towards Updates of Iterates: Diffeological Levi-Civita ConnectionandDiffeologicalRetraction 4.3 Formulation of Diffeological Optimization Algorithms andTheirApplication 5 Conclusion 6 Notations / This thesis is concerned with non-smoothness from two different points of view regarding shape optimization problems. On one hand this thesis considers shape optimization problems that are constrained by variational inequalities of the first kind, often known as obstacle-type problems. These problems find numerous applications when constructing a shape that must adhere to constraints imposed on the solution of a partial differential equation dependent on the geometry being optimized. Since those problems are non-smooth and non-convex optimization problems, they are often handled using several regularization methods. Besides the non-smoothness there are complementary aspects due to the variational inequalities as well as non-linear, non-convex and infinite-dimensional aspects due to the shapes. This complicates setting up an optimality system, and thus developing an efficient solution algorithm. This thesis is presenting a way to deal with the non-smoothness without the requirement of regularizations. Therefore, the Hadamard semiderivative setting is considered. After introducing the Hadamard semiderivative and considering the Hadamard shape calculus, the Hadamard adjoint is introduced. This allows us to handle shape optimization problems that are constrained by variational inequalities of the first kind without using any kind of regularization. On the other hand this thesis is confronted with the question of suitable shape spaces. What is a suitable space that contains all important shapes? This question is the motivation of the second part of this thesis. A common shape space usually contains some kind of smooth deformations of the sphere. Often those shape spaces are equipped with a suitable smooth structure. This can results in the neglection of shapes that have kinks and corners and are non-smooth. The construction of a shape space that includes non-smooth shapes is not a major challenge but working with such a space is. How do you optimize if your space is not a manifold? On answering this question lies the main focus of the second part of this thesis. Therefore, this thesis introduces so-called diffeological spaces. Diffeological spaces, firstly introduced by J.M. Souriau in the 1980s, are a natural generalization of smooth manifolds. To date, optimization techniques have primarily been developed on manifolds. Extending these methods to diffeological spaces presents a significant challenge for several reasons. One prominent obstacle is the existence of different definitions for tangent spaces that do not coincide with one another. Furthermore, the expansion necessitates the creation of a broader concept of a Riemannian structure to establish gradients, which are essential components for optimization strategies. The first major step is a suitable definition of a tangent space in view of optimization methods. This definition is then used in order to present a diffeological Riemannian space and a diffeological gradient, which this thesis needs to formulate an optimization algorithm on diffeological spaces. Moreover, this thesis presents a diffeological retraction and the Levi-Civita connection on diffeological spaces. As a result a diffeological version of the steepest decent method is obtained. This thesis gives examples for the novel objects and apply the presented diffeological algorithm (an algorithm for diffeological spaces) to an optimization problem.:Abstract Zusammenfassung Acknowledgments Preface 1 Introduction 1.1 Motivation,Aim,andScopeoftheThesis 1.2 StructureoftheThesis 2 Background Knowledge 2.1 DifferentialGeometry 2.2 ShapeOptimization 2.2.1 ABasicIntroduction 2.2.2 A Brief Overview of Variational Inequality Constraints 2.2.3 ShapeSpaces 3 A Hadamard Approach for Variational Inequality Constrains 3.1 HadamardSemiderivative 3.1.1 A Brief Introduction into Hadamard Semiderivatives 3.1.2 HadamardShapeDerivativeCalculus 3.2 HadamardOptimalitySystem 3.3 Application 4 Optimization on Diffeological Spaces 4.1 ABriefIntroductiontoDiffeologicalSpaces 4.2 Towards Optimization Algorithms on Diffeological Spaces 4.2.1 TangentSpace 4.2.2 Examples of Diffeological Tangent Spaces 4.2.3 DiffeologicalRiemannianSpace 4.2.4 Towards Updates of Iterates: Diffeological Levi-Civita ConnectionandDiffeologicalRetraction 4.3 Formulation of Diffeological Optimization Algorithms andTheirApplication 5 Conclusion 6 Notations