## 510 Mathematik

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We consider variational discretization of three different optimal control problems.
The first being a parabolic optimal control problem governed by space-time measure controls. This problem has a nice sparsity structure, which motivates our aim to achieve maximal sparsity on the discrete level. Due to the measures on the right hand side of the partial differential equation, we consider a very weak solution theory for the state equation and need an embedding into the continuous functions for the pairings to make sense. Furthermore, we employ Fenchel duality to formulate the predual problem and give results on solution theory of both the predual and the primal problem. Later on, the duality is also helpful for the derivation of algorithms, since the predual problem can be differentiated twice so that we can apply a semismooth Newton method. We then retrieve the optimal control by duality relations.
For the state discretization we use a Petrov-Galerkin method employing piecewise constant states and piecewise linear and continuous test functions in time. For the space discretization we choose piecewise linear and continuous functions. As a result the controls are composed of Dirac measures in space-time, centered at points on the discrete space-time grid. We prove that the optimal discrete states and controls converge strongly in L^q and weakly-* in Μ, respectively, to their smooth counterparts, where q ϵ (1,min{2,1+2/d}] is the spatial dimension. The variational discrete version of the state equation with the above choice of spaces yields a Crank-Nicolson time stepping scheme with half a Rannacher smoothing step.
Furthermore, we compare our approach to a full discretization of the corresponding control problem, precisely a discontinuous Galerkin method for the state discretization, where the discrete controls are piecewise constant in time and Dirac measures in space. Numerical experiments highlight the sparsity features of our discrete approach and verify the convergence results.
The second problem we analyze is a parabolic optimal control problem governed by bounded initial measure controls. Here, the cost functional consists of a tracking term corresponding to the observation of the state at final time. Instead of a regularization term for the control in the cost functional, we consider a bound on the measure norm of the initial control. As in the first problem we observe a sparsity structure, but here the control resides only in space at initial time, so we focus on the space discretization to achieve maximal sparsity of the control. Again, due to the initial measure in the partial differential equation, we rely on a very weak solution theory of the state equation.
We employ a dG(0) approximation of the state equation, i.e. we choose piecewise linear and continuous functions in space, which are piecewise constant in time for our ansatz and test space. Then, the variational discretization of the problem together with the optimality conditions induce maximal discrete sparsity of the initial control, i.e. Dirac measures in space. We present numerical experiments to illustrate our approach and investigate the sparsity structure
As third problem we choose an elliptic optimal control governed by functions of bounded variation (BV) in one space dimension. The cost functional consists of a tracking term for the state and a BV-seminorm in terms of the derivative of the control. We derive a sparsity structure for the derivative of the BV control. Additionally, we utilize the mixed formulation for the state equation.
A variational discretization approach with piecewise constant discretization of the state and piecewise linear and continuous discretization of the adjoint state yields that the derivative of the control is a sum of Dirac measures. Consequently the control is a piecewise constant function. Under a structural assumption we even get that the number of jumps of the control is finite. We prove error estimates for the variational discretization approach in combination with the mixed formulation of the state equation and confirm our findings in numerical experiments that display the convergence rate.
In summary we confirm the use of variational discretization for optimal control problems with measures that inherit a sparsity. We are able to preserve the sparsity on the discrete level without discretizing the control variable.

Gegeben sei eine Basis b>=10 und eine Ziffer a0 aus der Menge {0,..., b − 1}. Wir untersuchen, ob es unendlich viele Primzahlen gibt, die in ihrer b-adischen Zifferndarstellung die Ziffer a0 nicht besitzen. Ferner schätzen wir die Anzahl dieser Primzahlen, die kleiner sind als X = b^k, nach oben und unten ab.
Damit gelingt uns eine Verallgemeinerung von Maynards Beweis für den Fall b = 10 und wir nutzen hierzu auch die in seiner Arbeit verwendeten Werkzeuge. Unter Anderem benötigen wir die Hardy-Littlewoodsche Kreismethode sowie diverse Siebmethoden, um die Minor Arcs zu kontrollieren.
Schließlich sehen wir, dass wir Maynard's Aussage vor allem dann auf beliebige Basen b>= 10 und ausgeschlossene Ziffern a0 aus {0, ..., b − 1} übertragen können, wenn zwei betragsmäßig größte Eigenwerte von Matrizen, die von b und a0 parametrisiert werden, bestimmte Abschätzungen erfüllen. Dass diese Abschätzungen im Fall b>=102 erfüllt sind, beweisen wir im letzten Kapitel. Für die Fälle b = 10 und b = 11 liegt ebenfalls ein Mathematica-Code vor, der die Abschätzungen bestätigt.

In Part I: "The flow-decomposition problem", we introduce and discuss the flow-decomposition problem. Given a flow F, this problem consists of decomposing the flow into a set of paths optimizing specific properties of those paths. We introduce different types of decompositions, such as integer decompositions and alpha-decompositions, and provide two formulations of the set of feasible decompositions.
We show that the problem of minimizing the longest path in a decomposition is NP-hard, even for fractional solutions. Then we develop an algorithm based on column generation which is able to solve the problem.
Tight upper bounds on the optimal objective value help to improve the performance.
To find upper bounds on the optimal solution for the shortest longest path problem, we develop several heuristics and analyze their quality. On pearl graphs we prove a constant approximation ratio of 2 and 3 respectively for all heuristics. A numerical study on random pearl graphs shows that the solutions generated by the heuristics are usually much better than this worst-case bound.
In Part II: "Construction and analysis of evacuation models using flows over time", we consider two optimization models in the context of evacuation planning. The first model is a parameter-based quickest flow model with time-dependent supply values. We give a detailed description of the network construction and of how different scenarios are modeled by scenario parameters. In a second step we analyze the effect of the scenario parameters on the evacuation time. Understanding how the different parameters influence the evacuation time allows us to provide better advice for evacuation planning and allows us to predict evacuation times without solving additional optimization problems. To understand the effect of the time-dependent supply values, we consider the quickest path problem with time-dependent supply values and provide a solution algorithm. The results from this consideration are generalized to approximate the behavior of the evacuation times in the context of quickest flow problems.
The second model we consider is a path-based model for evacuation in the presence of a dynamic cost function. We discuss the challenges of this model and provide ideas for how to approach the problem from different angles. We relate the problem to the flow-decomposition problem and consider the computation of evacuation paths with dynamic costs for large capacities. For the latter method we provide heuristics to find paths and compare them to the optimal solutions by applying the methods to two evacuation scenarios. An analysis shows that the paths generated by the heuristic yield close to optimal solutions and in addition have several desirable properties for evacuation paths which are not given for the optimal solution.

The formulation of the decoding problem for linear block codes as an integer program (IP) with a rather tight linear programming (LP) relaxation has made a central part of channel coding accessible for the theory and methods of mathematical optimization, especially integer programming, polyhedral combinatorics and also algorithmic graph theory, since the important class of turbo codes exhibits an inherent graphical structure. We present several novel models, algorithms and theoretical results for error-correction decoding based on mathematical optimization. Our contribution includes a partly combinatorial LP decoder for turbo codes, a fast branch-and-cut algorithm for maximum-likelihood (ML) decoding of arbitrary binary linear codes, a theoretical analysis of the LP decoder's performance for 3-dimensional turbo codes, compact IP models for various heuristic algorithms as well as ML decoding in combination with higher-order modulation, and, finally, first steps towards an implementation of the LP decoder in specialized hardware. The scientific contributions are presented in the form of seven revised reprints of papers that appeared in peer-reviewed international journals or conference proceedings. They are accompanied by an extensive introductory part that reviews the basics of mathematical optimization, coding theory, and the previous results on LP decoding that we rely on afterwards.

Mathematical Modelling of GIS Tailored GUI Design with the Application of Spatial Fuzzy Logic
(2014)

This PhD thesis is situated within the framework of the Research-Group Learning and Neurosciences (ReGLaN)-Health and Logistics project. The goal of this project is the optimisation of health service delivery in the rural areas of South Africa. Cooperation takes place between ReGLaN-Health and Logistics and the South African Council for Scientific and Industrial Research (CSIR) Meraka Institute, with Prof Dr Dr Marlien Herselman of Pretoria, South Africa, as the central contact person. This thesis deals with the mathematical modelling of Geographic Information System (GIS)-tailoredrnGraphical User Interface (GUI) design with the application of spatial fuzzy logic. This thesis considers the mathematical visualisation of risk and resource maps for epidemiological issues using GIS and adaptive GUI design for an Open Source (OS) application for digital devices. The intention ofrnthis thesis is to provide spatial decision support tailored to different user groups. In order for the GUI elements to be evaluated and initialised, empirical teaching-learning-research on dealing with geomedia and GUI elements was conducted.

New media are continually gaining importance in society. This process also has an increasing influence on developments in the field of education. Due to the use of computers as an integral part of schooling, new possibilities with regard to the organisation of learning processes arise. In this context, it is of great significance that appropriate computer applications for the respective learning group be prepared, so that justifiable use of computers in lessons can take place. Furthermore, efficient integration of computers requires changes in spatial organisation, in teaching methodology and in the role of the teacher. Such reflection and re-orientation are the essential basis for meaningful usage of new media in teaching and learning processes. An initial aim of this thesis is an empirical analysis of the situation regarding the usage of computers in geometry lessons in primary schools, based on a regional survey. The evaluation gives information as to how intensively the computer is used in the learning process and shows us which factors determine the use of computers in geometry lessons.
The results are an empirical foundation for the development of a computer-based learning environment called "Geolizi" (the second aim of my study). Within this learning environment, the pupils should work independently on the topics "mirror-imaged figures" and "the construction of rectangles and squares", with the help of the computer. During this process, hands-on media, traditional drawing instruments and interactive worksheets are available to the pupils. The computer (with its appropriate applications) takes over different functions in this learning process. Testing of this learning environment ("Geolizi") took place in several primary school classes, within the scope of formative and summative evaluation. With the help of questionnaires filled in by the pupils, the usability of the individual elements was tested. Based on a pre-post-investigation design, an attempt has been made to discover possible changes in the attitude of teachers regarding the usage of computers in the teaching of elementary geometry.
The results of this test phase, together with the evaluation of the questionnaires, lead to the founded presumption that usage of the multimedia-based learning environment " Geolizi " could result in greater use of computers in geometry lessons. All in all, the developed learning environment demonstrates an interesting possibility of how to use computers in the teaching of geometry at primary schools, thus making an important contribution to an independent, individualised learning process.

This dissertation provides an interdisciplinary contribution to the project ReGLaN-Health & Logistics. ReGLaN-Health & Logistics, is an international cooperation deriving benefits from the capabilities of scientists working on different fields. The aim of the project is the development of a socalled SDSS that supports decision makers working within health systems with a special focus on rural areas. In this dissertation, one important component for the development of the DSS named EWARS is proposed and described in detail. This component called SPATTB is developed with the intention of dealing with spatial data, i.e. data with additional geocoded information with regard to the special requirements of the EWARS.rnrnAn important component in the process of developing the EWARS is the concept of GIS. Classically, geocoded information with a vectorial character numerically describing spatial phenomena is managed and processed in a GIS. For the development of the EWARS, the manageability of the type of data exemplarily given by (x,y,o) with coordinates x,y ) and Ozon-concentration o is not sufficient. It is described, that the manageable data has to be extended to data of type (x,y,f ), where (x,y) are the geocoded information, but where f is not only a numerical value but a functional description of a certain phenomenom. An example for the existence and appearance of that type of data is the geocoded information about the variation of the Ozon-concentration in time or depending on temperature. A knowledge-base as important subsystem of DSS containing expert knowledge is mentioned. This expert-knowledge can be made manageable when using methods from the field of fuzzy logic. Thereby mappings, socalled fuzzy-sets, are generated. Within the EWARS, these mappings will be used with respect to additional geocoded data. The knowledge about the geocoded mapping information only at a finite set of locations (x,y) associated with mapping information f is not sufficient in applications that need continuous statements in a certain geographical area. To provide a contribution towards solving this problem, methods from the field of computer geometry and CAD, so-called Bezier-methods, are used for interpolating this geocoded mapping information. Classically, these methods operates on vectors a the multidimensional vector-space whose elements contain real-valued components but in terms of dealing with mapping information, there has to be an extension on topological vector spaces since mapping spaces can be defined as such spaces. This builds a new perspective and possibility in the application of these methods. Therefore, the according algorithms have to be extended; this work is presented. The field of Artificial Neural Networks plays an important role for the processing and management of the data within the EWARS, where features of biological processes and structures are modeled and implemented as algorithms. Generally, the developed methods can be divided as usable in terms of interpolation or approximation functional coherences and in such being applicable to classification problems. In this dissertation one method from each type is regarded in more detailed. Thereby, the classical algorithms of the so-called Backpropagation-Networks for approximation and the Kohonen-Networks for classification are described. Within the thesis, an extension of these algorithms is then proposed using coherences from mathematical measure-theory and approximation theory. The mentioned extension of these algorithms is based on a preprocessing of the mapping data using integration methods from measure theory.