## Fachbereich Mathematik

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- 2012 (13) (remove)

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- Doctoral Thesis (13) (remove)

#### Keywords

- Transaction Costs (2)
- Cohen-Lenstra heuristic (1)
- Consistent Price Processes (1)
- Gyroscopic (1)
- Image restoration (1)
- No-Arbitrage (1)
- Poisson noise (1)
- Portfolio Optimization (1)
- QVIs (1)
- Stochastic Control (1)

- On the Second Class Group of Real Quadratic Number Fields (2012)
- This thesis generalizes the Cohen-Lenstra heuristic for the class groups of real quadratic number fields to higher class groups. A "good part" of the second class group is defined. In general this is a non abelian proper factor group of the second class group. Properties of those groups are described, a probability distribution on the set of those groups is in- troduced and proposed as generalization of the Cohen-Lenstra heuristic for real quadratic number fields. The calculation of number field tables which contain information about higher class groups is explained and the tables are compared to the heuristic. The agree- ment is close. A program which can create an internet database for number field tables is presented.

- Mathematical Models for 3-Phase Traffic Flow Theory (2012)
- In this thesis we outline the Kerner's 3-phase traffic flow theory, which states that the flow of vehicular traffic occur in three phases i.e. free flow, synchronized flow and wide moving jam phases. A macroscopic 3-phase traffic model of the Aw-Rascle type is derived from the microscopic Speed Adaptation 3-phase traffic model developed by Kerner and Klenov [J. Phys. A: Math. Gen., 39(2006), pp. 1775-1809 ]. We derive the same macroscopic model from the kinetic traffic flow model of Klar and Wegener [SIAM J. Appl. Math., 60(2000), pp. 1749-1766 ] as well as that of Illner, Klar and Materne [Comm. Math. Sci., 1(2003), pp. 1-12 ]. In the above stated derivations, the 3-phase traffic theory is constituted in the macroscopic model through a relaxation term. This serves as an incentive to modify the relaxation term of the `switching curve' model of Greenberg, Klar and Rascle [SIAM J. Appl. Math.,63(2003), pp.818-833 ] to obtain another macroscopic 3-phase traffic model, which is still of the Aw-Rascle type. By specifying the relaxation term differently we obtain three kinds of models, namely the macroscopic Speed Adaptation, the Switching Curve and the modified Switching Curve models. To demonstrate the capability of the derived macroscopic traffic models to reproduce the features of 3-phase traffic theory, we simulate a multi-lane road that has a bottleneck. We consider a stationary and a moving bottleneck. The results of the simulations for the three models are compared.

- On Gyroscopic Stabilization (2012)
- This thesis deals with systems of the form \( M\ddot x+D\dot x+Kx=0\;, \; x \in \mathbb R^n\;, \) with a positive definite mass matrix \(M\), a symmetric damping matrix \(D\) and a positive definite stiffness matrix \(K\). If the equilibrium in the system is unstable, a small disturbance is enough to set the system in motion again. The motion of the system sustains itself, an effect which is called self-excitation or self-induced vibration. The reason behind this effect is the presence of negative damping, which results for example from dry friction. Negative damping implies that the damping matrix \(D\) is indefinite or negative definite. Throughout our work, we assume \(D\) to be indefinite, and that the system possesses both stable and unstable modes and thus is unstable. It is now the idea of gyroscopic stabilization to mix the modes of a system with indefinite damping such that the system is stabilized without introducing further dissipation. This is done by adding gyroscopic forces \(G\dot x\) with a suitable skew-symmetric matrix \(G\) to the left-hand side. We call \(G=-G^T\in\mathbb R^{n\times n}\) a gyroscopic stabilizer for the unstable system, if \( M\ddot x+(D+ G)\dot x+Kx=0 \) is asymptotically stable. We show the existence of \(G\) in space dimensions three and four.

- Signature-based algorithms to compute standard bases (2012)
- Standard bases are one of the main tools in computational commutative algebra. In 1965 Buchberger presented a criterion for such bases and thus was able to introduce a first approach for their computation. Since the basic version of this algorithm is rather inefficient due to the fact that it processes lots of useless data during its execution, active research for improvements of those kind of algorithms is quite important. In this thesis we introduce the reader to the area of computational commutative algebra with a focus on so-called signature-based standard basis algorithms. We do not only present the basic version of Buchberger’s algorithm, but give an extensive discussion of different attempts optimizing standard basis computations, from several sorting algorithms for internal data up to different reduction processes. Afterwards the reader gets a complete introduction to the origin of signature-based algorithms in general, explaining the under- lying ideas in detail. Furthermore, we give an extensive discussion in terms of correctness, termination, and efficiency, presenting various different variants of signature-based standard basis algorithms. Whereas Buchberger and others found criteria to discard useless computations which are completely based on the polynomial structure of the elements considered, Faugère presented a first signature-based algorithm in 2002, the F5 Algorithm. This algorithm is famous for generating much less computational overhead during its execution. Within this thesis we not only present Faugère’s ideas, we also generalize them and end up with several different, optimized variants of his criteria for detecting redundant data. Being not completely focussed on theory, we also present information about practical aspects, comparing the performance of various implementations of those algorithms in the computer algebra system Singular over a wide range of example sets. In the end we give a rather extensive overview of recent research in this area of computational commutative algebra.

- Mathematical Modeling and Simulation of Two-Phase Flow in Porous Media with Application to the Pressing Section of a Paper Machine (2012)
- Paper production is a problem with significant importance for the society and it is a challenging topic for scientific investigations. This study is concerned with the simulations of the pressing section of a paper machine. We aim at the development of an advanced mathematical model of the pressing section, which is able to recover the behavior of the fluid flow within the paper felt sandwich obtained in laboratory experiments. From the modeling point of view the pressing of the paper-felt sandwich is a complex process since one has to deal with the two-phase flow in moving and deformable porous media. To account for the solid deformations, we use developments from the PhD thesis by S. Rief where the elasticity model is stated and discussed in detail. The flow model which accounts for the movement of water within the paper-felt sandwich is described with the help of two flow regimes: single-phase water flow and two-phase air-water flow. The model for the saturated flow is presented by the Darcy's law and the mass conservation. The second regime is described by the Richards' approach together with dynamic capillary effects. The model for the dynamic capillary pressure - saturation relation proposed by Hassanizadeh and Gray is adapted for the needs of the paper manufacturing process. We have started the development of the flow model with the mathematical modeling in one-dimensional case. The one-dimensional flow model is derived from a two-dimensional one by an averaging procedure in vertical direction. The model is numerically studied and verified in comparison with measurements. Some theoretical investigations are performed to prove the convergence of the discrete solution to the continuous one. For completeness of the studies, the models with the static and dynamic capillary pressure–saturation relations are considered. Existence, compactness and convergence results are obtained for both models. Then, a two-dimensional model is developed, which accounts for a multilayer computational domain and formation of the fully saturated zones. For discretization we use a non-orthogonal grid resolving the layer interfaces and the multipoint flux approximation O-method. The numerical experiments are carried out for parameters which are typical for the production process. The static and dynamic capillary pressure-saturation relations are tested to evaluate the influence of the dynamic capillary effect. The last part of the thesis is an investigation of the validity range of the Richards’ assumption for the two-dimensional flow model with the static capillary pressure-saturation relation. Numerical experiments show that the Richards’ assumption is not the best choice in simulating processes in the pressing section.

- Innovative Techniken und Algorithmen im Bereich Computational-Finance und Risikomanagement (2012)
- Diese Dissertation besteht aus zwei aktuellen Themen im Bereich Finanzmathematik, die voneinander unabhängig sind. Beim ersten Thema, "Flexible Algorithmen zur Bewertung komplexer Optionen mit mehreren Eigenschaften mittels der funktionalen Programmiersprache Haskell", handelt es sich um ein interdisziplinäres Projekt, in dem eine wissenschaftliche Brücke zwischen der Optionsbewertung und der funktionalen Programmierung geschlagen wurde. Im diesem Projekt wurde eine funktionale Bibliothek zur Konstruktion von Optionen entworfen, in dem es eine Reihe von grundlegenden Konstruktoren gibt, mit denen man verschiedene Optionen kombinieren kann. Im Rahmen der funktionalen Bibliothek wurde ein allgemeiner Algorithmus entwickelt, durch den die aus den Konstruktoren kombinierten Optionen bewertet werden können. Der mathematische Aspekt des Projekts besteht in der Entwicklung eines neuen Konzeptes zur Bewertung der Optionen. Dieses Konzept basiert auf dem Binomialmodell, welches in den letzten Jahren eine weite Verbreitung im Forschungsgebiet der Optionsbewertung fand. Der kerne Algorithmus des Konzeptes ist eine Kombination von mehreren sorgfältig ausgewählten numerischen Methoden in Bezug auf den Binomialbaum. Diese Kombination ist nicht trivial, sondern entwikelt sich nach bestimmten Regeln und ist eng mit den grundlegenden Konstruktoren verknüpft. Ein wichtiger Charakterzug des Projekts ist die funktionale Denkweise. D. h. der Algorithmus ließ sich mithilfe einer funktionalen Programmiersprache formulieren. In unserem Projekt wurde Haskell verwendet. Das zweite Thema, Monte-Carlo-Simulation des Deltas und (Cross-)Gammas von Bermuda-Swaptions im LIBOR-Marktmodell, bezieht sich auf ein zentrales Problem der Finanzmathematik, nämlich die Bestimmung der Risikoparameter komplexer Zinsderivate. In dieser Arbeit wurde die numerische Berechnung des Delta-Vektors einer Bermuda- Swaption ausführlich untersucht und die neue Herausforderung, die Gamma-Matrix einer Bermuda-Swaption exakt simulieren, erfolgreich gemeistert. Die beiden Risikoparameter spielen bei Handelsstrategien in Form des Delta-Hedgings und Gamma-Hedgings eine entscheidende Rolle. Das zugrunde liegende Zinsstrukturmodell ist das LIBORMarktmodell, welches in den letzten Jahren eine auffällige Entwicklung in der Finanzmathematik gemacht hat. Bei der Simulation und Anwendung des LIBOR-Marktmodells fällt die Monte-Carlo-Simulation ins Gewicht. Für die Berechung des Delta-Vektors einer Bermuda-Swaption wurden drei klassische und drei von uns entwickelte numerische Methoden vorgestellt und gegenübergestellt, welche fast alle vorhandenen Arten der Monte-Carlo-Simulation zur Berechnung des Delta-Vektors einer Bermuda-Swaption enthalten. Darüber hinaus gibt es in der Arbeit noch zwei neu entwickelte Methoden, um die Gamma-Matrix einer Bermuda-Swaption exakt zu berechnen, was völlig neu im Forschungsgebiet der Computational-Finance ist. Eine ist die modifizierte Finite-Differenzen-Methode. Die andere ist die reine Pathwise-Methode, die auf pfadweiser Differentialrechnung basiert und einem robusten und erwartungstreuen Simulationsverfahren entspricht.

- Anisotropic Smoothing and Image Restoration Facing Non-Gaussian Noise (2012)
- Image restoration and enhancement methods that respect important features such as edges play a fundamental role in digital image processing. In the last decades a large variety of methods have been proposed. Nevertheless, the correct restoration and preservation of, e.g., sharp corners, crossings or texture in images is still a challenge, in particular in the presence of severe distortions. Moreover, in the context of image denoising many methods are designed for the removal of additive Gaussian noise and their adaptation for other types of noise occurring in practice requires usually additional efforts. The aim of this thesis is to contribute to these topics and to develop and analyze new methods for restoring images corrupted by different types of noise: First, we present variational models and diffusion methods which are particularly well suited for the restoration of sharp corners and X junctions in images corrupted by strong additive Gaussian noise. For their deduction we present and analyze different tensor based methods for locally estimating orientations in images and show how to successfully incorporate the obtained information in the denoising process. The advantageous properties of the obtained methods are shown theoretically as well as by numerical experiments. Moreover, the potential of the proposed methods is demonstrated for applications beyond image denoising. Afterwards, we focus on variational methods for the restoration of images corrupted by Poisson and multiplicative Gamma noise. Here, different methods from the literature are compared and the surprising equivalence between a standard model for the removal of Poisson noise and a recently introduced approach for multiplicative Gamma noise is proven. Since this Poisson model has not been considered for multiplicative Gamma noise before, we investigate its properties further for more general regularizers including also nonlocal ones. Moreover, an efficient algorithm for solving the involved minimization problems is proposed, which can also handle an additional linear transformation of the data. The good performance of this algorithm is demonstrated experimentally and different examples with images corrupted by Poisson and multiplicative Gamma noise are presented. In the final part of this thesis new nonlocal filters for images corrupted by multiplicative noise are presented. These filters are deduced in a weighted maximum likelihood estimation framework and for the definition of the involved weights a new similarity measure for the comparison of data corrupted by multiplicative noise is applied. The advantageous properties of the new measure are demonstrated theoretically and by numerical examples. Besides, denoising results for images corrupted by multiplicative Gamma and Rayleigh noise show the very good performance of the new filters.

- Maximizing the Asymptotic Growth Rate under Fixed and Proportional Transaction Costs in a Financial Market with Jumps (2012)
- In this thesis we consider the problem of maximizing the growth rate with proportional and fixed costs in a framework with one bond and one stock, which is modeled as a jump diffusion with compound Poisson jumps. Following the approach from [1], we prove that in this framework it is optimal for an investor to follow a CB-strategy. The boundaries depend only on the parameters of the underlying stock and bond. Now it is natural to ask for the investor who follows a CB-strategy which is given by the stopping times \((\tau_i)_{i\in\mathbb N}\) and impulses \((\eta_i)_{i\in\mathbb N}\) how often he has to rebalance. In other words we want to obtain the limit of the inter trading times \[ \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n(\tau_{i+1}-\tau_{i}). \] We are able to obtain this limit which is given by the expected first exit time of the risky fraction process from some interval under the invariant measure of the Markov chain \((\eta_i)_{i\in\mathbb N}\) using the Ergodic Theorem from von Neumann and Birkhoff. In general, it is difficult to obtain the expectation of the first exit time for the process with jumps. Because of the jump part, when the process crosses the boundaries of the interval an overshoot may occur which makes it difficult to obtain the distribution. Nevertheless we can obtain the first exit time if the process has only negative jumps using scale functions. The main difficulty of this approach is that the scale functions are known only up to their Laplace transforms. In [2] and [3] the closed-form expression for the scale function of the Levy process with phase-type distributed jumps is obtained. Phase-type distributions build a rich class of positive-valued distributions: the exponential, hyperexponential, Erlang, hyper-Erlang and Coxian distributions. Since the scale function is given as a function in a closed form we can differentiate to obtain the expected first exit time using the fluctuation identities explicitly. [1] Irle, A. and Sass,J.: Optimal portfolio policies under fixed and proportional transaction costs, Advances in Applied Probability 38, 916-942. [2] Egami, M., Yamazaki, K.: On scale functions of spectrally negative Levy processes with phase-type jumps, working paper, July 3. [3]Egami, M., Yamazaki, K.: Precautionary measures for credit risk management in jump models, working paper, June 17.

- Utility-based proof for the existence of strictly consistent price processes under proportional transaction costs (2012)
- This thesis deals with the relationship between no-arbitrage and (strictly) consistent price processes for a financial market with proportional transaction costs in a discrete time model. The exact mathematical statement behind this relationship is formulated in the so-called Fundamental Theorem of Asset Pricing (FTAP). Among the many proofs of the FTAP without transaction costs there is also an economic intuitive utility-based approach. It relies on the economic intuitive fact that the investor can maximize his expected utility from terminal wealth. This approach is rather constructive since the equivalent martingale measure is then given by the marginal utility evaluated at the optimal terminal payoff. However, in the presence of proportional transaction costs such a utility-based approach for the existence of consistent price processes is missing in the literature. So far, rather deep methods from functional analysis or from the theory of random sets have been used to show the FTAP under proportional transaction costs. For the sake of existence of a utility-maximizing payoff we first concentrate on a generic single-period model with only one risky asset. The marignal utility evaluated at the optimal terminal payoff yields the first component of a consistent price process. The second component is given by the bid-ask prices depending on the investors optimal action. Even more is true: nearby this consistent price process there are many strictly consistent price processes. Their exact structure allows us to apply this utility-maximizing argument in a multi-period model. In a backwards induction we adapt the given bid-ask prices in such a way so that the strictly consistent price processes found from maximizing utility can be extended to terminal time. In addition possible arbitrage opportunities of the 2nd kind vanish which can present for the original bid-ask process. The notion of arbitrage opportunities of the 2nd kind has been so far investigated only in models with strict costs in every state. In our model transaction costs need not be present in every state. For a model with finitely many risky assets a similar idea is applicable. However, in the single-period case we need to develop new methods compared to the single-period case with only one risky asset. There are mainly two reasons for that. Firstly, it is not at all obvious how to get a consistent price process from the utility-maximizing payoff, since the consistent price process has to be found for all assets simultaneously. Secondly, we need to show directly that the so-called vector space property for null payoffs implies the robust no-arbitrage condition. Once this step is accomplished we can à priori use prices with a smaller spread than the original ones so that the consistent price process found from the utility-maximizing payoff is strictly consistent for the original prices. To make the results applicable for the multi-period case we assume that the prices are given by compact and convex random sets. Then the multi-period case is similar to the case with only one risky asset but more demanding with regard to technical questions.

- Effective mechanical properties of technical textile materials via asymptotic homogenization (2012)
- The goal of this work is to develop a simulation-based algorithm, allowing the prediction of the effective mechanical properties of textiles on the basis of their microstructure and corresponding properties of fibers. This method can be used for optimization of the microstructure, in order to obtain a better stiffness or strength of the corresponding fiber material later on. An additional aspect of the thesis is that we want to take into account the microcontacts between fibers of the textile. One more aspect of the thesis is the accounting for the thickness of thin fibers in the textile. An introduction of an additional asymptotics with respect to a small parameter, the relation between the thickness and the representative length of the fibers, allows a reduction of local contact problems between fibers to 1-dimensional problems, which reduces numerical computations significantly. A fiber composite material with periodic microstructure and multiple frictional microcontacts between fibers is studied. The textile is modeled by introducing small geometrical parameters: the periodicity of the microstructure and the characteristic diameter of fibers. The contact linear elasticity problem is considered. A two-scale approach is used for obtaining the effective mechanical properties. The algorithm using asymptotic two-scale homogenization for computation of the effective mechanical properties of textiles with periodic rod or fiber microstructure is proposed. The algorithm is based on the consequent passing to the asymptotics with respect to the in-plane period and the characteristic diameter of fibers. This allows to come to the equivalent homogenized problem and to reduce the dimension of the auxiliary problems. Further numerical simulations of the cell problems give the effective material properties of the textile. The homogenization of the boundary conditions on the vanishing out-of-plane interface of a textile or fiber structured layer has been studied. Introducing additional auxiliary functions into the formal asymptotic expansion for a heterogeneous plate, the corresponding auxiliary and homogenized problems for a nonhomogeneous Neumann boundary condition were deduced. It is incorporated into the right hand side of the homogenized problem via effective out-of-plane moduli. FiberFEM, a C++ finite element code for solving contact elasticity problems, is developed. The code is based on the implementation of the algorithm for the contact between fibers, proposed in the thesis. Numerical examples of homogenization of geotexiles and wovens are obtained in the work by implementation of the developed algorithm. The effective material moduli are computed numerically using the finite element solutions of the auxiliary contact problems obtained by FiberFEM.