## Fachbereich Mathematik

### Filtern

#### Fachbereich / Organisatorische Einheit

- Fachbereich Mathematik (217)
- Fraunhofer (ITWM) (2)

#### Erscheinungsjahr

#### Dokumenttyp

- Dissertation (217) (entfernen)

#### Schlagworte

- Modelling and Multicriteria Optimization of the Web Formation in a Spunbond Process (2008)
- This dissertation deals with the optimization of the web formation in a spunbond process for the production of artificial fabrics. A mathematical model of the process is presented. Based on the model, two kind of attributes to be optimized are considered, those related with the quality of the fabric and those describing the stability of the production process. The problem falls in the multicriteria and decision making framework. The functions involved on the model of the process are non linear, non convex and non differentiable. A strategy in two steps; exploration and continuation, is proposed to approximate numerically the Pareto frontier and alternative methods are proposed to navigate the set and support the decision making process. The proposed strategy is applied to a particular production process and numerical results are presented.

- Stochastic Optimization in Finance and Life Insurance: Applications of the Martingale Method (2008)
- 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.

- Hub Location Models in Public Transport Planning (2008)
- The dissertation deals with the application of Hub Location models in public transport planning. The author proposes new mathematical models along with different solution approaches to solve the instances. Moreover, a novel multi-period formulation is proposed as an extension to the general model. Due to its high complexity heuristic approaches are formulated to find a good solution within a reasonable amount of time.

- Nonlinear diffusion filtering of images using the topological gradient approach to edges detection (2007)
- In this thesis, the problem of nonlinear diffusion filtering of gray-scale images is theoretically and numerically investigated. In the first part of the thesis, we derive the topological asymptotic expansion of the Mumford-Shah like functional. We show that the dominant term of this expansion can be regarded as a criterion to edges detection in an image. In the numerical part, we propose the finite volume discretization for the Catté et al. and the Weickert diffusion filter models. The proposed discretization is based on the integro-interpolation method introduced by Samarskii. The numerical schemes are derived for the case of uniform and nonuniform cell-centered grids of the computational domain \(\Omega \subset \mathbb{R}^2\). In order to generate a nonuniform grid, the adaptive coarsening technique is proposed.

- Numerical Estimation of Surface Parameters by Level Set Methods (2008)
- 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.

- Fast Wavelet Transform by Biorthogonal Locally Supported Radial Basis Functions on Fixed Spherical Grids (2007)
- The thesis is concerned with multiscale approximation by means of radial basis functions on hierarchically structured spherical grids. A new approach is proposed to construct a biorthogonal system of locally supported zonal functions. By use of this biorthogonal system of locally supported zonal functions, a spherical fast wavelet transform (SFWT) is established. Finally, based on the wavelet analysis, geophysically and geodetically relevant problems involving rotation-invariant pseudodifferential operators are shown to be efficiently and economically solvable.

- Parallel tetrahedral mesh generation based on a-priori domain decomposition (2008)
- The desire to model in ever increasing detail geometrical and physical features has lead to a steady increase in the number of points used in field solvers. While many solvers have been ported to parallel machines, grid generators have left behind. Sequential generation of meshes of large size is extremely problematic both in terms of time and memory requirements. Therefore, the need for developing parallel mesh generation technique is well justified. In this work a novel algorithm is presented for automatic parallel generation of tetrahedral computational meshes based on geometrical domain decomposition. It has a potential to remove this bottleneck. Different domain decomposition approaches and criteria have been investigated. Questions regarding time and memory consumption, efficiency of computations and quality of generated surface and volume meshes have been considered. As a result of the work parTgen (partitioner and parallel tetrahedral mesh generator) software package based on the developed algorithm has been created. Several real-life examples of relatively complex structures involving large meshes (of order 10^7-10^8 elements) are given. It has been shown that high mesh quality is achieved. Memory and time consumption are reduced significantly, and parallel algorithm is efficient.

- A-infinity-bimodules and Serre A-infinity-functors (2007)
- This dissertation is intended to transport the theory of Serre functors into the context of A-infinity-categories. We begin with an introduction to multicategories and closed multicategories, which form a framework in which the theory of A-infinity-categories is developed. We prove that (unital) A-infinity-categories constitute a closed symmetric multicategory. We define the notion of A-infinity-bimodule similarly to Tradler and show that it is equivalent to an A-infinity-functor of two arguments which takes values in the differential graded category of complexes of k-modules, where k is a commutative ground ring. Serre A-infinity-functors are defined via A-infinity-bimodules following ideas of Kontsevich and Soibelman. We prove that a unital closed under shifts A-infinity-category over a field admits a Serre A-infinity-functor if and only if its homotopy category admits an ordinary Serre functor. The proof uses categories and Serre functors enriched in the homotopy category of complexes of k-modules. Another important ingredient is an A-infinity-version of the Yoneda Lemma.

- Vector Field Approximation on Regular Surfaces in Terms of Outer Harmonic Representations (2007)
- The present work deals with the (global and local) modeling of the windfield on the real topography of Rheinland-Pfalz. Thereby the focus is on the construction of a vectorial windfield from low, irregularly distributed data given on a topographical surface. The developed spline procedure works by means of vectorial (homogeneous, harmonic) polynomials (outer harmonics) which control the oscillation behaviour of the spline interpoland. In the process the characteristic of the spline curvature which defines the energy norm is assumed to be on a sphere inside the Earth interior and not on the Earth’s surface. The numerical advantage of this method arises from the maximum-minimum principle for harmonic functions.

- The difference of the solutions of the elastic and elastoplastic boundary value problem and an approach to multiaxial stress-strain correction (2007)
- In the theoretical part of this thesis, the difference of the solutions of the elastic and the elastoplastic boundary value problem is analysed, both for linear kinematic and combined linear kinematic and isotropic hardening material. We consider both models in their quasistatic, rate-independent formulation with linearised geometry. The main result of the thesis is, that the differences of the physical obervables (the stresses, strains and displacements) can be expressed as composition of some linear operators and play operators with respect to the exterior forces. Explicit homotopies between both solutions are presented. The main analytical devices are Lipschitz estimates for the stop and the play operator. We present some generalisations of the standard estimates. They allow different input functions, different initial memories and different scalar products. Thereby, the underlying time involving function spaces are the Sobolov spaces of first order with arbitrary integrability exponent between one and infinity. The main results can easily be generalised for the class of continuous functions with bounded total variation. In the practical part of this work, a method to correct the elastic stress tensor over a long time interval at some chosen points of the body is presented and analysed. In contrast to widespread uniaxial corrections (Neuber or ESED), our method takes multiaxiality phenomena like cyclic hardening/softening, ratchetting and non-masing behaviour into account using Jiang's model of elastoplasticity. It can be easily adapted to other constitutive elastoplastic material laws. The theory for our correction model is developped for linear kinematic hardening material, for which error estimated are derived. Our numerical algorithm is very fast and designed for the case that the elastic stress is piecewise linear. The results for the stresses can be significantly improved with Seeger's empirical strain constraint. For the improved model, a simple predictor-correcor algorithm for smooth input loading is established.