## Fachbereich Mathematik

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This thesis is devoted to deal with the stochastic optimization problems in various situations with the aid of the Martingale method. Chapter 2 discusses the Martingale method and its applications to the basic optimization problems, which are well addressed in the literature (for example, [15], [23] and [24]). In Chapter 3, we study the problem of maximizing expected utility of real terminal wealth in the presence of an index bond. Chapter 4, which is a modification of the original research paper joint with Korn and Ewald [39], investigates an optimization problem faced by a DC pension fund manager under inflationary risk. Although the problem is addressed in the context of a pension fund, it presents a way of how to deal with the optimization problem, in the case there is a (positive) endowment. In Chapter 5, we turn to a situation where the additional income, other than the income from returns on investment, is gained by supplying labor. Chapter 6 concerns a situation where the market considered is incomplete. A trick of completing an incomplete market is presented there. The general theory which supports the discussion followed is summarized in the first chapter.

We present a new efficient and robust algorithm for topology optimization of 3D cast parts. Special constraints are fulfilled to make possible the incorporation of a simulation of the casting process into the optimization: In order to keep track of the exact position of the boundary and to provide a full finite element model of the structure in each iteration, we use a twofold approach for the structural update. A level set function technique for boundary representation is combined with a new tetrahedral mesh generator for geometries specified by implicit boundary descriptions. Boundary conditions are mapped automatically onto the updated mesh. For sensitivity analysis, we employ the concept of the topological gradient. Modification of the level set function is reduced to efficient summation of several level set functions, and the finite element mesh is adapted to the modified structure in each iteration of the optimization process. We show that the resulting meshes are of high quality. A domain decomposition technique is used to keep the computational costs of remeshing low. The capabilities of our algorithm are demonstrated by industrial-scale optimization examples.

In this paper we develop a data-driven mixture of vector autoregressive models with exogenous components. The process is assumed to change regimes according to an underlying Markov process. In contrast to the hidden Markov setup, we allow the transition probabilities of the underlying Markov process to depend on past time series values and exogenous variables. Such processes have potential applications to modeling brain signals. For example, brain activity at time t (measured by electroencephalograms) will can be modeled as a function of both its past values as well as exogenous variables (such as visual or somatosensory stimuli). Furthermore, we establish stationarity, geometric ergodicity and the existence of moments for these processes under suitable conditions on the parameters of the model. Such properties are important for understanding the stability properties of the model as well as deriving the asymptotic behavior of various statistics and model parameter estimators.

In this dissertation we consider mesoscale based models for flow driven fibre orientation dynamics in suspensions. Models for fibre orientation dynamics are derived for two classes of suspensions. For concentrated suspensions of rigid fibres the Folgar-Tucker model is generalized by incorporating the excluded volume effect. For dilute semi-flexible fibre suspensions a novel moments based description of fibre orientation state is introduced and a model for the flow-driven evolution of the corresponding variables is derived together with several closure approximations. The equation system describing fibre suspension flows, consisting of the incompressible Navier-Stokes equation with an orientation state dependent non-Newtonian constitutive relation and a linear first order hyperbolic system for the fibre orientation variables, has been analyzed, allowing rather general fibre orientation evolution models and constitutive relations. The existence and uniqueness of a solution has been demonstrated locally in time for sufficiently small data. The closure relations for the semiflexible fibre suspension model are studied numerically. A finite volume based discretization of the suspension flow is given and the numerical results for several two and three dimensional domains with different parameter values are presented and discussed.

We study the complexity of finding extreme pure Nash equilibria in symmetric network congestion games and analyse how it depends on the graph topology and the number of users. In our context best and worst equilibria are those with minimum respectively maximum total latency. We establish that both problems can be solved by a Greedy algorithm with a suitable tie breaking rule on parallel links. On series-parallel graphs finding a worst Nash equilibrium is NP-hard for two or more users while finding a best one is solvable in polynomial time for two users and NP-hard for three or more. Additionally we establish NP-hardness in the strong sense for the problem of finding a worst Nash equilibrium on a general acyclic graph.

The goal of a multicriteria program is to explore different possibilities and their respective compromises which adequately represent the nondominated set. An exact description will in most cases fail because the number of efficient solutions is either too large or even infinite. We approximate the nondominated by computing a finite collection of nondominated points. Different ideas have been applied, including nonnegative weighted scalarization, Tchebycheff weighted scalarization, block norms and epsilon-constraints. Block norms are the building blocks for the inner and outer approximation algorithms proposed by Klamroth. We review these algorithms and propose three different variants. However, block norm based algorithms require to solve a sequence of subproblems, the number of subproblems becomes relatively high for six criteria and even intractable for real applications with nine criteria. Thus, we use bilevel linear programming to derive an approximation algorithm. We finally analyze and compare the approximation quality, running time and numerical convergence of the proposed methods.

Minimum Cut Tree Games
(2008)

In this paper we introduce a cooperative game based on the minimum cut tree problem which is also known as multi-terminal maximum flow problem. Minimum cut tree games are shown to be totally balanced and a solution in their core can be obtained in polynomial time. This special core allocation is closely related to the solution of the original graph theoretical problem. We give an example showing that the game is not supermodular in general, however, it is for special cases and for some of those we give an explicit formula for the calculation of the Shapley value.

In this thesis, we investigate a statistical model for precipitation time series recorded at a single site. The sequence of observations consists of rainfall amounts aggregated over time periods of fixed duration. As the properties of this sequence depend strongly on the length of the observation intervals, we follow the approach of Rodriguez-Iturbe et. al. [1] and use an underlying model for rainfall intensity in continuous time. In this idealized representation, rainfall occurs in clusters of rectangular cells, and each observations is treated as the sum of cell contributions during a given time period. Unlike the previous work, we use a multivariate lognormal distribution for the temporal structure of the cells and clusters. After formulating the model, we develop a Markov-Chain Monte-Carlo algorithm for fitting it to a given data set. A particular problem we have to deal with is the need to estimate the unobserved intensity process alongside the parameter of interest. The performance of the algorithm is tested on artificial data sets generated from the model. [1] I. Rodriguez-Iturbe, D. R. Cox, and Valerie Isham. Some models for rainfall based on stochastic point processes. Proc. R. Soc. Lond. A, 410:269-288, 1987.

In this thesis, the coupling of the Stokes equations and the Biot poroelasticity equations for fluid flow normal to porous media is investigated. For that purpose, the transmission conditions across the interfaces between the fluid regions and the porous domain are derived. A proper algorithm is formulated and numerical examples are presented. First, the transmission conditions for the coupling of various physical phenomena are reviewed. For the coupling of free flow with porous media, it has to be distinguished whether the fluid flows tangentially or perpendicularly to the porous medium. This plays an essential role for the formulation of the transmission conditions. In the thesis, the transmission conditions for the coupling of the Stokes equations and the Biot poroelasticity equations for fluid flow normal to the porous medium in one and three dimensions are derived. With these conditions, the continuous fully coupled system of equations in one and three dimensions is formulated. In the one dimensional case the extreme cases, i.e. fluid-fluid interface and fluid impermeable solid interface, are considered. Two chapters of the thesis are devoted to the discretisation of the fully coupled Biot-Stokes system for matching and non-matching grids, respectively. Therefor, operators are introduced that map the internal and boundary variables to the respective domains via Stokes equations, Biot equations and the transmission conditions. The matrix representation of some of these operators is shown. For the non-matching case, a cell-centred grid in the fluid region and a staggered grid in the porous domain are used. Hence, the discretisation is more difficult, since an additional grid on the interface has to be introduced. Corresponding matching functions are needed to transfer the values properly from one domain to the other across the interface. In the end, the iterative solution procedure for the Biot-Stokes system on non-matching grids is presented. For this purpose, a short review of domain decomposition methods is given, which are often the methods of choice for such coupled problems. The iterative solution algorithm is presented, including details like stopping criteria, choice and computation of parameters, formulae for non-dimensionalisation, software and so on. Finally, numerical results for steady state examples, depth filtration and cake filtration examples are presented.

A modular level set algorithm is developed to study the interface and its movement for free moving boundary problems. The algorithm is divided into three basic modules : initialization, propagation and contouring. Initialization is the process of finding the signed distance function from closed objects. We discuss here, a methodology to find an accurate signed distance function from a closed, simply connected surface discretized by triangulation. We compute the signed distance function using the direct method and it is stored efficiently in the neighborhood of the interface by a narrow band level set method. A novel approach is employed to determine the correct sign of the distance function at convex-concave junctions of the surface. The accuracy and convergence of the method with respect to the surface resolution is studied. It is shown that the efficient organization of surface and narrow band data structures enables the solution of large industrial problems. We also compare the accuracy of the signed distance function by direct approach with Fast Marching Method (FMM). It is found that the direct approach is more accurate than FMM. Contouring is performed through a variant of the marching cube algorithm used for the isosurface construction from volumetric data sets. The algorithm is designed to keep foreground and background information consistent, contrary to the neutrality principle followed for surface rendering in computer graphics. The algorithm ensures that the isosurface triangulation is closed, non-degenerate and non-ambiguous. The constructed triangulation has desirable properties required for the generation of good volume meshes. These volume meshes are used in the boundary element method for the study of linear electrostatics. For estimating surface properties like interface position, normal and curvature accurately from a discrete level set function, a method based on higher order weighted least squares is developed. It is found that least squares approach is more accurate than finite difference approximation. Furthermore, the method of least squares requires a more compact stencil than those of finite difference schemes. The accuracy and convergence of the method depends on the surface resolution and the discrete mesh width. This approach is used in propagation for the study of mean curvature flow and bubble dynamics. The advantage of this approach is that the curvature is not discretized explicitly on the grid and is estimated on the interface. The method of constant velocity extension is employed for the propagation of the interface. With least squares approach, the mean curvature flow has considerable reduction in mass loss compared to finite difference techniques. In the bubble dynamics, the modules are used for the study of a bubble under the influence of surface tension forces to validate Young-Laplace law. It is found that the order of curvature estimation plays a crucial role for calculating accurate pressure difference between inside and outside of the bubble. Further, we study the coalescence of two bubbles under surface tension force. The application of these modules to various industrial problems is discussed.