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Extreme value theory is concerned with the stochastic modeling of rare and extreme events. While fundamental theories of classical stochastics - such as the laws of small numbers or the central limit theorem - are used to investigate the asymptotic behavior of the sum of random variables, extreme value theory focuses on the maximum or minimum of a set of observations. The limit distribution of the normalized sample maximum among a sequence of independent and identically distributed random variables can be characterized by means of so-called max-stable distributions.
This dissertation concerns with different aspects of the theory of max-stable random vectors and stochastic processes. In particular, the concept of 'differentiability in distribution' of a max-stable process is introduced and investigated. Moreover, 'generalized max-linear models' are introduced in order to interpolate a known max-stable random vector by a max-stable process. Further, the connection between extreme value theory and multivariate records is established. In particular, so-called 'complete' and 'simple' records are introduced as well as it is examined their asymptotic behavior.

In this paper, convex approximation methods, suclt as CONLIN, the method of moving asymptotes (MMA) and a stabilized version of MMA (Sequential Convex Programming), are discussed with respect to their convergence behaviour. In an extensive numerical study they are :finally compared with other well-known optimization methods at 72 examples of sizing problems.

This thesis is devoted to Bernoulli Stochastics, which was initiated by Jakob Bernoulli more than 300 years ago by his master piece 'Ars conjectandi', which can be translated as 'Science of Prediction'. Thus, Jakob Bernoulli's Stochastics focus on prediction in contrast to the later emerging disciplines probability theory, statistics and mathematical statistics. Only recently Jakob Bernoulli's focus was taken up von Collani, who developed a unified theory of uncertainty aiming at making reliable and accurate predictions. In this thesis, teaching material as well as a virtual classroom are developed for fostering ideas and techniques initiated by Jakob Bernoulli and elaborated by Elart von Collani. The thesis is part of an extensively construed project called 'Stochastikon' aiming at introducing Bernoulli Stochastics as a unified science of prediction and measurement under uncertainty. This ambitious aim shall be reached by the development of an internet-based comprehensive system offering the science of Bernoulli Stochastics on any level of application. So far it is planned that the 'Stochastikon' system (http://www.stochastikon.com/) will consist of five subsystems. Two of them are developed and introduced in this thesis. The first one is the e-learning programme 'Stochastikon Magister' and the second one 'Stochastikon Graphics' that provides the entire Stochastikon system with graphical illustrations. E-learning is the outcome of merging education and internet techniques. E-learning is characterized by the facts that teaching and learning are independent of place and time and of the availability of specially trained teachers. Knowledge offering as well as knowledge transferring are realized by using modern information technologies. Nowadays more and more e-learning environments are based on the internet as the primary tool for communication and presentation. E-learning presentation tools are for instance text-files, pictures, graphics, audio and videos, which can be networked with each other. There could be no limit as to the access to teaching contents. Moreover, the students can adapt the speed of learning to their individual abilities. E-learning is particularly appropriate for newly arising scientific and technical disciplines, which generally cannot be presented by traditional learning methods sufficiently well, because neither trained teachers nor textbooks are available. The first part of this dissertation introduces the state of the art of e-learning in statistics, since statistics and Bernoulli Stochastics are both based on probability theory and exhibit many similar features. Since Stochastikon Magister is the first e-learning programme for Bernoulli Stochastics, the educational statistics systems is selected for the purpose of comparison and evaluation. This makes sense as both disciplines are an attempt to handle uncertainty and use methods that often can be directly compared. The second part of this dissertation is devoted to Bernoulli Stochastics. This part aims at outlining the content of two courses, which have been developed for the anticipated e-learning programme Stochastikon Magister in order to show the difficulties in teaching, understanding and applying Bernoulli Stochastics. The third part discusses the realization of the e-learning programme Stochastikon Magister, its design and implementation, which aims at offering a systematic learning of principles and techniques developed in Bernoulli Stochastics. The resulting e-learning programme differs from the commonly developed e-learning programmes as it is an attempt to provide a virtual classroom that simulates all the functions of real classroom teaching. This is in general not necessary, since most of the e-learning programmes aim at supporting existing classroom teaching. The forth part presents two empirical evaluations of Stochastikon Magister. The evaluations are performed by means of comparisons between traditional classroom learning in statistics and e-learning of Bernoulli Stochastics. The aim is to assess the usability and learnability of Stochastikon Magister. Finally, the fifth part of this dissertation is added as an appendix. It refers to Stochastikon Graphics, the fifth component of the entire Stochastikon system. Stochastikon Graphics provides the other components with graphical representations of concepts, procedures and results obtained or used in the framework of Bernoulli Stochastics. The primary aim of this thesis is the development of an appropriate software for the anticipated e-learning environment meant for Bernoulli Stochastics, while the preparation of the necessary teaching material constitutes only a secondary aim used for demonstrating the functionality of the e-learning platform and the scientific novelty of Bernoulli Stochastics. To this end, a first version of two teaching courses are developed, implemented and offered on-line in order to collect practical experiences. The two courses, which were developed as part of this projects are submitted as a supplement to this dissertation. For the time being the first experience with the e-learning programme Stochastikon Magister has been made. Students of different faculties of the University of Würzburg, as well as researchers and engineers, who are involved in the Stochastikon project have obtained access to Stochastikon Magister via internet. They have registered for Stochastikon Magister and participated in the course programme. This thesis reports on two assessments of these first experiences and the results will lead to further improvements with respect to content and organization of Stochastikon Magister.

This work is concerned with the numerical approximation of solutions to models that are used to describe atmospheric or oceanographic flows. In particular, this work concen- trates on the approximation of the Shallow Water equations with bottom topography and the compressible Euler equations with a gravitational potential. Numerous methods have been developed to approximate solutions of these models. Of specific interest here are the approximations of near equilibrium solutions and, in the case of the Euler equations, the low Mach number flow regime. It is inherent in most of the numerical methods that the quality of the approximation increases with the number of degrees of freedom that are used. Therefore, these schemes are often run in parallel on big computers to achieve the best pos- sible approximation. However, even on those big machines, the desired accuracy can not be achieved by the given maximal number of degrees of freedom that these machines allow. The main focus in this work therefore lies in the development of numerical schemes that give better resolution of the resulting dynamics on the same number of degrees of freedom, compared to classical schemes.
This work is the result of a cooperation of Prof. Klingenberg of the Institute of Mathe- matics in Wu¨rzburg and Prof. R¨opke of the Astrophysical Institute in Wu¨rzburg. The aim of this collaboration is the development of methods to compute stellar atmospheres. Two main challenges are tackled in this work. First, the accurate treatment of source terms in the numerical scheme. This leads to the so called well-balanced schemes. They allow for an accurate approximation of near equilibrium dynamics. The second challenge is the approx- imation of flows in the low Mach number regime. It is known that the compressible Euler equations tend towards the incompressible Euler equations when the Mach number tends to zero. Classical schemes often show excessive diffusion in that flow regime. The here devel- oped scheme falls into the category of an asymptotic preserving scheme, i.e. the numerical scheme reflects the behavior that is computed on the continuous equations. Moreover, it is shown that the diffusion of the numerical scheme is independent of the Mach number.
In chapter 3, an HLL-type approximate Riemann solver is adapted for simulations of the Shallow Water equations with bottom topography to develop a well-balanced scheme. In the literature, most schemes only tackle the equilibria when the fluid is at rest, the so called Lake at rest solutions. Here a scheme is developed to accurately capture all the equilibria of the Shallow Water equations. Moreover, in contrast to other works, a second order extension is proposed, that does not rely on an iterative scheme inside the reconstruction procedure, leading to a more efficient scheme.
In chapter 4, a Suliciu relaxation scheme is adapted for the resolution of hydrostatic equilibria of the Euler equations with a gravitational potential. The hydrostatic relations are underdetermined and therefore the solutions to that equations are not unique. However, the scheme is shown to be well-balanced for a wide class of hydrostatic equilibria. For specific classes, some quadrature rules are computed to ensure the exact well-balanced property. Moreover, the scheme is shown to be robust, i.e. it preserves the positivity of mass and energy, and stable with respect to the entropy. Numerical results are presented in order to investigate the impact of the different quadrature rules on the well-balanced property.
In chapter 5, a Suliciu relaxation scheme is adapted for the simulations of low Mach number flows. The scheme is shown to be asymptotic preserving and not suffering from excessive diffusion in the low Mach number regime. Moreover, it is shown to be robust under certain parameter combinations and to be stable from an Chapman-Enskog analysis.
Numerical results are presented in order to show the advantages of the new approach.
In chapter 6, the schemes developed in the chapters 4 and 5 are combined in order to investigate the performance of the numerical scheme in the low Mach number regime in a gravitational stratified atmosphere. The scheme is shown the be well-balanced, robust and stable with respect to a Chapman-Enskog analysis. Numerical tests are presented to show the advantage of the newly proposed method over the classical scheme.
In chapter 7, some remarks on an alternative way to tackle multidimensional simulations are presented. However no numerical simulations are performed and it is shown why further research on the suggested approach is necessary.

This thesis deals with the hp-ﬁnite element method (FEM) for linear quadratic optimal control problems. Here, a tracking type functional with control costs as regularization shall be minimized subject to an elliptic partial diﬀerential equation. In the presence of control constraints, the ﬁrst order necessary conditions, which are typically used to ﬁnd optimal solutions numerically, can be formulated as a semi-smooth projection formula. Consequently, optimal solutions may be non-smooth as well. The hp-discretization technique considers this fact and approximates rough functions on ﬁne meshes while using higher order ﬁnite elements on domains where the solution is smooth.
The ﬁrst main achievement of this thesis is the successful application of hp-FEM to two related problem classes: Neumann boundary and interface control problems. They are solved with an a-priori reﬁnement strategy called boundary concentrated (bc) FEM and interface concentrated (ic) FEM, respectively. These strategies generate grids that are heavily reﬁned towards the boundary or interface. We construct an elementwise interpolant that allows to prove algebraic decay of the approximation error for both techniques. Additionally, a detailed analysis of global and local regularity of solutions, which is critical for the speed of convergence, is included. Since the bc- and ic-FEM retain small polynomial degrees for elements touching the boundary and interface, respectively, we are able to deduce novel error estimates in the L2- and L∞-norm. The latter allows an a-priori strategy for updating the regularization parameter in the objective functional to solve bang-bang problems.
Furthermore, we apply the traditional idea of the hp-FEM, i.e., grading the mesh geometrically towards vertices of the domain, for solving optimal control problems (vc-FEM). In doing so, we obtain exponential convergence with respect to the number of unknowns. This is proved with a regularity result in countably normed spaces for the variables of the coupled optimality system.
The second main achievement of this thesis is the development of a fully adaptive hp-interior point method that can solve problems with distributed or Neumann control. The underlying barrier problem yields a non-linear optimality system, which poses a numerical challenge: the numerically stable evaluation of integrals over possibly singular functions in higher order elements. We successfully overcome this diﬃculty by monitoring the control variable at the integration points and enforcing feasibility in an additional smoothing step. In this work, we prove convergence of an interior point method with smoothing step and derive a-posteriori error estimators. The adaptive mesh reﬁnement is based on the expansion of the solution in a Legendre series. The decay of the coeﬃcients serves as an indicator for smoothness that guides between h- and p-reﬁnement.

The investigation of interacting multi-agent models is a new field of mathematical research with application to the study of behavior in groups of animals or community of people. One interesting feature of multi-agent systems is collective behavior. From the mathematical point of view, one of the challenging issues considering with these dynamical models is development of control mechanisms that are able to influence the time evolution of these systems.
In this thesis, we focus on the study of controllability, stabilization and optimal control problems for multi-agent systems considering three models as follows: The first one is the Hegselmann Krause opinion formation (HK) model. The HK dynamics describes how individuals' opinions are changed by the interaction with others taking place in a bounded domain of confidence. The study of this model focuses on determining feedback controls in order to drive the agents' opinions to reach a desired agreement. The second model is the Heider social balance (HB) model. The HB dynamics explains the evolution of relationships in a social network. One purpose of studying this system is the construction of control function in oder to steer the relationship to reach a friendship state. The third model that we discuss is a flocking model describing collective motion observed in biological systems. The flocking model under consideration includes self-propelling, friction, attraction, repulsion, and alignment features. We investigate a control for steering the flocking system to track a desired trajectory. Common to all these systems is our strategy to add a leader agent that interacts with all other members of the system and includes the control mechanism.
Our control through leadership approach is developed using classical theoretical control methods and a model predictive control (MPC) scheme. To apply the former method, for each model the stability of the corresponding linearized system near consensus is investigated. Further, local controllability is examined. However, only in the
Hegselmann-Krause opinion formation model, the feedback control is determined in order to steer agents' opinions to globally converge to a desired agreement. The MPC approach is an optimal control strategy based on numerical optimization. To apply the MPC scheme, optimal control problems for each model are formulated where the objective functions are different depending on the desired objective of the problem. The first-oder necessary optimality conditions for each problem are presented. Moreover for the numerical treatment, a sequence of open-loop discrete optimality systems is solved by accurate Runge-Kutta schemes, and in the optimization procedure, a nonlinear conjugate gradient solver is implemented. Finally, numerical experiments are performed to investigate the properties of the multi-agent models and demonstrate the ability of the proposed control strategies to drive multi-agent systems to attain a desired consensus and to track a given trajectory.

The classification of isoparametric hypersurfaces in spheres with a homogeneous focal manifold is a project that has been started by Linus Kramer. It extends results by E. Cartan and Hsiang and Lawson. Kramer does most part of this classification in his Habilitationsschrift. In particular he obtains a classification for the cases where the homogeneous focal manifold is at least 2-connected. Results of E. Cartan, Dorfmeister and Neher, and Takagi also solve parts of the classification problem. This thesis completes the classification. We classify all closed isoparametric hypersurfaces in spheres with g>2 distinct principal curvatures one of whose multiplicities is 2 such that the lower dimensional focal manifold is homogeneous. The methods are essentially the same as in Kramer's 'Habilitationsschrift'. The cohomology of the focal manifolds in question is known. This leads to two topological classification problems, which are also solved in this thesis. We classify simply connected homogeneous spaces of compact Lie groups with the same integral cohomology ring as a product of spheres S^2 x S^m and m odd on the one hand and a truncated polynomial ring Q[a]/(a^m) with one generator of even degree and m > 1 as its rational cohomology ring on the other hand.

It is well known, that the least squares estimator performs poorly in the presence of multicollinearity. One way to overcome this problem is using biased estimators, e.g. ridge regression estimators. In this study an estimation procedure is proposed based on adding a small quantity omega on some or each regressor. The resulting biased estimator is described in dependence of omega and furthermore it is shown that its mean squared error is smaller than the one corresponding to the least squares estimator in the case of highly correlated regressors.

In der vorliegenden Arbeit werden lineare Systeme elliptischer partieller Differentialgleichungen in schwacher Formulierung auf konischen Gebieten untersucht. Auf einem zunächst unbeschränkten Kegelgebiet betrachten wir den Fall beschränkter und nur von den Winkelvariablen abhängiger Koeffizientenfunktionen. Die durch selbige definierte Bilinearform genüge einer Gårdingschen Ungleichung. In gewichteten Sobolevräumen werden Existenz- und Eindeutigkeitsfragen geklärt, wobei das Problem mittels Fouriertransformation auf eine von einem komplexen Parameter abhängige Familie T(·) von Fredholmoperatoren zurückgeführt wird. Unter Anwendung des Residuenkalküls gewinnen wir eine Darstellung der Lösung in Form einer Zerlegung in einen glatten Anteil einerseits sowie eine endliche Summe von Singulärfunktionen andererseits. Durch Abschneidetechniken werden die gewonnenen Erkenntnisse auf den Fall schwach formulierter elliptischer Systeme auf beschränkten Kegelgebieten unter Formulierung in gewöhnlichen, nicht-gewichteten Sobolevräumen angewendet. Die für Regularitätsfragen maßgeblichen Eigenwerte der Operatorfunktion T mit minimalem positiven Imaginärteil werden im letzten Kapitel der Arbeit am Beispiel der ebenen elastischen Gleichungen numerisch bestimmt.