Although universality has fascinated over the last decades, there are still numerous open questions in this field that require further investigation. In this work, we will mainly focus on classes of functions whose Fourier series are universal in the sense that they allow us to approximate uniformly any continuous function defined on a suitable subset of the unit circle. The structure of this thesis is as follows. In the first chapter, we will initially introduce the most important notation which is needed for our following discussion. Subsequently, after recalling the notion of universality in a general context, we will revisit significant results concerning universality of Taylor series. The focus here is particularly on universality with respect to uniform convergence and convergence in measure. By a result of Menshov, we will transition to universality of Fourier series which is the central object of study in this work. In the second chapter, we recall spaces of holomorphic functions which are characterized by the growth of their coefficients. In this context, we will derive a relationship to functions on the unit circle via an application of the Fourier transform. In the second part of the chapter, our attention is devoted to the $\mathcal{D}_{\textup{harm}}^p$ spaces which can be viewed as the set of harmonic functions contained in the $W^{1,p}(\D)$ Sobolev spaces. In this context, we will also recall the Bergman projection. Thanks to the intensive study of the latter in relation to Sobolev spaces, we can derive a decomposition of $\mathcal{D}_{\textup{harm}}^p$ spaces which may be seen as analogous to the Riesz projection for $L^p$ spaces. Owing to this result, we are able to provide a link between $\mathcal{D}_{\textup{harm}}^p$ spaces and spaces of holomorphic functions on $\mathbb{C}_\infty \setminus \s$ which turns out to be a crucial step in determining the dual of $\mathcal{D}_{\textup{harm}}^p$ spaces. The last section of this chapter deals with the Cauchy dual which has a close connection to the Fantappié transform. As an application, we will determine the Cauchy dual of the spaces $D_\alpha$ and $D_{\textup{harm}}^p$, two results that will prove to be very helpful later on. Finally, we will provide a useful criterion that establishes a connection between the density of a set in the direct sum $X \oplus Y$ and the Cauchy dual of the intersection of the respective spaces. The subsequent chapter will delve into the theory of capacities and, consequently, potential theory which will prove to be essential in formulating our universality results. In addition to introducing further necessary terminologies, we will define capacities in the first section following [16], however in the frame of separable metric spaces, and revisit the most important results about them. Simultaneously, we make preparations that allow us to define the $\mathrm{Li}_\alpha$-capacity which will turn out to be equivalent to the classical Riesz $\alpha$-capacity. The $\mathrm{Li}_\alpha$-capacity proves to be more adapted to the $D_\alpha$ spaces. It becomes apparent in the course of our discussion that the $\mathrm{Li}_\alpha$-capacity is essential to prove uniqueness results for the class $D_\alpha$. This leads to the centerpiece of this chapter which forms the energy formula for the $\mathrm{Li}_\alpha$-capacity on the unit circle. More precisely, this identity establishes a connection between the energy of a measure and its corresponding Fourier coefficients. We will briefly deal with the complement-equivalence of capacities before we revisit the concept of Bessel and Riesz capacities, this time, however, in a much more general context, where we will mainly rely on [1]. Since we defined capacities on separable metric spaces in the first section, we can draw a connection between Bessel capacities and $\mathrm{Li}_\alpha$-capacities. To conclude this chapter, we would like to take a closer look at the geometric meaning of capacities. Here, we will point out a connection between the Hausdorff dimension and the polarity of a set, and transfer it to the $\mathrm{Li}_\alpha$-capacity. Another aspect will be the comparison of Bessel capacities across different dimensions, in which the theory of Wolff potentials crystallizes as a crucial auxiliary tool. In the fourth chapter of this thesis, we will turn our focus to the theory of sets of uniqueness, a subject within the broader field of harmonic analysis. This theory has a close relationship with sets of universality, a connection that will be further elucidated in the upcoming chapter. The initial section of this chapter will be dedicated to the notion of sets of uniqueness that is specifically adapted to our current context. Building on this concept, we will recall some of the fundamental results of this theory. In the subsequent section, we will primarily rely on techniques from previous chapters to determine the closed sets of uniqueness for the class $\mathcal{D}_{\alpha}$. The proofs we will discuss are largely influenced by [16, p.\ 178] and [9, pp.\ 82]. One more time, it will become evident that the introduction of the $\mathrm{Li}_\alpha$-capacity in the third chapter and the closely associated energy formula on the unit circle, were the pivotal factors that enabled us to carry out these proofs. In the final chapter of our discourse, we will present our results on universality. To begin, we will recall a version of the universality criterion which traces back to the work of Grosse-Erdmann (see [26]). Coupled with an outcome from the second chapter, we will prove a result that allows us to obtain the universality of a class using the technique of simultaneous approximation. This tool will play a key role in the proof of our universality results which will follow hereafter. Our attention will first be directed toward the class $D_\alpha$ with $\alpha$ in the interval $(0,1]$. Here, we summarize that universality with respect to uniform convergence occurs on closed and $\alpha$-polar sets $E \subset \s$. Thanks to results of Carleson and further considerations, which particularly rely on the favorable behavior of the $\mathrm{Li}_\alpha$-kernel, we also find that this result is sharp. In particular, it may be seen as a generalization of the universality result for the harmonic Dirichlet space. Following this, we will investigate the same class, however, this time for $\alpha \in [-1,0)$. In this case, it turns out that universality with respect to uniform convergence occurs on closed and $(-\alpha)$-complement-polar sets $E \subset \s$. In particular, these sets of universality can have positive arc measure. In the final section, we will focus on the class $D_{\textup{harm}}^p$. Here, we manage to prove that universality occurs on closed and $(1,p)$-polar sets $E \subset \s$. Through results of Twomey [68] combined with an observation by Girela and Pélaez [23], as well as the decomposition of $D_{\textup{harm}}^p$, we can deduce that the closed sets of universality with respect to uniform convergence of the class $D_{\textup{harm}}^p$ are characterized by $(1,p)$-polarity. We conclude our work with an application of the latter result to the class $D^p$. We will show that the closed sets of divergence for the class $D^p$ are given by the $(1,p)$-polar sets.

In this thesis we study structure-preserving model reduction methods for the efficient and reliable approximation of dynamical systems. A major focus is the approximation of a nonlinear flow problem on networks, which can, e.g., be used to describe gas network systems. Our proposed approximation framework guarantees so-called port-Hamiltonian structure and is general enough to be realizable by projection-based model order reduction combined with complexity reduction. We divide the discussion of the flow problem into two parts, one concerned with the linear damped wave equation and the other one with the general nonlinear flow problem on networks. The study around the linear damped wave equation relies on a Galerkin framework, which allows for convenient network generalizations. Notable contributions of this part are the profound analysis of the algebraic setting after space-discretization in relation to the infinite dimensional setting and its implications for model reduction. In particular, this includes the discussion of differential-algebraic structures associated to the network-character of our problem and the derivation of compatibility conditions related to fundamental physical properties. Amongst the different model reduction techniques, we consider the moment matching method to be a particularly well-suited choice in our framework. The Galerkin framework is then appropriately extended to our general nonlinear flow problem. Crucial supplementary concepts are required for the analysis, such as the partial Legendre transform and a more careful discussion of the underlying energy-based modeling. The preservation of the port-Hamiltonian structure after the model-order- and complexity-reduction-step represents a major focus of this work. Similar as in the analysis of the model order reduction, compatibility conditions play a crucial role in the analysis of our complexity reduction, which relies on a quadrature-type ansatz. Furthermore, energy-stable time-discretization schemes are derived for our port-Hamiltonian approximations, as structure-preserving methods from literature are not applicable due to our rather unconventional parametrization of the solution. Apart from the port-Hamiltonian approximation of the flow problem, another topic of this thesis is the derivation of a new extension of moment matching methods from linear systems to quadratic-bilinear systems. Most system-theoretic reduction methods for nonlinear systems rely on multivariate frequency representations. Our approach instead uses univariate frequency representations tailored towards user-defined families of inputs. Then moment matching corresponds to a one-dimensional interpolation problem rather than to a multi-dimensional interpolation as for the multivariate approaches, i.e., it involves fewer interpolation frequencies to be chosen. The notion of signal-generator-driven systems, variational expansions of the resulting autonomous systems as well as the derivation of convenient tensor-structured approximation conditions are the main ingredients of this part. Notably, our approach allows for the incorporation of general input relations in the state equations, not only affine-linear ones as in existing system-theoretic methods.

An important field of applied mathematics is the simulation of complex financial, mechanical, chemical, physical or medical processes with mathematical models. In addition to the pure modeling of the processes, the simultaneous optimization of an objective function by changing the model parameters is often the actual goal. Models in fields such as finance, biology or medicine benefit from this optimization step. While many processes can be modeled using an ordinary differential equation (ODE), partial differential equations (PDEs) are needed to optimize heat conduction and flow characteristics, spreading of tumor cells in tissue as well as option prices. A partial integro-differential equation (PIDE) is a parital differential equation involving an integral operator, e.g., the convolution of the unknown function with a given kernel function. PIDEs occur for example in models that simulate adhesive forces between cells or option prices with jumps. In each of the two parts of this thesis, a certain PIDE is the main object of interest. In the first part, we study a semilinear PIDE-constrained optimal control problem with the aim to derive necessary optimality conditions. In the second, we analyze a linear PIDE that includes the convolution of the unknown function with the Gaussian kernel.

We will consider discrete dynamical systems (X,T) which consist of a state space X and a linear operator T acting on X. Given a state x in X at time zero, its state at time n is determined by the n-th iteration T^n(x). We are interested in the long-term behaviour of this system, that means we want to know how the sequence (T^n (x))_(n in N) behaves for increasing n and x in X. In the first chapter, we will sum up the relevant definitions and results of linear dynamics. In particular, in topological dynamics the notions of hypercyclic, frequently hypercyclic and mixing operators will be presented. In the setting of measurable dynamics, the most important definitions will be those of weakly and strongly mixing operators. If U is an open set in the (extended) complex plane containing 0, we can define the Taylor shift operator on the space H(U) of functions f holomorphic in U as Tf(z) = (f(z)- f(0))/z if z is not equal to 0 and otherwise Tf(0) = f'(0). In the second chapter, we will start examining the Taylor shift on H(U) endowed with the topology of locally uniform convergence. Depending on the choice of U, we will study whether or not the Taylor shift is weakly or strongly mixing in the Gaussian sense. Next, we will consider Banach spaces of functions holomorphic on the unit disc D. The first section of this chapter will sum up the basic properties of Bergman and Hardy spaces in order to analyse the dynamical behaviour of the Taylor shift on these Banach spaces in the next part. In the third section, we study the space of Cauchy transforms of complex Borel measures on the unit circle first endowed with the quotient norm of the total variation and then with a weak-* topology. While the Taylor shift is not even hypercyclic in the first case, we show that it is mixing for the latter case. In Chapter 4, we will first introduce Bergman spaces A^p(U) for general open sets and provide approximation results which will be needed in the next chapter where we examine the Taylor shift on these spaces on its dynamical properties. In particular, for 1<=p<2 we will find sufficient conditions for the Taylor shift to be weakly mixing or strongly mixing in the Gaussian sense. For p>=2, we consider specific Cauchy transforms in order to determine open sets U such that the Taylor shift is mixing on A^p(U). In both sections, we will illustrate the results with appropriate examples. Finally, we apply our results to universal Taylor series. The results of Chapter 5 about the Taylor shift allow us to consider the behaviour of the partial sums of the Taylor expansion of functions in general Bergman spaces outside its disc of convergence.

Industrial companies mainly aim for increasing their profit. That is why they intend to reduce production costs without sacrificing the quality. Furthermore, in the context of the 2020 energy targets, energy efficiency plays a crucial role. Mathematical modeling, simulation and optimization tools can contribute to the achievement of these industrial and environmental goals. For the process of white wine fermentation, there exists a huge potential for saving energy. In this thesis mathematical modeling, simulation and optimization tools are customized to the needs of this biochemical process and applied to it. Two different models are derived that represent the process as it can be observed in real experiments. One model takes the growth, division and death behavior of the single yeast cell into account. This is modeled by a partial integro-differential equation and additional multiple ordinary integro-differential equations showing the development of the other substrates involved. The other model, described by ordinary differential equations, represents the growth and death behavior of the yeast concentration and development of the other substrates involved. The more detailed model is investigated analytically and numerically. Thereby existence and uniqueness of solutions are studied and the process is simulated. These investigations initiate a discussion regarding the value of the additional benefit of this model compared to the simpler one. For optimization, the process is described by the less detailed model. The process is identified by a parameter and state estimation problem. The energy and quality targets are formulated in the objective function of an optimal control or model predictive control problem controlling the fermentation temperature. This means that cooling during the process of wine fermentation is controlled. Parameter and state estimation with nonlinear economic model predictive control is applied in two experiments. For the first experiment, the optimization problems are solved by multiple shooting with a backward differentiation formula method for the discretization of the problem and a sequential quadratic programming method with a line search strategy and a Broyden-Fletcher-Goldfarb-Shanno update for the solution of the constrained nonlinear optimization problems. Different rounding strategies are applied to the resulting post-fermentation control profile. Furthermore, a quality assurance test is performed. The outcomes of this experiment are remarkable energy savings and tasty wine. For the next experiment, some modifications are made, and the optimization problems are solved by using direct transcription via orthogonal collocation on finite elements for the discretization and an interior-point filter line-search method for the solution of the constrained nonlinear optimization problems. The second experiment verifies the results of the first experiment. This means that by the use of this novel control strategy energy conservation is ensured and production costs are reduced. From now on tasty white wine can be produced at a lower price and with a clearer conscience at the same time.

This thesis is divided into three main parts: The description of the calibration problem, the numerical solution of this problem and the connection to optimal stochastic control problems. Fitting model prices to given market prices leads to an abstract least squares formulation as calibration problem. The corresponding option price can be computed by solving a stochastic differential equation via the Monte-Carlo method which seems to be preferred by most practitioners. Due to the fact that the Monte-Carlo method is expensive in terms of computational effort and requires memory, more sophisticated stochastic predictor-corrector schemes are established in this thesis. The numerical advantage of these predictor-corrector schemes ispresented and discussed. The adjoint method is applied to the calibration. The theoretical advantage of the adjoint method is discussed in detail. It is shown that the computational effort of gradient calculation via the adjoint method is independent of the number of calibration parameters. Numerical results confirm the theoretical results and summarize the computational advantage of the adjoint method. Furthermore, provides the connection to optimal stochastic control problems is proven in this thesis.

In dieser Arbeit untersuchen wir das Optimierungsproblem der optimalen Materialausrichtung orthotroper Materialien in der Hülle von dreidimensionalen Schalenkonstruktionen. Ziel der Optimierung ist dabei die Minimierung der Gesamtnachgiebigkeit der Konstruktion, was der Suche nach einem möglichst steifen Design entspricht. Sowohl die mathematischen als auch die mechanischen Grundlagen werden in kompakter Form zusammengetragen und basierend darauf werden sowohl gradientenbasierte als auch auf mechanischen Prinzipien beruhende, neue Erweiterungen punktweise formulierter Optimierungsverfahren entwickelt und implementiert. Die vorgestellten Verfahren werden anhand des Beispiels des Modells einer Flugzeugtragfläche mit praxisrelevanter Problemgröße getestet und verglichen. Schließlich werden die untersuchten Methoden in ihrer Koppelung mit einem Verfahren zur Topologieoptimierung, basierend auf dem topologischen Gradienten untersucht.

Shape optimization is of interest in many fields of application. In particular, shape optimization problems arise frequently in technological processes which are modelled by partial differential equations (PDEs). In a lot of practical circumstances, the shape under investigation is parametrized by a finite number of parameters, which, on the one hand, allows the application of standard optimization approaches, but, on the other hand, unnecessarily limits the space of reachable shapes. Shape calculus presents a way to circumvent this dilemma. However, so far shape optimization based on shape calculus is mainly performed using gradient descent methods. One reason for this is the lack of symmetry of second order shape derivatives or shape Hessians. A major difference between shape optimization and the standard PDE constrained optimization framework is the lack of a linear space structure on shape spaces. If one cannot use a linear space structure, then the next best structure is a Riemannian manifold structure, in which one works with Riemannian shape Hessians. They possess the often sought property of symmetry, characterize well-posedness of optimization problems and define sufficient optimality conditions. In general, shape Hessians are used to accelerate gradient-based shape optimization methods. This thesis deals with shape optimization problems constrained by PDEs and embeds these problems in the framework of optimization on Riemannian manifolds to provide efficient techniques for PDE constrained shape optimization problems on shape spaces. A Lagrange-Newton and a quasi-Newton technique in shape spaces for PDE constrained shape optimization problems are formulated. These techniques are based on the Hadamard-form of shape derivatives, i.e., on the form of integrals over the surface of the shape under investigation. It is often a very tedious, not to say painful, process to derive such surface expressions. Along the way, volume formulations in the form of integrals over the entire domain appear as an intermediate step. This thesis couples volume integral formulations of shape derivatives with optimization strategies on shape spaces in order to establish efficient shape algorithms reducing analytical effort and programming work. In this context, a novel shape space is proposed.

Matching problems with additional resource constraints are generalizations of the classical matching problem. The focus of this work is on matching problems with two types of additional resource constraints: The couple constrained matching problem and the level constrained matching problem. The first one is a matching problem which has imposed a set of additional equality constraints. Each constraint demands that for a given pair of edges either both edges are in the matching or none of them is in the matching. The second one is a matching problem which has imposed a single equality constraint. This constraint demands that an exact number of edges in the matching are so-called on-level edges. In a bipartite graph with fixed indices of the nodes, these are the edges with end-nodes that have the same index. As a central result concerning the couple constrained matching problem we prove that this problem is NP-hard, even on bipartite cycle graphs. Concerning the complexity of the level constrained perfect matching problem we show that it is polynomially equivalent to three other combinatorial optimization problems from the literature. For different combinations of fixed and variable parameters of one of these problems, the restricted perfect matching problem, we investigate their effect on the complexity of the problem. Further, the complexity of the assignment problem with an additional equality constraint is investigated. In a central part of this work we bring couple constraints into connection with a level constraint. We introduce the couple and level constrained matching problem with on-level couples, which is a matching problem with a special case of couple constraints together with a level constraint imposed on it. We prove that the decision version of this problem is NP-complete. This shows that the level constraint can be sufficient for making a polynomially solvable problem NP-hard when being imposed on that problem. This work also deals with the polyhedral structure of resource constrained matching problems. For the polytope corresponding to the relaxation of the level constrained perfect matching problem we develop a characterization of its non-integral vertices. We prove that for any given non-integral vertex of the polytope a corresponding inequality which separates this vertex from the convex hull of integral points can be found in polynomial time. Regarding the calculation of solutions of resource constrained matching problems, two new algorithms are presented. We develop a polynomial approximation algorithm for the level constrained matching problem on level graphs, which returns solutions whose size is at most one less than the size of an optimal solution. We then describe the Objective Branching Algorithm, a new algorithm for exactly solving the perfect matching problem with an additional equality constraint. The algorithm makes use of the fact that the weighted perfect matching problem without an additional side constraint is polynomially solvable. In the Appendix, experimental results of an implementation of the Objective Branching Algorithm are listed.

In recent years, the study of dynamical systems has developed into a central research area in mathematics. Actually, in combination with keywords such as "chaos" or "butterfly effect", parts of this theory have been incorporated in other scientific fields, e.g. in physics, biology, meteorology and economics. In general, a discrete dynamical system is given by a set X and a self-map f of X. The set X can be interpreted as the state space of the system and the function f describes the temporal development of the system. If the system is in state x ∈ X at time zero, its state at time n ∈ N is denoted by f^n(x), where f^n stands for the n-th iterate of the map f. Typically, one is interested in the long-time behaviour of the dynamical system, i.e. in the behaviour of the sequence (f^n(x)) for an arbitrary initial state x ∈ X as the time n increases. On the one hand, it is possible that there exist certain states x ∈ X such that the system behaves stably, which means that f^n(x) approaches a state of equilibrium for n→∞. On the other hand, it might be the case that the system runs unstably for some initial states x ∈ X so that the sequence (f^n(x)) somehow shows chaotic behaviour. In case of a non-linear entire function f, the complex plane always decomposes into two disjoint parts, the Fatou set F_f of f and the Julia set J_f of f. These two sets are defined in such a way that the sequence of iterates (f^n) behaves quite "wildly" or "chaotically" on J_f whereas, on the other hand, the behaviour of (f^n) on F_f is rather "nice" and well-understood. However, this nice behaviour of the iterates on the Fatou set can "change dramatically" if we compose the iterates from the left with just one other suitable holomorphic function, i.e. if we consider sequences of the form (g∘f^n) on D, where D is an open subset of F_f with f(D)⊂ D and g is holomorphic on D. The general aim of this work is to study the long-time behaviour of such modified sequences. In particular, we will prove the existence of holomorphic functions g on D having the property that the behaviour of the sequence of compositions (g∘f^n) on the set D becomes quite similarly chaotic as the behaviour of the sequence (f^n) on the Julia set of f. With this approach, we immerse ourselves into the theory of universal families and hypercyclic operators, which itself has developed into an own branch of research. In general, for topological spaces X, Y and a family {T_i: i ∈ I} of continuous functions T_i:X→Y, an element x ∈ X is called universal for the family {T_i: i ∈ I} if the set {T_i(x): i ∈ I} is dense in Y. In case that X is a topological vector space and T is a continuous linear operator on X, a vector x ∈ X is called hypercyclic for T if it is universal for the family {T^n: n ∈ N}. Thus, roughly speaking, universality and hypercyclicity can be described via the following two aspects: There exists a single object which allows us, via simple analytical operations, to approximate every element of a whole class of objects. In the above situation, i.e. for a non-linear entire function f and an open subset D of F_f with f(D)⊂ D, we endow the space H(D) of holomorphic functions on D with the topology of locally uniform convergence and we consider the map C_f:H(D)→H(D), C_f(g):=g∘f|_D, which is called the composition operator with symbol f. The transform C_f is a continuous linear operator on the Fréchet space H(D). In order to show that the above-mentioned "nice" behaviour of the sequence of iterates (f^n) on the set D ⊂ F_f can "change dramatically" if we compose the iterates from the left with another suitable holomorphic function, our aim consists in finding functions g ∈ H(D) which are hypercyclic for C_f. Indeed, for each hypercyclic function g for C_f, the set of compositions {g∘f^n|_D: n ∈ N} is dense in H(D) so that the sequence of compositions (g∘f^n|_D) is kind of "maximally divergent" " meaning that each function in H(D) can be approximated locally uniformly on D via subsequences of (g∘f^n|_D). This kind of behaviour stands in sharp contrast to the fact that the sequence of iterates (f^n) itself converges, behaves like a rotation or shows some "wandering behaviour" on each component of F_f. To put it in a nutshell, this work combines the theory of non-linear complex dynamics in the complex plane with the theory of dynamics of continuous linear operators on spaces of holomorphic functions. As far as the author knows, this approach has not been investigated before.