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.

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.

Diese Dissertation entsteht im Rahmen des Projekts Research-Group Learning and Neurosciences (ReGLaN)-Health and Logistics, welches die Optimierung der Gesundheitsversorgung in ländlichen Gebieten Südafrikas zum Ziel hat. Es besteht dabei eine Kooperation mit dem Council for Scientific and Industrial Research (CSIR) Meraka Institute mit Prof. Dr. Dr. Marlien Herselman, Pretoria, Südafrika, als zentrale Ansprechpartnerin. Die Dissertation befasst sich mit der mathematischen Modellierung für adaptive graphische Benutzerschnittstellen (GUI), die ein angepasstes Verhalten in Abhängigkeit von Geographischen Informationssystemen (GIS) besitzen und durch räumliche Fuzzy-Logik gesteuert werden. Innerhalb der Arbeit geht es um die mathematische Visualisierung von maßgeschneiderten Risiko- und Ressourcenkarten für epidemiologische Fragestellungen mit GIS und adaptives GUI-Design für eine Open Source (OS)-Anwendung für digitale Endgeräte zur räumlichen Entscheidungsunterstützung zugeschnitten auf unterschiedliche Benutzergruppen. Zur Evaluation und Initialisierung der GUI-Elemente wurde empirische Lehr-Lern-Forschung zum Umgang mit Geomedien und GUI-Elementen eingesetzt.

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.