## Fachbereich Mathematik

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- Fachbereich Mathematik (227)
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This dissertation is intended to give a systematic treatment of hypersurface singularities in arbitrary characteristic which provides the necessary tools, theoretically and computationally, for the purpose of classification. This thesis consists of five chapters: In chapter 1, we introduce the background on isolated hypersurface singularities needed for our work. In chapter 2, we formalize the notions of piecewise-homogeneous grading and we discuss thoroughly non-degeneracy in arbitrary characteristic. Chapter 3 is devoted to determinacy and normal forms of isolated hypersurface singularities. In the first part, we give finite determinacy theorems in arbitrary characteristic with respect to right respectively contact equivalence. Furthermore, we show that "isolated" and finite determinacy properties are equivalent. In the second part, we formalize Arnol'd's key ideas for the computation of normal forms an define the conditions (AA) and (AAC). The last part of Chapter 3 is devoted to the study of normal forms in the general setting of hypersurface singularities imposing neither condition (A) nor Newton-Nondegeneracy. In Chapter 4, we present algorithms which we implement in Singular for the purpose of explicit computation of regular bases and normal forms. In chapter 5, we transfer some classical results on invariants over the field C of complex numbers to algebraically closed fields of characteristic zero known as Lefschetz principle.

In der Automobilindustrie muss der Nachweis von Bauteilzuverlässigkeiten auf statistischen Verfahren basieren, da die Bauteilfestigkeit und Kundenbeanspruchung streuen. Die bisherigen Vorgehensweisen der Tests führen häufig Fehlentscheidungen bzgl. der Freigabe, was unnötige Design-Änderungen und somit hohe Kosten bedeuten kann. In vorliegender Arbeit wird der Ansatz der partiellen Durchläuferzählung entwickelt, welche die statische Güte der bisherigen Testverfahren (Success Runs) erhöht.

Limit theorems constitute a classical and important field in probability theory. In several applications, in particular in demographic or medical contexts, killed Markov processes suggest themselves as models for populations undergoing culling by mortality or other processes. In these situations mathematical research features a general interest in the observable distribution of survivors, which is known as Yaglom limit or quasi-stationary distribution. Previous work often focuses on discrete state spaces, commonly birth-death processes (or with some more flexible localization of the transitions), with killing only on the boundary. The central concerns of this thesis are to describe, for a given class of one dimensional diffusion processes, the quasistationary distributions (if any), and to describe the convergence (or not) of the process conditioned on survival to one of these quasistationary distributions. Rather general diffusion processes on the half-line are considered, where 0 is allowed to be regular or an exit boundary. Very similar techniques are applied in this work in order to derive results on the large time behavior of an exotic measure valued process, which is closely related to so-called point interactions, which have been widely studied in the mathematical physics literature.

This thesis deals with 3 important aspects of optimal investment in real-world financial markets: taxes, crashes, and illiquidity. An introductory chapter reviews the portfolio problem in its historical context and motivates the theme of this work: We extend the standard modelling framework to include specific real-world features and evaluate their significance. In the first chapter, we analyze the optimal portfolio problem with capital gains taxes, assuming that taxes are deferred until the end of the investment horizon. The problem is solved with the help of a modification of the classical martingale method. The second chapter is concerned with optimal asset allocation under the threat of a financial market crash. The investor takes a worst-case attitude towards the crash, so her investment objective is to be best off in the most adverse crash scenario. We first survey the existing literature on the worst-case approach to optimal investment and then present in detail the novel martingale approach to worst-case portfolio optimization. The first part of this chapter is based on joint work with Ralf Korn. In the last chapter, we investigate optimal portfolio decisions in the presence of illiquidity. Illiquidity is understood as a period in which it is impossible to trade on financial markets. We use dynamic programming techniques in combination with abstract convergence results to solve the corresponding optimal investment problem. This chapter is based on joint work with Holger Kraft and Peter Diesinger.

The thesis at hand deals with the numerical solution of multiscale problems arising in the modeling of processes in fluid and thermo dynamics. Many of these processes, governed by partial differential equations, are relevant in engineering, geoscience, and environmental studies. More precisely, this thesis discusses the efficient numerical computation of effective macroscopic thermal conductivity tensors of high-contrast composite materials. The term "high-contrast" refers to large variations in the conductivities of the constituents of the composite. Additionally, this thesis deals with the numerical solution of Brinkman's equations. This system of equations adequately models viscous flows in (highly) permeable media. It was introduced by Brinkman in 1947 to reduce the deviations between the measurements for flows in such media and the predictions according to Darcy's model.

This thesis is devoted to two main topics (accordingly, there are two chapters): In the first chapter, we establish a tropical intersection theory with analogue notions and tools as its algebro-geometric counterpart. This includes tropical cycles, rational functions, intersection products of Cartier divisors and cycles, morphisms, their functors and the projection formula, rational equivalence. The most important features of this theory are the following: - It unifies and simplifies many of the existing results of tropical enumerative geometry, which often contained involved ad-hoc computations. - It is indispensable to formulate and solve further tropical enumerative problems. - It shows deep relations to the intersection theory of toric varieties and connected fields. - The relationship between tropical and classical Gromov-Witten invariants found by Mikhalkin is made plausible from inside tropical geometry. - It is interesting on its own as a subfield of convex geometry. In the second chapter, we study tropical gravitational descendants (i.e. Gromov-Witten invariants with incidence and "Psi-class" factors) and show that many concepts of the classical Gromov-Witten theory such as the famous WDVV equations can be carried over to the tropical world. We use this to extend Mikhalkin's results to a certain class of gravitational descendants, i.e. we show that many of the classical gravitational descendants of P^2 and P^1 x P^1 can be computed by counting tropical curves satisfying certain incidence conditions and with prescribed valences of their vertices. Moreover, the presented theory is not restricted to plane curves and therefore provides an important tool to derive similar results in higher dimensions. A more detailed chapter synopsis can be found at the beginning of each individual chapter.

Zusammenfassung. In dieser Arbeit werden Probleme der numerischen Lösung finiter Differenzenverfahren partieller Differentialgleichungen in einem algebraischen Ansatz behandelt. Es werden sowohl theoretische Ergebnisse präsentiert als auch die praktische Implementierung mithilfe der Systeme SINGULAR und QEPCAD vorgeführt. Dabei beziehen sich die algebraischen Methoden auf zwei unterschiedliche Aspekte bei finiten Differenzenverfahren: die Erzeugung von Schemata mithilfe von Gröbnerbasen und die darauf folgende Stabilitätsanalyse mittels Quantorenelimination durch algebraische zylindrische Dekomposition. Beim Aufbau der Arbeit werden in den ersten drei Kapiteln in einer Rückschau die nötigen Begriffe aus der Computeralgebra gelegt, die Grundzüge der numerischen Konvergenztheorie finiter Differenzenschemata erklärt sowie die Anwendung des CAD-Algorithmus zur Quantorenelimierung skizziert. Das Kapitel 4 entwickelt ausgehend vom zugrunde liegenden Kontext die Formulierung und die dafür nötigen Bedingungen an Differenzenschemata, die algebraisch nach Definition ein Ideal in einem Polynomring darstellen. Neben der praktischen Handhabbarkeit der Objekte liegt die Betonung auf größtmöglicher Allgemeinheit in den Definitionen der Begriffe. Es werden äquivalente Wege der Erzeugung sowie Eigenschaften der Eindeutigkeit unter sehr speziellen Bedingungen an die verwendeten Approximationen gezeigt. Die Anwendung des CAD-Algorithmus auf die Abschätzung des Symbols eines Schemas wird erläutert. Das fünfte Kapitel beschreibt die SINGULAR-Bibliothek findiff.lib, welche das Zusammenspiel von SINGULAR und QEPCAD garantiert und eine vollständige Automatisierung der Erzeugung und Stabilitätsanalyse eines finiten Differenzenverfahrens ermöglicht.

In the classical Merton investment problem of maximizing the expected utility from terminal wealth and intermediate consumption stock prices are independent of the investor who is optimizing his investment strategy. This is reasonable as long as the considered investor is small and thus does not influence the asset prices. However for an investor whose actions may affect the financial market the framework of the classical investment problem turns out to be inappropriate. In this thesis we provide a new approach to the field of large investor models. We study the optimal investment problem of a large investor in a jump-diffusion market which is in one of two states or regimes. The investor’s portfolio proportions as well as his consumption rate affect the intensity of transitions between the different regimes. Thus the investor is ’large’ in the sense that his investment decisions are interpreted by the market as signals: If, for instance, the large investor holds 25% of his wealth in a certain asset then the market may regard this as evidence for the corresponding asset to be priced incorrectly, and a regime shift becomes likely. More specifically, the large investor as modeled here may be the manager of a big mutual fund, a big insurance company or a sovereign wealth fund, or the executive of a company whose stocks are in his own portfolio. Typically, such investors have to disclose their portfolio allocations which impacts on market prices. But even if a large investor does not disclose his portfolio composition as it is the case of several hedge funds then the other market participants may speculate about the investor’s strategy which finally could influence the asset prices. Since the investor’s strategy only impacts on the regime shift intensities the asset prices do not necessarily react instantaneously. Our model is a generalization of the two-states version of the Bäuerle-Rieder model. Hence as the Bäuerle-Rieder model it is suitable for long investment periods during which market conditions could change. The fact that the investor’s influence enters the intensities of the transitions between the two states enables us to solve the investment problem of maximizing the expected utility from terminal wealth and intermediate consumption explicitly. We present the optimal investment strategy for a large investor with CRRA utility for three different kinds of strategy-dependent regime shift intensities – constant, step and affine intensity functions. In each case we derive the large investor’s optimal strategy in explicit form only dependent on the solution of a system of coupled ODEs of which we show that it admits a unique global solution. The thesis is organized as follows. In Section 2 we repeat the classical Merton investment problem of a small investor who does not influence the market. Further the Bäuerle-Rieder investment problem in which the market states follow a Markov chain with constant transition intensities is discussed. Section 3 introduces the aforementioned investment problem of a large investor. Besides the mathematical framework and the HJB-system we present a verification theorem that is necessary to verify the optimality of the solutions to the investment problem that we derive later on. The explicit derivation of the optimal investment strategy for a large investor with power utility is given in Section 4. For three kinds of intensity functions – constant, step and affine – we give the optimal solution and verify that the corresponding ODE-system admits a unique global solution. In case of the strategy-dependent intensity functions we distinguish three particular kinds of this dependency – portfolio-dependency, consumption-dependency and combined portfolio- and consumption-dependency. The corresponding results for an investor having logarithmic utility are shown in Section 5. In the subsequent Section 6 we consider the special case of a market consisting of only two correlated stocks besides the money market account. We analyze the investor’s optimal strategy when only the position in one of those two assets affects the market state whereas the position in the other asset is irrelevant for the regime switches. Various comparisons of the derived investment problems are presented in Section 7. Besides the comparisons of the particular problems with each other we also dwell on the sensitivity of the solution concerning the parameters of the intensity functions. Finally we consider the loss the large investor had to face if he neglected his influence on the market. In Section 8 we conclude the thesis.

A classical conjecture in the representation theory of finite groups, the McKay conjecture, states that for any finite group and prime number p the number of complex irreducible characters of degree prime to p is equal to the number of complex irreducible characters of degree prime to p of the normalizer of a p-Sylow subgroup. Recently a reduction theorem was proved by Isaacs, Malle and Navarro: If all simple groups are “good”, then the McKay conjecture holds. In this work we are concerned with the problem of goodness for finite groups of Lie type in their defining characteristic. A simple group is called “good” if certain equivariant bijections between the involved character sets exist. We present a structural approach to the construction of such a bijection by utilizing the so-called “Steinberg-Map”. This yields very natural bijections and we prove most of the desired properties.

Die vorliegende Dissertation besteht aus zwei Hauptteilen: Neue Ergebnisse aus der Gaußchen Analysis und ihre Anwendung auf die Theorie der Pfadintegrale. Das zentrale Resultat des ersten Teils ist die Charakterisierung aller regulären Distributionen die man mit Donsker's Delta multiplizieren kann. Dabei wird eine explizite Formel für solche Produkte, die sogenannte Wick-Formel, angegeben. Im Anwendungsteil dieser Arbeit wird zunächst eine komplex skalierte Feynman-Kac-Formel und ihre zugehörigen Kerne mit Hilfe dieser Wick-Formel gezeigt. Desweiteren werden Feynman Integranden für neue Klassen von Potentialen als White Noise Distributionen konstruiert.

The purpose of Exploration in Oil Industry is to "discover" an oil-containing geological formation from exploration data. In the context of this PhD project this oil-containing geological formation plays the role of a geometrical object, which may have any shape. The exploration data may be viewed as a "cloud of points", that is a finite set of points, related to the geological formation surveyed in the exploration experiment. Extensions of topological methodologies, such as homology, to point clouds are helpful in studying them qualitatively and capable of resolving the underlying structure of a data set. Estimation of topological invariants of the data space is a good basis for asserting the global features of the simplicial model of the data. For instance the basic statistical idea, clustering, are correspond to dimension of the zero homology group of the data. A statistics of Betti numbers can provide us with another connectivity information. In this work represented a method for topological feature analysis of exploration data on the base of so called persistent homology. Loosely, this is the homology of a growing space that captures the lifetimes of topological attributes in a multiset of intervals called a barcode. Constructions from algebraic topology empowers to transform the data, to distillate it into some persistent features, and to understand then how it is organized on a large scale or at least to obtain a low-dimensional information which can point to areas of interest. The algorithm for computing of the persistent Betti numbers via barcode is realized in the computer algebra system "Singular" in the scope of the work.

This thesis deals with the solution of special problems arising in financial engineering or financial mathematics. The main focus lies on commodity indices. Chapter 1 addresses the important issue for the financial engineering practice of developing well-suited models for certain assets (here: commodity indices). Descriptive analysis of the Dow Jones-UBS commodity index compared to the Standard & Poor 500 stock index provides us with first insights of some features of the corresponding distributions. Statistical tests of normality and mean reversion then helps us in setting up a model for commodity indices. Additionally, chapter 1 encompasses a thorough introduction to commodity investment, history of commodities trading and the most important derivatives, namely futures and European options on futures. Chapter 2 proposes a model for commodity indices and derives fair prices for the most important derivatives in the commodity markets. It is a Heston model supplemented with a stochastic convenience yield. The Heston model belongs to the model class of stochastic volatility models and is currently widely used in stock markets. For the application in the commodity markets the stochastic convenience yield is included in the drift of the instantaneous spot return process. Motivated by the results of chapter 1 it seems reasonable to model the convenience yield by a mean reverting Ornstein-Uhlenbeck process. Since trading desks only apply and consider models with closed form solutions for options I derive such formulas for commodity futures by solving the corresponding partial differential equation. Additionally, semi-closed form formulas for European options on futures are determined. The Cauchy problem with respect to these options is more challenging than the first one. A solution can be provided. Unlike equities, which typically entitle the holder to a continuing stake in a corporation, commodity futures contracts normally specify a certain date for the delivery of the underlying physical commodity. In order to avoid the delivery process and maintain a futures position, nearby contracts must be sold and contracts that have not yet reached the delivery period must be purchased (so called rolling). Optimal trading days for selling and buying futures are determined by applying statistical tests for stochastic dominance. Besides the optimization of the rolling procedure for commodity futures we dedicate ourselves in chapter 3 with the optimization of the weightings of the commodity futures that make up the index. To this end, I apply the Markowitz approach or mean-variance optimization. The mean-variance optimization penalizes up-side and down-side risk equally, whereas most investors do not mind up-side risk. To overcome this, I consider in the next step other risk measures, namely Value-at-Risk and Conditional Value-at-Risk. The Conditional Value-at-Risk is generalized to discontinuous cumulative distribution functions of the loss. For continuous loss distributions, the Conditional Value-at-Risk at a given confidence level is defined as the expected loss exceeding the Value-at-Risk. Loss distributions associated with finite sampling or scenario modeling are, however, discontinuous. Various risk measures involving discontinuous loss distributions shall be introduced and compared. I then apply the theoretical results to the field of portfolio optimization with commodity indices. Furthermore, I uncover graphically the behavior of these risk measures. For this purpose, I consider the risk measures as a function of the confidence level. Based on a special discrete loss distribution, the graphs demonstrate the different properties of these risk measures. The goal of the first section of chapter 4 is to apply the mathematical concept of excursions for the creation of optimal highly automated or algorithmic trading strategies. The idea is to consider the gain of the strategy and the excursion time it takes to realize the gain. In this section I calculate formulas for the Ornstein-Uhlenbeck process. I show that the corresponding formulas can be calculated quite fast since the only function appearing in the formulas is the so called imaginary error function. This function is already implemented in many programs, such as in Maple. My main contribution of this topic is the optimization of the trading strategy for Ornstein-Uhlenbeck processes via the Banach fixed-point theorem. The second section of chapter 4 deals with statistical arbitrage strategies, a long horizon trading opportunity that generates a riskless profit. The results of this section provide an investor with a tool to investigate empirically if some strategies (for example momentum strategies) constitute statistical arbitrage opportunities or not.

In this work, we develop a framework for analyzing an executive’s own- company stockholding and work effort preferences. The executive, character- ized by risk aversion and work effectiveness parameters, invests his personal wealth without constraint in the financial market, including the stock of his own company whose value he can directly influence with work effort. The executive’s utility-maximizing personal investment and work effort strategy is derived in closed form for logarithmic and power utility and for exponential utility for the case of zero interest rates. Additionally, a utility indifference rationale is applied to determine his fair compensation. Being unconstrained by performance contracting, the executive’s work effort strategy establishes a base case for theoretical or empirical assessment of the benefits or otherwise of constraining executives with performance contracting. Further, we consider a highly-qualified individual with respect to her choice between two distinct career paths. She can choose between a mid-level management position in a large company and an executive position within a smaller listed company with the possibility to directly affect the company’s share price. She invests in the financial market including the share of the smaller listed company. The utility maximizing strategy from consumption, investment, and work effort is derived in closed form for logarithmic utility and power utility. Conditions for the individual to pursue her career with the smaller listed company are obtained. The participation constraint is formulated in terms of the salary differential between the two positions. The smaller listed company can offer less salary. The salary shortfall is offset by the possibilityto benefit from her work effort by acquiring own-company shares. This givesinsight into aspects of optimal contract design. Our framework is applicable to the pharmaceutical and financial industry, as well as the IT sector.

This thesis deals with the numerical study of multiscale problems arising in the modelling of processes of the flow of fluid in plain and porous media. Many of these processes, governed by partial differential equations, are relevant in engineering, industry, and environmental studies. The overall task of modelling and simulating the filtration-related multiscale processes becomes interdisciplinary as it employs physics, mathematics and computer programming to reach its aim. Keeping the challenges in mind, the main focus is to overcome the limitations of accuracy, speed and memory and to develop novel efficient numerical algorithms which could, in part or whole, be utilized by those working in the field of porous media. This work has essentially four parts. A single grid basic algorithm and a corresponding parallel algorithm to solve the macroscopic Navier-Stokes-Brinkmann model is discussed. An upscaling subgrid algorithm is derived and numerically tested for the same model. Moving a step further in the line of multiscale methods, an iterative Mutliscale Finite Volume (iMSFV) method is developed for the Stokes-Darcy system. Additionally, the last part of the thesis deals with ways to incorporate changes occurring at different (meso) scale level. The flow equations are coupled with the Convection-Diffusion-Reaction (CDR) equation, which models the transport and capturing of particle concentrations. By employing the numerical method for the coupled flow and transport problem, we understand the interplay between the flow velocity and filtration.

Tropical intersection theory
(2010)

This thesis consists of five chapters: Chapter 1 contains the basics of the theory and is essential for the rest of the thesis. Chapters 2-5 are to a large extent independent of each other and can be read separately. - Chapter 1: Foundations of tropical intersection theory In this first chapter we set up the foundations of a tropical intersection theory covering many concepts and tools of its counterpart in algebraic geometry such as affine tropical cycles, Cartier divisors, morphisms of tropical cycles, pull-backs of Cartier divisors, push-forwards of cycles and an intersection product of Cartier divisors and cycles. Afterwards, we generalize these concepts to abstract tropical cycles and introduce a concept of rational equivalence. Finally, we set up an intersection product of cycles and prove that every cycle is rationally equivalent to some affine cycle in the special case that our ambient cycle is R^n. We use this result to show that rational and numerical equivalence agree in this case and prove a tropical Bézout's theorem. - Chapter 2: Tropical cycles with real slopes and numerical equivalence In this chapter we generalize our definitions of tropical cycles to polyhedral complexes with non-rational slopes. We use this new definition to show that if our ambient cycle is a fan then every subcycle is numerically equivalent to some affine cycle. Finally, we restrict ourselves to cycles in R^n that are "generic" in some sense and study the concept of numerical equivalence in more detail. - Chapter 3: Tropical intersection products on smooth varieties We define an intersection product of tropical cycles on tropical linear spaces L^n_k and on other, related fans. Then, we use this result to obtain an intersection product of cycles on any "smooth" tropical variety. Finally, we use the intersection product to introduce a concept of pull-backs of cycles along morphisms of smooth tropical varieties and prove that this pull-back has all expected properties. - Chapter 4: Weil and Cartier divisors under tropical modifications First, we introduce "modifications" and "contractions" and study their basic properties. After that, we prove that under some further assumptions a one-to-one correspondence of Weil and Cartier divisors is preserved by modifications. In particular we can prove that on any smooth tropical variety we have a one-to-one correspondence of Weil and Cartier divisors. - Chapter 5: Chern classes of tropical vector bundles We give definitions of tropical vector bundles and rational sections of tropical vector bundles. We use these rational sections to define the Chern classes of such a tropical vector bundle. Moreover, we prove that these Chern classes have all expected properties. Finally, we classify all tropical vector bundles on an elliptic curve up to isomorphisms.

Mrázek et al. [25] proposed a unified approach to curve estimation which combines localization and regularization. Franke et al. [10] used that approach to discuss the case of the regularized local least-squares (RLLS) estimate. In this thesis we will use the unified approach of Mrázek et al. to study some asymptotic properties of local smoothers with regularization. In particular, we shall discuss the Huber M-estimate and its limiting cases towards the L2 and the L1 cases. For the regularization part, we will use quadratic regularization. Then, we will define a more general class of regularization functions. Finally, we will do a Monte Carlo simulation study to compare different types of estimates.

The interest of the exploration of new hydrocarbon fields as well as deep geothermal reservoirs is permanently growing. The analysis of seismic data specific for such exploration projects is very complex and requires the deep knowledge in geology, geophysics, petrology, etc from interpreters, as well as the ability of advanced tools that are able to recover some particular properties. There again the existing wavelet techniques have a huge success in signal processing, data compression, noise reduction, etc. They enable to break complicate functions into many simple pieces at different scales and positions that makes detection and interpretation of local events significantly easier.
In this thesis mathematical methods and tools are presented which are applicable to the seismic data postprocessing in regions with non-smooth boundaries. We provide wavelet techniques that relate to the solutions of the Helmholtz equation. As application we are interested in seismic data analysis. A similar idea to construct wavelet functions from the limit and jump relations of the layer potentials was first suggested by Freeden and his Geomathematics Group.
The particular difficulty in such approaches is the formulation of limit and
jump relations for surfaces used in seismic data processing, i.e., non-smooth
surfaces in various topologies (for example, uniform and
quadratic). The essential idea is to replace the concept of parallel surfaces known for a smooth regular surface by certain appropriate substitutes for non-smooth surfaces.
By using the jump and limit relations formulated for regular surfaces, Helmholtz wavelets can be introduced that recursively approximate functions on surfaces with edges and corners. The exceptional point is that the construction of wavelets allows the efficient implementation in form of
a tree algorithm for the fast numerical computation of functions on the boundary.
In order to demonstrate the
applicability of the Helmholtz FWT, we study a seismic image obtained by the reverse time migration which is based on a finite-difference implementation. In fact, regarding the requirements of such migration algorithms in filtering and denoising the wavelet decomposition is successfully applied to this image for the attenuation of low-frequency
artifacts and noise. Essential feature is the space localization property of
Helmholtz wavelets which numerically enables to discuss the velocity field in
pointwise dependence. Moreover, the multiscale analysis leads us to reveal additional geological information from optical features.

In this work two main approaches for the evaluation of credit derivatives are analyzed: the copula based approach and the Markov Chain based approach. This work gives the opportunity to use the advantages and avoid disadvantages of both approaches. For example, modeling of contagion effects, i.e. modeling dependencies between counterparty defaults, is complicated under the copula approach. One remedy is to use Markov Chain, where it can be done directly. The work consists of five chapters. The first chapter of this work extends the model for the pricing of CDS contracts presented in the paper by Kraft and Steffensen (2007). In the widely used models for CDS pricing it is assumed that only borrower can default. In our model we assume that each of the counterparties involved in the contract may default. Calculated contract prices are compared with those calculated under usual assumptions. All results are summarized in the form of numerical examples and plots. In the second chapter the copula and its main properties are described. The methods of constructing copulas as well as most common copulas families and its properties are introduced. In the third chapter the method of constructing a copula for the existing Markov Chain is introduced. The cases with two and three counterparties are considered. Necessary relations between the transition intensities are derived to directly find some copula functions. The formulae for default dependencies like Spearman's rho and Kendall's tau for defined copulas are derived. Several numerical examples are presented in which the copulas are built for given Markov Chains. The fourth chapter deals with the approximation of copulas if for a given Markov Chain a copula cannot be provided explicitly. The fifth chapter concludes this thesis.

Mrázek et al. [14] proposed a unified approach to curve estimation which combines
localization and regularization. In this thesis we will use their approach to study
some asymptotic properties of local smoothers with regularization. In Particular, we
shall discuss the regularized local least squares (RLLS) estimate with correlated errors
(more precisely with stationary time series errors), and then based on this approach
we will discuss the case when the kernel function is dirac function and compare our
smoother with the spline smoother. Finally, we will do some simulation study.

Intersection Theory on Tropical Toric Varieties and Compactifications of Tropical Parameter Spaces
(2011)

We study toric varieties over the tropical semifield. We define tropical cycles inside these toric varieties and extend the stable intersection of tropical cycles in R^n to these toric varieties. In particular, we show that every tropical cycle can be degenerated into a sum of torus-invariant cycles. This allows us to tropicalize algebraic cycles of toric varieties over an algebraically closed field with non-Archimedean valuation. We see that the tropicalization map is a homomorphism on cycles and an isomorphism on cycle classes. Furthermore, we can use projective toric varieties to compactify known tropical varieties and study their combinatorics. We do this for the tropical Grassmannian in the Plücker embedding and compactify the tropical parameter space of rational degree d curves in tropical projective space using Chow quotients of the tropical Grassmannian.

A Multi-Phase Flow Model Incorporated with Population Balance Equation in a Meshfree Framework
(2011)

This study deals with the numerical solution of a meshfree coupled model of Computational Fluid Dynamics (CFD) and Population Balance Equation (PBE) for liquid-liquid extraction columns. In modeling the coupled hydrodynamics and mass transfer in liquid extraction columns one encounters multidimensional population balance equation that could not be fully resolved numerically within a reasonable time necessary for steady state or dynamic simulations. For this reason, there is an obvious need for a new liquid extraction model that captures all the essential physical phenomena and still tractable from computational point of view. This thesis discusses a new model which focuses on discretization of the external (spatial) and internal coordinates such that the computational time is drastically reduced. For the internal coordinates, the concept of the multi-primary particle method; as a special case of the Sectional Quadrature Method of Moments (SQMOM) is used to represent the droplet internal properties. This model is capable of conserving the most important integral properties of the distribution; namely: the total number, solute and volume concentrations and reduces the computational time when compared to the classical finite difference methods, which require many grid points to conserve the desired physical quantities. On the other hand, due to the discrete nature of the dispersed phase, a meshfree Lagrangian particle method is used to discretize the spatial domain (extraction column height) using the Finite Pointset Method (FPM). This method avoids the extremely difficult convective term discretization using the classical finite volume methods, which require a lot of grid points to capture the moving fronts propagating along column height.

A main result of this thesis is a conceptual proof of the fact that the weighted number of tropical curves of given degree and genus, which pass through the right number of general points in the plane (resp., which pass through general points in R^r and represent a given point in the moduli space of genus g curves) is independent of the choices of points. Another main result is a new correspondence theorem between plane tropical cycles and plane elliptic algebraic curves.

The goal of this thesis is to find ways to improve the analysis of hyperspectral Terahertz images. Although it would be desirable to have methods that can be applied on all spectral areas, this is impossible. Depending on the spectroscopic technique, the way the data is acquired differs as well as the characteristics that are to be detected. For these reasons, methods have to be developed or adapted to be especially suitable for the THz range and its applications. Among those are particularly the security sector and the pharmaceutical industry.
Due to the fact that in many applications the volume of spectra to be organized is high, manual data processing is difficult. Especially in hyperspectral imaging, the literature is concerned with various forms of data organization such as feature reduction and classification. In all these methods, the amount of necessary influence of the user should be minimized on the one hand and on the other hand the adaption to the specific application should be maximized.
Therefore, this work aims at automatically segmenting or clustering THz-TDS data. To achieve this, we propose a course of action that makes the methods adaptable to different kinds of measurements and applications. State of the art methods will be analyzed and supplemented where necessary, improvements and new methods will be proposed. This course of action includes preprocessing methods to make the data comparable. Furthermore, feature reduction that represents chemical content in about 20 channels instead of the initial hundreds will be presented. Finally the data will be segmented by efficient hierarchical clustering schemes. Various application examples will be shown.
Further work should include a final classification of the detected segments. It is not discussed here as it strongly depends on specific applications.

In the first part of the thesis we develop the theory of standard bases in free modules over (localized) polynomial rings. Given that linear equations are solvable in the coefficients of the polynomials, we introduce an algorithm to compute standard bases with respect to arbitrary (module) monomial orderings. Moreover, we take special care to principal ideal rings, allowing zero divisors. For these rings we design modified algorithms which are new and much faster than the general ones. These algorithms were motivated by current limitations in formal verification of microelectronic System-on-Chip designs. We show that our novel approach using computational algebra is able to overcome these limitations in important classes of applications coming from industrial challenges.
The second part is based on research in collaboration with Jason Morton, Bernd Sturmfels and Anne Shiu. We devise a general method to describe and compute a certain class of rank tests motivated by statistics. The class of rank tests may loosely be described as being based on computing the number of linear extensions to given partial orders. In order to apply these tests to actual data we developed two algorithms and used our implementations to apply the methodology to gene expression data created at the Stowers Institute for Medical Research. The dataset is concerned with the development of the vertebra. Our rankings proved valuable to the biologists.

For computational reasons, the spline interpolation of the Earth's gravitational potential is usually done in a spherical framework. In this work, however, we investigate a spline method with respect to the real Earth. We are concerned with developing the real Earth oriented strategies and methods for the Earth's gravitational potential determination. For this purpose we introduce the reproducing kernel Hilbert space of Newton potentials on and outside given regular surface with reproducing kernel defined as a Newton integral over it's interior. We first give an overview of thus far achieved results considering approximations on regular surfaces using surface potentials (Chapter 3). The main results are contained in the fourth chapter where we give a closer look to the Earth's gravitational potential, the Newton potentials and their characterization in the interior and the exterior space of the Earth. We also present the L2-decomposition for regions in R3 in terms of distributions, as a main strategy to impose the Hilbert space structure on the space of potentials on and outside a given regular surface. The properties of the Newton potential operator are investigated in relation to the closed subspace of harmonic density functions. After these preparations, in the fifth chapter we are able to construct the reproducing kernel Hilbert space of Newton potentials on and outside a regular surface. The spline formulation for the solution to interpolation problems, corresponding to a set of bounded linear functionals is given, and corresponding convergence theorems are proven. The spline formulation reflects the specifics of the Earth's surface, due to the representation of the reproducing kernel (of the solution space) as a Newton integral over the inner space of the Earth. Moreover, the approximating potential functions have the same domain of harmonicity as the actual Earth's gravitational potential, i.e., they are harmonic outside and continuous on the Earth's surface. This is a step forward in comparison to the spherical harmonic spline formulation involving functions harmonic down to the Runge sphere. The sixth chapter deals with the representation of the used kernel in the spherical case. It turns out that in the case of the spherical Earth, this kernel can be considered a kind of generalization to spherically oriented kernels, such as Abel-Poisson or the singularity kernel. We also investigate the existence of the closed expression of the kernel. However, at this point it remains to be unknown to us. So, in Chapter 7, we are led to consider certain discretization methods for integrals over regions in R3, in connection to theory of the multidimensional Euler summation formula for the Laplace operator. We discretize the Newton integral over the real Earth (representing the spline function) and give a priori estimates for approximate integration when using this discretization method. The last chapter summarizes our results and gives some directions for the future research.

Numerical Algorithms in Algebraic Geometry with Implementation in Computer Algebra System SINGULAR
(2011)

Polynomial systems arise in many applications: robotics, kinematics, chemical kinetics,
computer vision, truss design, geometric modeling, and many others. Many polynomial
systems have solutions sets, called algebraic varieties, having several irreducible
components. A fundamental problem of the numerical algebraic geometry is to decompose
such an algebraic variety into its irreducible components. The witness point sets are
the natural numerical data structure to encode irreducible algebraic varieties.
Sommese, Verschelde and Wampler represented the irreducible algebraic decomposition of
an affine algebraic variety \(X\) as a union of finite disjoint sets \(\cup_{i=0}^{d}W_i=\cup_{i=0}^{d}\left(\cup_{j=1}^{d_i}W_{ij}\right)\) called numerical irreducible decomposition. The \(W_i\) correspond to the pure i-dimensional components, and the \(W_{ij}\) represent the i-dimensional irreducible components. The numerical irreducible decomposition is implemented in BERTINI.
We modify this concept using partially Gröbner bases, triangular sets, local dimension, and
the so-called zero sum relation. We present in the second chapter the corresponding
algorithms and their implementations in SINGULAR. We give some examples and timings,
which show that the modified algorithms are more efficient if the number of variables is not
too large. For a large number of variables BERTINI is more efficient.
Leykin presented an algorithm to compute the embedded components of an algebraic variety
based on the concept of the deflation of an algebraic variety.
Depending on the modified algorithm mentioned above, we will present in the third chapter an
algorithm and its implementation in SINGULAR to compute the embedded components.
The irreducible decomposition of algebraic varieties allows us to formulate in the fourth
chapter some numerical algebraic algorithms.
In the last chapter we present two SINGULAR libraries. The first library is used to compute
the numerical irreducible decomposition and the embedded components of an algebraic variety.
The second library contains the procedures of the algorithms in the last Chapter to test
inclusion, equality of two algebraic varieties, to compute the degree of a pure i-dimensional
component, and the local dimension.

The various uses of fiber-reinforced composites, for example in the enclosures of planes, boats and cars, generates the demand for a detailed analysis of these materials. The final goal is to optimize fibrous materials by the means of “virtual material design”. New fibrous materials are virtually created as realizations of a stochastic model and evaluated with physical simulations. In that way, materials can be optimized for specific use cases, without constructing expensive prototypes or performing mechanical experiments. In order to design a practically fabricable material, the stochastic model is first adapted to an existing material and then slightly modified. The virtual reconstruction of the existing material requires a precise knowledge of the geometry of its microstructure. The first part of this thesis describes a fiber quantification method by the means of local measurements of the fiber radius and orientation. The combination of a sparse chord length transform and inertia moments leads to an efficient and precise new algorithm. It outperforms existing approaches with the possibility to treat different fiber radii within one sample, with high precision in continuous space and comparably fast computing time. This local quantification method can be directly applied on gray value images by adapting the directional distance transforms on gray values. In this work, several approaches of this kind are developed and evaluated. Further characterization of the fiber system requires a segmentation of each single fiber. Using basic morphological operators with specific structuring elements, it is possible to derive a probability for each pixel describing if the pixel belongs to a fiber core in a region without overlapping fibers. Tracking high probabilities leads to a partly reconstruction of the fiber cores in non crossing regions. These core parts are then reconnected over critical regions, if they fulfill certain conditions ensuring the affiliation to the same fiber. In the second part of this work, we develop a new stochastic model for dense systems of non overlapping fibers with a controllable level of bending. Existing approaches in the literature have at least one weakness in either achieving high volume fractions, producing non overlapping fibers, or controlling the bending or the orientation distribution. This gap can be bridged by our stochastic model, which operates in two steps. Firstly, a random walk with the multivariate von Mises-Fisher orientation distribution defines bent fibers. Secondly, a force-biased packing approach arranges them in a non overlapping configuration. Furthermore, we provide the estimation of all parameters needed for the fitting of this model to a real microstructure. Finally, we simulate the macroscopic behavior of different microstructures to derive their mechanical and thermal properties. This part is mostly supported by existing software and serves as a summary of physical simulation applied to random fiber systems. The application on a glass fiber reinforced polymer proves the quality of the reconstruction by our stochastic model, as the effective properties match for both the real microstructure and the realizations of the fitted model. This thesis includes all steps to successfully perform virtual material design on various data sets. With novel and efficient algorithms it contributes to the science of analysis and modeling of fiber reinforced materials.

We discuss some first steps towards experimental design for neural network regression which, at present, is too complex to treat fully in general. We encounter two difficulties: the nonlinearity of the models together with the high parameter dimension on one hand, and the common misspecification of the models on the other hand.
Regarding the first problem, we restrict our consideration to neural networks with only one and two neurons in the hidden layer and a univariate input variable. We prove some results regarding locally D-optimal designs, and present a numerical study using the concept of maximin optimal designs.
In respect of the second problem, we have a look at the effects of misspecification on optimal experimental designs.

This thesis is devoted to constructive module theory of polynomial
graded commutative algebras over a field.
It treats the theory of Groebner bases (GB), standard bases (SB) and syzygies as well as algorithms
and their implementations.
Graded commutative algebras naturally unify exterior and commutative polynomial algebras.
They are graded non-commutative, associative unital algebras over fields and may contain zero-divisors.
In this thesis
we try to make the most use out of _a priori_ knowledge about
their characteristic (super-commutative) structure
in developing direct symbolic methods, algorithms and implementations,
which are intrinsic to graded commutative algebras and practically efficient.
For our symbolic treatment we represent them as polynomial algebras
and redefine the product rule in order to allow super-commutative structures
and, in particular, to allow zero-divisors.
Using this representation we give a nice characterization
of a GB and an algorithm for its computation.
We can also tackle central localizations of graded commutative algebras by allowing commutative variables to be _local_,
generalizing Mora algorithm (in a similar fashion as G.M.Greuel and G.Pfister by allowing local or mixed monomial orderings)
and working with SBs.
In this general setting we prove a generalized Buchberger's criterion,
which shows that syzygies of leading terms play the utmost important role
in SB and syzygy module computations.
Furthermore, we develop a variation of the La Scala-Stillman free resolution algorithm,
which we can formulate particularly close to our implementation.
On the implementation side
we have further developed the Singular non-commutative subsystem Plural
in order to allow polynomial arithmetic
and more involved non-commutative basic Computer Algebra computations (e.g. S-polynomial, GB)
to be easily implementable for specific algebras.
At the moment graded commutative algebra-related algorithms
are implemented in this framework.
Benchmarks show that our new algorithms and implementation are practically efficient.
The developed framework has a lot of applications in various
branches of mathematics and theoretical physics.
They include computation of sheaf cohomology, coordinate-free verification of affine geometry
theorems and computation of cohomology rings of p-groups, which are partially described in this thesis.

Filtering, Approximation and Portfolio Optimization for Shot-Noise Models and the Heston Model
(2012)

We consider a continuous time market model in which stock returns satisfy a stochastic differential equation with stochastic drift, e.g. following an Ornstein-Uhlenbeck process. The driving noise of the stock returns consists not only of Brownian motion but also of a jump part (shot noise or compound Poisson process). The investor's objective is to maximize expected utility of terminal wealth under partial information which means that the investor only observes stock prices but does not observe the drift process. Since the drift of the stock prices is unobservable, it has to be estimated using filtering techniques. E.g., if the drift follows an Ornstein-Uhlenbeck process and without
jump part, Kalman filtering can be applied and optimal strategies can be computed explicitly. Also in other cases, like for an underlying
Markov chain, finite-dimensional filters exist. But for certain jump processes (e.g. shot noise) or certain nonlinear drift dynamics explicit computations, based on discrete observations, are no longer possible or existence of finite dimensional filters is no longer valid. The same
computational difficulties apply to the optimal strategy since it depends on the filter. In this case the model may be approximated by
a model where the filter is known and can be computed. E.g., we use statistical linearization for non-linear drift processes, finite-state-Markov chain approximations for the drift process and/or diffusion approximations for small jumps in the noise term.
In the approximating models, filters and optimal strategies can often be computed explicitly. We analyze and compare different approximation methods, in particular in view of performance of the corresponding utility maximizing strategies.

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.

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.

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.

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.

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.

This thesis is devoted to furthering the tropical intersection theory as well as to applying the
developed theory to gain new insights about tropical moduli spaces.
We use piecewise polynomials to define tropical cocycles that generalise the notion of tropical Cartier divisors to higher codimensions, introduce an intersection product of cocycles with tropical cycles and use the connection to toric geometry to prove a Poincaré duality for certain cases. Our
main application of this Poincaré duality is the construction of intersection-theoretic fibres under a
large class of tropical morphisms.
We construct an intersection product of cycles on matroid varieties which are a natural
generalisation of tropicalisations of classical linear spaces and the local blocks of smooth tropical
varieties. The key ingredient is the ability to express a matroid variety contained in another matroid variety by a piecewise polynomial that is given in terms of the rank functions of the corresponding
matroids. In particular, this enables us to intersect cycles on the moduli spaces of n-marked abstract
rational curves. We also construct a pull-back of cycles along morphisms of smooth varieties, relate
pull-backs to tropical modifications and show that every cycle on a matroid variety is rationally
equivalent to its recession cycle and can be cut out by a cocycle.
Finally, we define families of smooth rational tropical curves over smooth varieties and construct a tropical fibre product in order to show that every morphism of a smooth variety to the moduli space of abstract rational tropical curves induces a family of curves over the domain of the morphism.
This leads to an alternative, inductive way of constructing moduli spaces of rational curves.

The main topic of this thesis is to define and analyze a multilevel Monte Carlo algorithm for path-dependent functionals of the solution of a stochastic differential equation (SDE) which is driven by a square integrable, \(d_X\)-dimensional Lévy process \(X\). We work with standard Lipschitz assumptions and denote by \(Y=(Y_t)_{t\in[0,1]}\) the \(d_Y\)-dimensional strong solution of the SDE.
We investigate the computation of expectations \(S(f) = \mathrm{E}[f(Y)]\) using randomized algorithms \(\widehat S\). Thereby, we are interested in the relation of the error and the computational cost of \(\widehat S\), where \(f:D[0,1] \to \mathbb{R}\) ranges in the class \(F\) of measurable functionals on the space of càdlàg functions on \([0,1]\), that are Lipschitz continuous with respect to the supremum norm.
We consider as error \(e(\widehat S)\) the worst case of the root mean square error over the class of functionals \(F\). The computational cost of an algorithm \(\widehat S\), denoted \(\mathrm{cost}(\widehat S)\), should represent the runtime of the algorithm on a computer. We work in the real number model of computation and further suppose that evaluations of \(f\) are possible for piecewise constant functions in time units according to its number of breakpoints.
We state strong error estimates for an approximate Euler scheme on a random time discretization. With this strong error estimates, the multilevel algorithm leads to upper bounds for the convergence order of the error with respect to the computational cost. The main results can be summarized in terms of the Blumenthal-Getoor index of the driving Lévy process, denoted by \(\beta\in[0,2]\). For \(\beta <1\) and no Brownian component present, we almost reach convergence order \(1/2\), which means, that there exists a sequence of multilevel algorithms \((\widehat S_n)_{n\in \mathbb{N}}\) with \(\mathrm{cost}(\widehat S_n) \leq n\) such that \( e(\widehat S_n) \precsim n^{-1/2}\). Here, by \( \precsim\), we denote a weak asymptotic upper bound, i.e. the inequality holds up to an unspecified positive constant. If \(X\) has a Brownian component, the order has an additional logarithmic term, in which case, we reach \( e(\widehat S_n) \precsim n^{-1/2} \, (\log(n))^{3/2}\).
For the special subclass of $Y$ being the Lévy process itself, we also provide a lower bound, which, up to a logarithmic term, recovers the order \(1/2\), i.e., neglecting logarithmic terms, the multilevel algorithm is order optimal for \( \beta <1\).
An empirical error analysis via numerical experiments matches the theoretical results and completes the analysis.

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.

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.

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.

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.

This thesis is separated into three main parts: Development of Gaussian and White Noise Analysis, Hamiltonian Path Integrals as White Noise Distributions, Numerical methods for polymers driven by fractional Brownian motion.
Throughout this thesis the Donsker's delta function plays a key role. We investigate this generalized function also in Chapter 2. Moreover we show by giving a counterexample, that the general definition for complex kernels is not true.
In Chapter 3 we take a closer look to generalized Gauss kernels and generalize these concepts to the case of vector-valued White Noise. These results are the basis for Hamiltonian path integrals of quadratic type. The core result of this chapter gives conditions under which pointwise products of generalized Gauss kernels and certain Hida distributions have a mathematical rigorous meaning as distributions in the Hida space.
In Chapter 4 we discuss operators which are related to applications for Feynman Integrals as differential operators, scaling, translation and projection. We show the relation of these operators to differential operators, which leads to the well-known notion of so called convolution operators. We generalize the central homomorphy theorem to regular generalized functions.
We generalize the concept of complex scaling to scaling with bounded operators and discuss the relation to generalized Radon-Nikodym derivatives. With the help of this we consider products of generalized functions in chapter 5. We show that the projection operator from the Wick formula for products with Donsker's deltais not closable on the square-integrable functions..
In Chapter 5 we discuss products of generalized functions. Moreover the Wick formula is revisited. We investigate under which conditions and on which spaces the Wick formula can be generalized to. At the end of the chapter we consider the products of Donsker's delta function with a generalized function with help of a measure transformation. Here also problems as measurability are concerned.
In Chapter 6 we characterize Hamiltonian path integrands for the free particle, the harmonic oscillator and the charged particle in a constant magnetic field as Hida distributions. This is done in terms of the T-transform and with the help of the results from chapter 3. For the free particle and the harmonic oscillator we also investigate the momentum space propagators. At the same time, the $T$-transform of the constructed Feynman integrands provides us with their generating functional. In Chapter 7, we can show that the generalized expectation (generating functional at zero) gives the Greens function to the corresponding Schrödinger equation.
Moreover, with help of the generating functional we can show that the canonical commutation relations for the free particle and the harmonic oscillator in phase space are fulfilled. This confirms on a mathematical rigorous level the heuristics developed by Feynman and Hibbs.
In Chapter 8 we give an outlook, how the scaling approach which is successfully applied in the Feynman integral setting can be transferred to the phase space setting. We give a mathematical rigorous meaning to an analogue construction to the scaled Feynman-Kac kernel. It is open if the expression solves the Schrödinger equation. At least for quadratic potentials we can get the right physics.
In the last chapter, we focus on the numerical analysis of polymer chains driven by fractional Brownian motion. Instead of complicated lattice algorithms, our discretization is based on the correlation matrix. Using fBm one can achieve a long-range dependence of the interaction of the monomers inside a polymer chain. Here a Metropolis algorithm is used to create the paths of a polymer driven by fBm taking the excluded volume effect in account.

Efficient time integration and nonlinear model reduction for incompressible hyperelastic materials
(2013)

This thesis deals with the time integration and nonlinear model reduction of nearly incompressible materials that have been discretized in space by mixed finite elements. We analyze the structure of the equations of motion and show that a differential-algebraic system of index 1 with a singular perturbation term needs to be solved. In the limit case the index may jump to index 3 and thus renders the time integration into a difficult problem. For the time integration we apply Rosenbrock methods and study their convergence behavior for a test problem, which highlights the importance of the well-known Scholz conditions for this problem class. Numerical tests demonstrate that such linear-implicit methods are an attractive alternative to established time integration methods in structural dynamics. In the second part we combine the simulation of nonlinear materials with a model reduction step. We use the method of proper orthogonal decomposition and apply it to the discretized system of second order. For a nonlinear model reduction to be efficient we approximate the nonlinearity by following the lookup approach. In a practical example we show that large CPU time savings can achieved. This work is in order to prepare the ground for including such finite element structures as components in complex vehicle dynamics applications.

The application behind the subject of this thesis are multiscale simulations on highly heterogeneous particle-reinforced composites with large jumps in their material coefficients. Such simulations are used, e.g., for the prediction of elastic properties. As the underlying microstructures have very complex geometries, a discretization by means of finite elements typically involves very fine resolved meshes. The latter results in discretized linear systems of more than \(10^8\) unknowns which need to be solved efficiently. However, the variation of the material coefficients even on very small scales reveals the failure of most available methods when solving the arising linear systems. While for scalar elliptic problems of multiscale character, robust domain decomposition methods are developed, their extension and application to 3D elasticity problems needs to be further established.
The focus of the thesis lies in the development and analysis of robust overlapping domain decomposition methods for multiscale problems in linear elasticity. The method combines corrections on local subdomains with a global correction on a coarser grid. As the robustness of the overall method is mainly determined by how well small scale features of the solution can be captured on the coarser grid levels, robust multiscale coarsening strategies need to be developed which properly transfer information between fine and coarse grids.
We carry out a detailed and novel analysis of two-level overlapping domain decomposition methods for the elasticity problems. The study also provides a concept for the construction of multiscale coarsening strategies to robustly solve the discretized linear systems, i.e. with iteration numbers independent of variations in the Young's modulus and the Poisson ratio of the underlying composite. The theory also captures anisotropic elasticity problems and allows applications to multi-phase elastic materials with non-isotropic constituents in two and three spatial dimensions.
Moreover, we develop and construct new multiscale coarsening strategies and show why they should be preferred over standard ones on several model problems. In a parallel implementation (MPI) of the developed methods, we present applications to real composites and robustly solve discretized systems of more than \(200\) million unknowns.

This thesis deals with generalized inverses, multivariate polynomial interpolation and approximation of scattered data. Moreover, it covers the lifting scheme, which basically links the aforementioned topics. For instance, determining filters for the lifting scheme is connected to multivariate polynomial interpolation. More precisely, sets of interpolation sites are required that can be interpolated by a unique polynomial of a certain degree. In this thesis a new class of such sets is introduced and elements from this class are used to construct new and computationally more efficient filters for the lifting scheme.
Furthermore, a method to approximate multidimensional scattered data is introduced which is based on the lifting scheme. A major task in this method is to solve an ordinary linear least squares problem which possesses a special structure. Exploiting this structure yields better approximations and therefore this particular least squares problem is analyzed in detail. This leads to a characterization of special generalized inverses with partially prescribed image spaces.

The use of trading stops is a common practice in financial markets for a variety of reasons: it provides a simple way to control losses on a given trade, while also ensuring that profit-taking is not deferred indefinitely; and it allows opportunities to consider reallocating resources to other investments. In this thesis, it is explained why the use of stops may be desirable in certain cases.
This is done by proposing a simple objective to be optimized. Some simple and commonly-used rules for the placing and use of stops are investigated; consisting of fixed or moving barriers, with fixed transaction costs. It is shown how to identify optimal levels at which to set stops, and the performances of different rules and strategies are compared. Thereby, uncertainty and altering of the drift parameter of the investment are incorporated.

The main purpose of the study was to improve the physical properties of the modelling of compressed materials, especially fibrous materials. Fibrous materials are finding increasing application in the industries. And most of the materials are compressed for different applications. For such situation, we are interested in how the fibre arranged, e.g. with which distribution. For given materials it is possible to obtain a three-dimensional image via micro computed tomography. Since some physical parameters, e.g. the fibre lengths or the directions for points in the fibre, can be checked under some other methods from image, it is beneficial to improve the physical properties by changing the parameters in the image.
In this thesis, we present a new maximum-likelihood approach for the estimation of parameters of a parametric distribution on the unit sphere, which is various as some well known distributions, e.g. the von-Mises Fisher distribution or the Watson distribution, and for some models better fit. The consistency and asymptotic normality of the maximum-likelihood estimator are proven. As the second main part of this thesis, a general model of mixtures of these distributions on a hypersphere is discussed. We derive numerical approximations of the parameters in an Expectation Maximization setting. Furthermore we introduce a non-parametric estimation of the EM algorithm for the mixture model. Finally, we present some applications to the statistical analysis of fibre composites.

This thesis is concerned with tropical moduli spaces, which are an important tool in tropical enumerative geometry. The main result is a construction of tropical moduli spaces of rational tropical covers of smooth tropical curves and of tropical lines in smooth tropical surfaces. The construction of a moduli space of tropical curves in a smooth tropical variety is reduced to the case of smooth fans. Furthermore, we point out relations to intersection theory on suitable moduli spaces on algebraic curves.

Factorization of multivariate polynomials is a cornerstone of many applications in computer algebra. To compute it, one uses an algorithm by Zassenhaus who used it in 1969 to factorize univariate polynomials over \(\mathbb{Z}\). Later Musser generalized it to the multivariate case. Subsequently, the algorithm was refined and improved.
In this work every step of the algorithm is described as well as the problems that arise in these steps.
In doing so, we restrict to the coefficient domains \(\mathbb{F}_{q}\), \(\mathbb{Z}\), and \(\mathbb{Q}(\alpha)\) while focussing on a fast implementation. The author has implemented almost all algorithms mentioned in this work in the C++ library factory which is part of the computer algebra system Singular.
Besides, a new bound on the coefficients of a factor of a multivariate polynomial over \(\mathbb{Q}(\alpha)\) is proven which does not require \(\alpha\) to be an algebraic integer. This bound is used to compute Hensel lifting and recombination of factors in a modular fashion. Furthermore, several sub-steps are improved.
Finally, an overview on the capability of the implementation is given which includes benchmark examples as well as random generated input which is supposed to give an impression of the average performance.

Constructing accurate earth models from seismic data is a challenging task. Traditional methods rely on ray based approximations of the wave equation and reach their limit in geologically complex areas. Full waveform inversion (FWI) on the other side seeks to minimize the misﬁt between modeled and observed data without such approximation.
While superior in accuracy, FWI uses a gradient based iterative scheme that makes it also very computationally expensive. In this thesis we analyse and test an Alternating Direction Implicit (ADI) scheme in order to reduce the costs of the two dimensional time domain algorithm for solving the acoustic wave equation. The ADI scheme can be seen as an intermediate between explicit and implicit ﬁnite diﬀerence modeling schemes. Compared to full implicit schemes the ADI scheme only requires the solution of much smaller matrices and is thus less computationally demanding. Using ADI we can handle coarser discretization compared to an explicit method. Although order of convergence and CFL conditions for the examined explicit method and ADI scheme are comparable, we observe that the ADI scheme is less prone to dispersion. Furhter, our algorithm is eﬃciently parallelized with vectorization and threading techniques. In a numerical comparison, we can demonstrate a runtime advantage of the ADI scheme over an explicit method of the same accuracy.
With the modeling in place, we test and compare several inverse schemes in the second part of the thesis. With the goal of avoiding local minima and improving speed of convergence, we use diﬀerent minimization functions and hierarchical approaches. In several tests, we demonstrate superior results of the L1 norm compared to the L2 norm – especially in the presence of noise. Furthermore we show positive eﬀects for applying three diﬀerent multiscale approaches to the inverse problem. These methods focus on low frequency, early recording, or far oﬀset during early iterations of the minimization and then proceed iteratively towards the full problem. We achieve best results with the frequency based multiscale scheme, for which we also provide a heuristical method of choosing iteratively increasing frequency bands.
Finally, we demonstrate the eﬀectiveness of the diﬀerent methods ﬁrst on the Marmousi model and then on an extract of the 2004 BP model, where we are able to recover both high contrast top salt structures and lower contrast inclusions accurately.

Many real life problems have multiple spatial scales. In addition to the multiscale nature one has to take uncertainty into account. In this work we consider multiscale problems with stochastic coefficients.
We combine multiscale methods, e.g., mixed multiscale finite elements or homogenization, which are used for deterministic problems with stochastic methods, such as multi-level Monte Carlo or polynomial chaos methods.
The work is divided into three parts.
In the first two parts we study homogenization with different stochastic methods. Therefore we consider elliptic stationary diffusion equations with stochastic coefficients.
The last part is devoted to the study of mixed multiscale finite elements in combination with multi-level Monte Carlo methods. In the third part we consider multi-phase flow and transport equations.

In the last few years a lot of work has been done in the investigation of Brownian motion with point interaction(s) in one and higher dimensions. Roughly speaking a Brownian motion with point interaction is nothing else than a Brownian motion whose generator is disturbed by a measure supported in just one point.
The purpose of the present work is the introducing of curve interactions of the two dimensional Brownian motion for a closed curve \(\mathcal{C}\). We will understand a curve interaction as a self-adjoint extension of the restriction of the Laplacian to the set of infinitely often continuously differentiable functions with compact support in \(\mathbb{R}^{2}\) which are constantly 0 at the closed curve. We will give a full description of all these self-adjoint extensions.
In the second chapter we will prove a generalization of Tanaka's formula to \(\mathbb{R}^{2}\). We define \(g\) to be a so-called harmonic single layer with continuous layer function \(\eta\) in \(\mathbb{R}^{2}\). For such a function \(g\) we prove
\begin{align}
g\left(B_{t}\right)=g\left(B_{0}\right)+\int\limits_{0}^{t}{\nabla g\left(B_{s}\right)\mathrm{d}B_{s}}+\int\limits_{0}^{t}\eta\left(B_{s}\right)\mathrm{d}L\left(s,\mathcal{C}\right)
\end{align}
where \(B_{t}\) is just the usual Brownian motion in \(\mathbb{R}^{2}\) and \(L\left(t,\mathcal{C}\right)\) is the connected unique local time process of \(B_{t}\) on the closed curve \(\mathcal{C}\).
We will use the generalized Tanaka formula in the following chapter to construct classes of processes related to curve interactions. In a first step we get the generalization of point interactions in a second step we get processes which behaves like a Brownian motion in the complement of \(\mathcal{C}\) and has an additional movement along the curve in the time- scale of \(L\left(t,\mathcal{C}\right)\). Such processes do not exist in the one point case since there we cannot move when the Brownian motion is in the point.
By establishing an approximation of a curve interaction by operators of the form Laplacian \(+V_{n}\) with "nice" potentials \(V_{n}\) we are able to deduce the existence of superprocesses related to curve interactions.
The last step is to give an approximation of these superprocesses by a sytem of branching particles. This approximation gives a better understanding of the related mass creation.

This thesis is divided into two parts. Both cope with multi-class image segmentation and utilize
non-smooth optimization algorithms.
The topic of the first part, namely unsupervised segmentation, is the application of clustering
to image pixels. Therefore, we start with an introduction of the biconvex center-based clustering
algorithms c-means and fuzzy c-means, where c denotes the number of classes. We show that
fuzzy c-means can be seen as an approximation of c-means in terms of power means.
Since noise is omnipresent in our image data, these simple clustering models are not suitable
for its segmentation. To this end, we introduce a general and finite dimensional segmentation
model that consists of a data term stemming from the aforementioned clustering models plus a
continuous regularization term. We tackle this optimization model via an alternating minimiza-
tion approach called regularized c-centers (RcC). Thereby, we fix the centers and optimize the
segment membership of the pixels and vice versa. In this general setting, we prove convergence
in the sense of set-valued algorithms using Zangwill’s Theory [172].
Further, we present a segmentation model with a total variation regularizer. While updating
the cluster centers is straightforward for fixed segment memberships of the pixels, updating the
segment membership can be solved iteratively via non-smooth, convex optimization. Thereby,
we do not iterate a convex optimization algorithm until convergence. Instead, we stop as soon as
we have a certain amount of decrease in the objective functional to increase the efficiency. This
algorithm is a particular implementation of RcC providing also the corresponding convergence
theory. Moreover, we show the good performance of our method in various examples such as
simulated 2d images of brain tissue and 3d volumes of two materials, namely a multi-filament
composite superconductor and a carbon fiber reinforced silicon carbide ceramics. Thereby, we
exploit the property of the latter material that two components have no common boundary in
our adapted model.
The second part of the thesis is concerned with supervised segmentation. We leave the area
of center based models and investigate convex approaches related to graph p-Laplacians and
reproducing kernel Hilbert spaces (RKHSs). We study the effect of different weights used to
construct the graph. In practical experiments we show on the one hand image types that
are better segmented by the p-Laplacian model and on the other hand images that are better
segmented by the RKHS-based approach. This is due to the fact that the p-Laplacian approach
provides smoother results, while the RKHS approach provides often more accurate and detailed
segmentations. Finally, we propose a novel combination of both approaches to benefit from the
advantages of both models and study the performance on challenging medical image data.

This thesis, whose subject is located in the field of algorithmic commutative algebra and algebraic geometry, consists of three parts.
The first part is devoted to parallelization, a technique which allows us to take advantage of the computational power of modern multicore processors. First, we present parallel algorithms for the normalization of a reduced affine algebra A over a perfect field. Starting from the algorithm of Greuel, Laplagne, and Seelisch, we propose two approaches. For the local-to-global approach, we stratify the singular locus Sing(A) of A, compute the normalization locally at each stratum and finally reconstruct the normalization of A from the local results. For the second approach, we apply modular methods to both the global and the local-to-global normalization algorithm.
Second, we propose a parallel version of the algorithm of Gianni, Trager, and Zacharias for primary decomposition. For the parallelization of this algorithm, we use modular methods for the computationally hardest steps, such as for the computation of the associated prime ideals in the zero-dimensional case and for the standard bases computations. We then apply an innovative fast method to verify that the result is indeed a primary decomposition of the input ideal. This allows us to skip the verification step at each of the intermediate modular computations.
The proposed parallel algorithms are implemented in the open-source computer algebra system SINGULAR. The implementation is based on SINGULAR's new parallel framework which has been developed as part of this thesis and which is specifically designed for applications in mathematical research.
In the second part, we propose new algorithms for the computation of syzygies, based on an in-depth analysis of Schreyer's algorithm. Here, the main ideas are that we may leave out so-called "lower order terms" which do not contribute to the result of the algorithm, that we do not need to order the terms of certain module elements which occur at intermediate steps, and that some partial results can be cached and reused.
Finally, the third part deals with the algorithmic classification of singularities over the real numbers. First, we present a real version of the Splitting Lemma and, based on the classification theorems of Arnold, algorithms for the classification of the simple real singularities. In addition to the algorithms, we also provide insights into how real and complex singularities are related geometrically. Second, we explicitly describe the structure of the equivalence classes of the unimodal real singularities of corank 2. We prove that the equivalences are given by automorphisms of a certain shape. Based on this theorem, we explain in detail how the structure of the equivalence classes can be computed using SINGULAR and present the results in concise form. The probably most surprising outcome is that the real singularity type \(J_{10}^-\) is actually redundant.

Die Dissertation "Portfoliooptimierung im Binomialmodell" befasst sich mit der Frage, inwieweit
das Problem der optimalen Portfolioauswahl im Binomialmodell lösbar ist bzw. inwieweit
die Ergebnisse auf das stetige Modell übertragbar sind. Dabei werden neben dem
klassischen Modell ohne Kosten und ohne Veränderung der Marktsituation auch Modellerweiterungen
untersucht.

In the theory of option pricing one is usually concerned with evaluating expectations under the risk-neutral measure in a continuous-time model.
However, very often these values cannot be calculated explicitly and numerical methods need to be applied to approximate the desired quantity. Monte Carlo simulations, numerical methods for PDEs and the lattice approach are the methods typically employed. In this thesis we consider the latter approach, with the main focus on binomial trees.
The binomial method is based on the concept of weak convergence. The discrete-time model is constructed so as to ensure convergence in distribution to the continuous process. This means that the expectations calculated in the binomial tree can be used as approximations of the option prices in the continuous model. The binomial method is easy to implement and can be adapted to options with different types of payout structures, including American options. This makes the approach very appealing. However, the problem is that in many cases, the convergence of the method is slow and highly irregular, and even a fine discretization does not guarantee accurate price approximations. Therefore, ways of improving the convergence properties are required.
We apply Edgeworth expansions to study the convergence behavior of the lattice approach. We propose a general framework, that allows to obtain asymptotic expansion for both multinomial and multidimensional trees. This information is then used to construct advanced models with superior convergence properties.
In binomial models we usually deal with triangular arrays of lattice random vectors. In this case the available results on Edgeworth expansions for lattices are not directly applicable. Therefore, we first present Edgeworth expansions, which are also valid for the binomial tree setting. We then apply these result to the one-dimensional and multidimensional Black-Scholes models. We obtain third order expansions
for general binomial and trinomial trees in the 1D setting, and construct advanced models for digital, vanilla and barrier options. Second order expansion are provided for the standard 2D binomial trees and advanced models are constructed for the two-asset digital and the two-asset correlation options. We also present advanced binomial models for a multidimensional setting.

This thesis focuses on dealing with some new aspects of continuous time portfolio optimization by using the stochastic control method.
First, we extend the Busch-Korn-Seifried model for a large investor by using the Vasicek model for the short rate, and that problem is solved explicitly for two types of intensity functions.
Next, we justify the existence of the constant proportion portfolio insurance (CPPI) strategy in a framework containing a stochastic short rate and a Markov switching parameter. The effect of Vasicek short rate on the CPPI strategy has been studied by Horsky (2012). This part of the thesis extends his research by including a Markov switching parameter, and the generalization is based on the B\"{a}uerle-Rieder investment problem. The explicit solutions are obtained for the portfolio problem without the Money Market Account as well as the portfolio problem with the Money Market Account.
Finally, we apply the method used in Busch-Korn-Seifried investment problem to explicitly solve the portfolio optimization with a stochastic benchmark.

Multilevel Constructions
(2014)

The thesis consists of the two chapters.
The first chapter is addressed to make a deep investigation of the MLMC method. In particular we take an optimisation view at the estimate. Rather than fixing the number of discretisation points \(n_i\) to be a geometric sequence, we are trying to find an optimal set up for \(n_i\) such that for a fixed error the estimate can be computed within a minimal time.
In the second chapter we propose to enhance the MLMC estimate with the weak extrapolation technique. This technique helps to improve order of a weak convergence of a scheme and as a result reduce CC of an estimate. In particular we study high order weak extrapolation approach, which is know not be inefficient in the standard settings. However, a combination of the MLMC and the weak extrapolation yields an improvement of the MLMC.

In this thesis, we combine Groebner basis with SAT Solver in different manners.
Both SAT solvers and Groebner basis techniques have their own strength and weakness.
Combining them could fix their weakness.
The first combination is using Groebner techniques to learn additional binary clauses for SAT solver from a selection of clauses. This combination is first proposed by Zengler and Kuechlin.
However, in our experiments, about 80 percent Groebner basis computations give no new binary clauses.
By selecting smaller and more compact input for Groebner basis computations, we can significantly
reduce the number of inefficient Groebner basis computations, learn much more binary clauses. In addition,
the new strategy can reduce the solving time of a SAT Solver in general, especially for large and hard problems.
The second combination is using all-solution SAT solver and interpolation to compute Boolean Groebner bases of Boolean elimination ideals of a given ideal. Computing Boolean Groebner basis of the given ideal is an inefficient method in case we want to eliminate most of the variables from a big system of Boolean polynomials.
Therefore, we propose a more efficient approach to handle such cases.
In this approach, the given ideal is translated to the CNF formula. Then an all-solution SAT Solver is used to find the projection of all solutions of the given ideal. Finally, an algorithm, e.g. Buchberger-Moeller Algorithm, is used to associate the reduced Groebner basis to the projection.
We also optimize the Buchberger-Moeller Algorithm for lexicographical ordering and compare it with Brickenstein's interpolation algorithm.
Finally, we combine Groebner basis and abstraction techniques to the verification of some digital designs that contain complicated data paths.
For a given design, we construct an abstract model.
Then, we reformulate it as a system of polynomials in the ring \({\mathbb Z}_{2^k}[x_1,\dots,x_n]\).
The variables are ordered in a way such that the system has already been a Groebner basis w.r.t lexicographical monomial ordering.
Finally, the normal form is employed to prove the desired properties.
To evaluate our approach, we verify the global property of a multiplier and a FIR filter using the computer algebra system Singular. The result shows that our approach is much faster than the commercial verification tool from Onespin on these benchmarks.

Das zinsoptimierte Schuldenmanagement hat zum Ziel, eine möglichst effiziente Abwägung zwischen den erwarteten Finanzierungskosten einerseits und den Risiken für den Staatshaushalt andererseits zu finden. Um sich diesem Spannungsfeld zu nähern, schlagen wir erstmals die Brücke zwischen den Problemstellungen des Schuldenmanagements und den Methoden der zeitkontinuierlichen, dynamischen Portfoliooptimierung.
Das Schlüsselelement ist dabei eine neue Metrik zur Messung der Finanzierungskosten, die Perpetualkosten. Diese spiegeln die durchschnittlichen zukünftigen Finanzierungskosten wider und beinhalten sowohl die bereits bekannten Zinszahlungen als auch die noch unbekannten Kosten für notwendige Anschlussfinanzierungen. Daher repräsentiert die Volatilität der Perpetualkosten auch das Risiko einer bestimmten Strategie; je langfristiger eine Finanzierung ist, desto kleiner ist die Schwankungsbreite der Perpetualkosten.
Die Perpetualkosten ergeben sich als Produkt aus dem Barwert eines Schuldenportfolios und aus der vom Portfolio unabhängigen Perpetualrate. Für die Modellierung des Barwertes greifen wir auf das aus der dynamischen Portfoliooptimierung bekannte Konzept eines selbstfinanzierenden Bondportfolios zurück, das hier auf einem mehrdimensionalen affin-linearen Zinsmodell basiert. Das Wachstum des Schuldenportfolios wird dabei durch die Einbeziehung des Primärüberschusses des Staates gebremst bzw. verhindert, indem wir diesen als externen Zufluss in das selbstfinanzierende Modell aufnehmen.
Wegen der Vielfältigkeit möglicher Finanzierungsinstrumente wählen wir nicht deren Wertanteile als Kontrollvariable, sondern kontrollieren die Sensitivitäten des Portfolios gegenüber verschiedenen Zinsbewegungen. Aus optimalen Sensitivitäten können in einem nachgelagerten Schritt dann optimale Wertanteile für verschiedenste Finanzierungsinstrumente abgeleitet werden. Beispielhaft demonstrieren wir dies mittels Rolling-Horizon-Bonds unterschiedlicher Laufzeit.
Schließlich lösen wir zwei Optimierungsprobleme mit Methoden der stochastischen Kontrolltheorie. Dabei wird stets der erwartete Nutzen der Perpetualkosten maximiert. Die Nutzenfunktionen sind jeweils an das Schuldenmanagement angepasst und zeichnen sich insbesondere dadurch aus, dass höhere Kosten mit einem niedrigeren Nutzen einhergehen. Im ersten Problem betrachten wir eine Potenznutzenfunktion mit konstanter relativer Risikoaversion, im zweiten wählen wir eine Nutzenfunktion, welche die Einhaltung einer vorgegebenen Schulden- bzw. Kostenobergrenze garantiert.

In automotive testrigs we apply load time series to components such that the outcome is as close as possible to some reference data. The testing procedure should in general be less expensive and at the same time take less time for testing. In my thesis, I propose a testrig damage optimization problem (WSDP). This approach improves upon the testrig stress optimization problem (TSOP) used as a state of the art by industry experts.
In both (TSOP) and (WSDP), we optimize the load time series for a given testrig configuration. As the name suggests, in (TSOP) the reference data is the stress time series. The detailed behaviour of the stresses as functions of time are sometimes not the most important topic. Instead the damage potential of the stress signals are considered. Since damage is not part of the objectives in the (TSOP) the total damage computed from the optimized load time series is not optimal with respect to the reference damage. Additionally, the load time series obtained is as long as the reference stress time series and the total damage computation needs cycle counting algorithms and Goodmann corrections. The use of cycle counting algorithms makes the computation of damage from load time series non-differentiable.
To overcome the issues discussed in the previous paragraph this thesis uses block loads for the load time series. Using of block loads makes the damage differentiable with respect to the load time series. Additionally, in some special cases it is shown that damage is convex when block loads are used and no cycle counting algorithms are required. Using load time series with block loads enables us to use damage in the objective function of the (WSDP).
During every iteration of the (WSDP), we have to find the maximum total damage over all plane angles. The first attempt at solving the (WSDP) uses discretization of the interval for plane angle to find the maximum total damage at each iteration. This is shown to give unreliable results and makes maximum total damage function non-differentiable with respect to the plane angle. To overcome this, damage function for a given surface stress tensor due to a block load is remodelled by Gaussian functions. The parameters for the new model are derived.
When we model the damage by Gaussian function, the total damage is computed as a sum of Gaussian functions. The plane with the maximum damage is similar to the modes of the Gaussian Mixture Models (GMM), the difference being that the Gaussian functions used in GMM are probability density functions which is not the case in the damage approximation presented in this work. We derive conditions for a single maximum for Gaussian functions, similar to the ones given for the unimodality of GMM by Aprausheva et al. in [1].
By using the conditions for a single maximum we give a clustering algorithm that merges the Gaussian functions in the sum as clusters. Each cluster obtained through clustering is such that it has a single maximum in the absence of other Gaussian functions of the sum. The approximate point of the maximum of each cluster is used as the starting point for a fixed point equation on the original damage function to get the actual maximum total damage at each iteration.
We implement the method for the (TSOP) and the two methods (with discretization and with clustering) for (WSDP) on two example problems. The results obtained from the (WSDP) using discretization is shown to be better than the results obtained from the (TSOP). Furthermore we show that, (WSDP) using clustering approach to finding the maximum total damage, takes less number of iterations and is more reliable than using discretization.

Monte Carlo simulation is one of the commonly used methods for risk estimation on financial markets, especially for option portfolios, where any analytical approximation is usually too inaccurate. However, the usually high computational effort for complex portfolios with a large number of underlying assets motivates the application of variance reduction procedures. Variance reduction for estimating the probability of high portfolio losses has been extensively studied by Glasserman et al. A great variance reduction is achieved by applying an exponential twisting importance sampling algorithm together with stratification. The popular and much faster Delta-Gamma approximation replaces the portfolio loss function in order to guide the choice of the importance sampling density and it plays the role of the stratification variable. The main disadvantage of the proposed algorithm is that it is derived only in the case of Gaussian and some heavy-tailed changes in risk factors.
Hence, our main goal is to keep the main advantage of the Monte Carlo simulation, namely its ability to perform a simulation under alternative assumptions on the distribution of the changes in risk factors, also in the variance reduction algorithms. Step by step, we construct new variance reduction techniques for estimating the probability of high portfolio losses. They are based on the idea of the Cross-Entropy importance sampling procedure. More precisely, the importance sampling density is chosen as the closest one to the optimal importance sampling density (zero variance estimator) out of some parametric family of densities with respect to Kullback - Leibler cross-entropy. Our algorithms are based on the special choices of the parametric family and can now use any approximation of the portfolio loss function. A special stratification is developed, so that any approximation of the portfolio loss function under any assumption of the distribution of the risk factors can be used. The constructed algorithms can easily be applied for any distribution of risk factors, no matter if light- or heavy-tailed. The numerical study exhibits a greater variance reduction than of the algorithm from Glasserman et al. The use of a better approximation may improve the performance of our algorithms significantly, as it is shown in the numerical study.
The literature on the estimation of the popular market risk measures, namely VaR and CVaR, often refers to the algorithms for estimating the probability of high portfolio losses, describing the corresponding transition process only briefly. Hence, we give a consecutive discussion of this problem. Results necessary to construct confidence intervals for both measures under the mentioned variance reduction procedures are also given.

This thesis is devoted to the modeling and simulation of Asymmetric Flow Field Flow Fractionation, which is a technique for separating particles of submicron scale. This process is a part of large family of Field Flow Fractionation techniques and has a very broad range of industrial applications, e. g. in microbiology, chemistry, pharmaceutics, environmental analysis.
Mathematical modeling is crucial for this process, as due to the own nature of the process, lab ex- periments are difficult and expensive to perform. On the other hand, there are several challenges for the mathematical modeling: huge dominance (up to 106 times) of the flow over the diffusion, highly stretched geometry of the device. This work is devoted to developing fast and efficient algorithms, which take into the account the challenges, posed by the application, and provide reliable approximations for the quantities of interest.
We present a new Multilevel Monte Carlo method for estimating the distribution functions on a compact interval, which are of the main interest for Asymmetric Flow Field Flow Fractionation. Error estimates for this method in terms of computational cost are also derived.
We optimize the flow control at the Focusing stage under the given constraints on the flow and present an important ingredients for the further optimization, such as two-grid Reduced Basis method, specially adapted for the Finite Volume discretization approach.

In 2006 Jeffrey Achter proved that the distribution of divisor class groups of degree 0 of function fields with a fixed genus and the distribution of eigenspaces in symplectic similitude groups are closely related to each other. Gunter Malle proposed that there should be a similar correspondence between the distribution of class groups of number fields and the distribution of eigenspaces in ceratin matrix groups. Motivated by these results and suggestions we study the distribution of eigenspaces corresponding to the eigenvalue one in some special subgroups of the general linear group over factor rings of rings of integers of number fields and derive some conjectural statements about the distribution of \(p\)-parts of class groups of number fields over a base field \(K_{0}\). Where our main interest lies in the case that \(K_{0}\) contains the \(p\)th roots of unity, because in this situation the \(p\)-parts of class groups seem to behave in an other way like predicted by the popular conjectures of Henri Cohen and Jacques Martinet. In 2010 based on computational data Malle has succeeded in formulating a conjecture in the spirit of Cohen and Martinet for this case. Here using our investigations about the distribution in matrixgroups we generalize the conjecture of Malle to a more abstract level and establish a theoretical backup for these statements.

This thesis is devoted to the computational aspects of intersection theory and enumerative geometry. The first results are a Sage package Schubert3 and a Singular library schubert.lib which both provide the key functionality necessary for computations in intersection theory and enumerative geometry. In particular, we describe an alternative method for computations in Schubert calculus via equivariant intersection theory. More concretely, we propose an explicit formula for computing the degree of Fano schemes of linear subspaces on hypersurfaces. As a special case, we also obtain an explicit formula for computing the number of linear subspaces on a general hypersurface when this number is finite. This leads to a much better performance than classical Schubert calculus.
Another result of this thesis is related to the computation of Gromov-Witten invariants. The most powerful method for computing Gromov-Witten invariants is the localization of moduli spaces of stable maps. This method was introduced by Kontsevich in 1995. It allows us to compute Gromov-Witten invariants via Bott's formula. As an insightful application, we computed the numbers of rational curves on general complete intersection Calabi-Yau threefolds in projective spaces up to degree six. The results are all in agreement with predictions made from mirror symmetry.

In the first part of this thesis we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus is the study of moduli spaces, where the underlying combinatorics of the varieties involved allow a much more efficient way of computing certain tropical cycles. The algorithms discussed here have been implemented in an extension for polymake, a software for polyhedral computations.
In the second part we apply the algorithmic toolkit developed in the first part to the study of tropical double Hurwitz cycles. Hurwitz cycles are a higher-dimensional generalization of Hurwitz numbers, which count covers of \(\mathbb{P}^1\) by smooth curves of a given genus with a certain fixed ramification behaviour. Double Hurwitz numbers provide a strong connection between various mathematical disciplines, including algebraic geometry, representation theory and combinatorics. The tropical cycles have a rather complex combinatorial nature, so it is very difficult to study them purely "by hand". Being able to compute examples has been very helpful
in coming up with theoretical results. Our main result states that all marked and unmarked Hurwitz cycles are connected in codimension one and that for a generic choice of simple ramification points the marked cycle is a multiple of an irreducible cycle. In addition we provide computational examples to show that this is the strongest possible statement.

Safety analysis is of ultimate importance for operating Nuclear Power Plants (NPP). The overall
modeling and simulation of physical and chemical processes occuring in the course of an accident
is an interdisciplinary problem and has origins in fluid dynamics, numerical analysis, reactor tech-
nology and computer programming. The aim of the study is therefore to create the foundations
of a multi-dimensional non-isothermal fluid model for a NPP containment and software tool based
on it. The numerical simulations allow to analyze and predict the behavior of NPP systems under
different working and accident conditions, and to develop proper action plans for minimizing the
risks of accidents, and/or minimizing the consequences of possible accidents. A very large number
of scenarios have to be simulated, and at the same time acceptable accuracy for the critical param-
eters, such as radioactive pollution, temperature, etc., have to be achieved. The existing software
tools are either too slow, or not accurate enough. This thesis deals with developing customized al-
gorithm and software tools for simulation of isothermal and non-isothermal flows in a containment
pool of NPP. Requirements to such a software are formulated, and proper algorithms are presented.
The goal of the work is to achieve a balance between accuracy and speed of calculation, and to
develop customized algorithm for this special case. Different discretization and solution approaches
are studied and those which correspond best to the formulated goal are selected, adjusted, and when
possible, analysed. Fast directional splitting algorithm for Navier-Stokes equations in complicated
geometries, in presence of solid and porous obstales, is in the core of the algorithm. Developing
suitable pre-processor and customized domain decomposition algorithms are essential part of the
overall algorithm amd software. Results from numerical simulations in test geometries and in real
geometries are presented and discussed.

Pedestrian Flow Models
(2014)

There have been many crowd disasters because of poor planning of the events. Pedestrian models are useful in analysing the behavior of pedestrians in advance to the events so that no pedestrians will be harmed during the event. This thesis deals with pedestrian flow models on microscopic, hydrodynamic and scalar scales. By following the Hughes' approach, who describes the crowd as a thinking fluid, we use the solution of the Eikonal equation to compute the optimal path for pedestrians. We start with the microscopic model for pedestrian flow and then derive the hydrodynamic and scalar models from it. We use particle methods to solve the governing equations. Moreover, we have coupled a mesh free particle method to the fixed grid for solving the Eikonal equation. We consider an example with a large number of pedestrians to investigate our models for different settings of obstacles and for different parameters. We also consider the pedestrian flow in a straight corridor and through T-junction and compare our numerical results with the experiments. A part of this work is devoted for finding a mesh free method to solve the Eikonal equation. Most of the available methods to solve the Eikonal equation are restricted to either cartesian grid or triangulated grid. In this context, we propose a mesh free method to solve the Eikonal equation, which can be applicable to any arbitrary grid and useful for the complex geometries.

This thesis is concerned with stochastic control problems under transaction costs. In particular, we consider a generalized menu cost problem with partially controlled regime switching, general multidimensional running cost problems and the maximization of long-term growth rates in incomplete markets. The first two problems are considered under a general cost structure that includes a fixed cost component, whereas the latter is analyzed under proportional and Morton-Pliska
transaction costs.
For the menu cost problem and the running cost problem we provide an equivalent characterization of the value function by means of a generalized version of the Ito-Dynkin formula instead of the more restrictive, traditional approach via the use of quasi-variational inequalities (QVIs). Based on the finite element method and weak solutions of QVIs in suitable Sobolev spaces, the value function is constructed iteratively. In addition to the analytical results, we study a novel application of the menu cost problem in management science. We consider a company that aims to implement an optimal investment and marketing strategy and must decide when to issue a new version of a product and when and how much
to invest into marketing.
For the long-term growth rate problem we provide a rigorous asymptotic analysis under both proportional and Morton-Pliska transaction costs in a general incomplete market that includes, for instance, the Heston stochastic volatility model and the Kim-Omberg stochastic excess return model as special cases. By means of a dynamic programming approach leading-order optimal strategies are constructed
and the leading-order coefficients in the expansions of the long-term growth rates are determined. Moreover, we analyze the asymptotic performance of Morton-Pliska strategies in settings with proportional transaction costs. Finally, pathwise optimality of the constructed strategies is established.

This work aims at including nonlinear elastic shell models in a multibody framework. We focus our attention to Kirchhoff-Love shells and explore the benefits of an isogeometric approach, the latest development in finite element methods, within a multibody system. Isogeometric analysis extends isoparametric finite elements to more general functions such as B-Splines and Non-Uniform Rational B-Splines (NURBS) and works on exact geometry representations even at the coarsest level of discretizations. Using NURBS as basis functions, high regularity requirements of the shell model, which are difficult to achieve with standard finite elements, are easily fulfilled. A particular advantage is the promise of simplifying the mesh generation step, and mesh refinement is easily performed by eliminating the need for communication with the geometry representation in a Computer-Aided Design (CAD) tool.
Quite often the domain consists of several patches where each patch is parametrized by means of NURBS, and these patches are then glued together by means of continuity conditions. Although the techniques known from domain decomposition can be carried over to this situation, the analysis of shell structures is substantially more involved as additional angle preservation constraints between the patches might arise. In this work, we address this issue in the stationary and transient case and make use of the analogy to constrained mechanical systems with joints and springs as interconnection elements. Starting point of our work is the bending strip method which is a penalty approach that adds extra stiffness to the interface between adjacent patches and which is found to lead to a so-called stiff mechanical system that might suffer from ill-conditioning and severe stepsize restrictions during time integration. As a remedy, an alternative formulation is developed that improves the condition number of the system and removes the penalty parameter dependence. Moreover, we study another alternative formulation with continuity constraints applied to triples of control points at the interface. The approach presented here to tackle stiff systems is quite general and can be applied to all penalty problems fulfilling some regularity requirements.
The numerical examples demonstrate an impressive convergence behavior of the isogeometric approach even for a coarse mesh, while offering substantial savings with respect to the number of degrees of freedom. We show a comparison between the different multipatch approaches and observe that the alternative formulations are well conditioned, independent of any penalty parameter and give the correct results. We also present a technique to couple the isogeometric shells with multibody systems using a pointwise interaction.

In this thesis we present a new method for nonlinear frequency response analysis of mechanical vibrations.
For an efficient spatial discretization of nonlinear partial differential equations of continuum mechanics we employ the concept of isogeometric analysis. Isogeometric finite element methods have already been shown to possess advantages over classical finite element discretizations in terms of exact geometry representation and higher accuracy of numerical approximations using spline functions.
For computing nonlinear frequency response to periodic external excitations, we rely on the well-established harmonic balance method. It expands the solution of the nonlinear ordinary differential equation system resulting from spatial discretization as a truncated Fourier series in the frequency domain.
A fundamental aspect for enabling large-scale and industrial application of the method is model order reduction of the spatial discretization of the equation of motion. Therefore we propose the utilization of a modal projection method enhanced with modal derivatives, providing second-order information. We investigate the concept of modal derivatives theoretically and using computational examples we demonstrate the applicability and accuracy of the reduction method for nonlinear static computations and vibration analysis.
Furthermore, we extend nonlinear vibration analysis to incompressible elasticity using isogeometric mixed finite element methods.

In some processes for spinning synthetic fibers the filaments are exposed to highly turbulent air flows to achieve a high degree of stretching (elongation). The quality of the resulting filaments, namely thickness and uniformity, is thus determined essentially by the aerodynamic force coming from the turbulent flow. Up to now, there is a gap between the elongation measured in experiments and the elongation obtained by numerical simulations available in the literature.
The main focus of this thesis is the development of an efficient and sufficiently accurate simulation algorithm for the velocity of a turbulent air flow and the application in turbulent spinning processes.
In stochastic turbulence models the velocity is described by an \(\mathbb{R}^3\)-valued random field. Based on an appropriate description of the random field by Marheineke, we have developed an algorithm that fulfills our requirements of efficiency and accuracy. Applying a resulting stochastic aerodynamic drag force on the fibers then allows the simulation of the fiber dynamics modeled by a random partial differential algebraic equation system as well as a quantization of the elongation in a simplified random ordinary differential equation model for turbulent spinning. The numerical results are very promising: whereas the numerical results available in the literature can only predict elongations up to order \(10^4\) we get an order of \(10^5\), which is closer to the elongations of order \(10^6\) measured in experiments.

Das Ziel dieser Dissertation ist die Entwicklung und Implementation eines Algorithmus zur Berechnung von tropischen Varietäten über allgemeine bewertete Körper. Die Berechnung von tropischen Varietäten über Körper mit trivialer Bewertung ist ein hinreichend gelöstes Problem. Hierfür kombinieren die Autoren Bogart, Jensen, Speyer, Sturmfels und Thomas eindrucksvoll klassische Techniken der Computeralgebra mit konstruktiven Methoden der konvexer Geometrie.
Haben wir allerdings einen Grundkörper mit nicht-trivialer Bewertung, wie zum Beispiel den Körper der \(p\)-adischen Zahlen \(\mathbb{Q}_p\), dann stößt die konventionelle Gröbnerbasentheorie scheinbar an ihre Grenzen. Die zugrundeliegenden Monomordnungen sind nicht geeignet um Problemstellungen zu untersuchen, die von einer nicht-trivialen Bewertung auf den Koeffizienten abhängig sind. Dies führte zu einer Reihe von Arbeiten, welche die gängige Gröbnerbasentheorie modifizieren um die Bewertung des Grundkörpers einzubeziehen.\[\phantom{newline}\]
In dieser Arbeit präsentieren wir einen alternativen Ansatz und zeigen, wie sich die Bewertung mittels einer speziell eingeführten Variable emulieren lässt, so dass eine Modifikation der klassischen Werkzeuge nicht notwendig ist.
Im Rahmen dessen wird Theorie der Standardbasen auf Potenzreihen über einen Koeffizientenring verallgemeinert. Hierbei wird besonders Wert darauf gelegt, dass alle Algorithmen bei polynomialen Eingabedaten mit ihren klassischen Pendants übereinstimmen, sodass für praktische Zwecke auf bereits etablierte Softwaresysteme zurückgegriffen werden kann. Darüber hinaus wird die Konstruktion des Gröbnerfächers sowie die Technik des Gröbnerwalks für leicht inhomogene Ideale eingeführt. Dies ist notwendig, da bei der Einführung der neuen Variable die Homogenität des Ausgangsideal gebrochen wird.\[\phantom{newline}\]
Alle Algorithmen wurden in Singular implementiert und sind als Teil der offiziellen Distribution erhältlich. Es ist die erste Implementation, welches in der Lage ist tropische Varietäten mit \(p\)-adischer Bewertung auszurechnen. Im Rahmen der Arbeit entstand ebenfalls ein Singular Paket für konvexe Geometrie, sowie eine Schnittstelle zu Polymake.

In this dissertation, we discuss how to price American-style options. Our aim is to study and improve the regression-based Monte Carlo methods. In order to have good benchmarks to compare with them, we also study the tree methods.
In the second chapter, we investigate the tree methods specifically. We do research firstly within the Black-Scholes model and then within the Heston model. In the Black-Scholes model, based on Müller's work, we illustrate how to price one dimensional and multidimensional American options, American Asian options, American lookback options, American barrier options and so on. In the Heston model, based on Sayer's research, we implement his algorithm to price one dimensional American options. In this way, we have good benchmarks of various American-style options and put them all in the appendix.
In the third chapter, we focus on the regression-based Monte Carlo methods theoretically and numerically. Firstly, we introduce two variations, the so called "Tsitsiklis-Roy method" and the "Longstaff-Schwartz method". Secondly, we illustrate the approximation of American option by its Bermudan counterpart. Thirdly we explain the source of low bias and high bias. Fourthly we compare these two methods using in-the-money paths and all paths. Fifthly, we examine the effect using different number and form of basis functions. Finally, we study the Andersen-Broadie method and present the lower and upper bounds.
In the fourth chapter, we study two machine learning techniques to improve the regression part of the Monte Carlo methods: Gaussian kernel method and kernel-based support vector machine. In order to choose a proper smooth parameter, we compare fixed bandwidth, global optimum and suboptimum from a finite set. We also point out that scaling the training data to [0,1] can avoid numerical difficulty. When out-of-sample paths of stock prices are simulated, the kernel method is robust and even performs better in several cases than the Tsitsiklis-Roy method and the Longstaff-Schwartz method. The support vector machine can keep on improving the kernel method and needs less representations of old stock prices during prediction of option continuation value for a new stock price.
In the fifth chapter, we switch to the hardware (FGPA) implementation of the Longstaff-Schwartz method and propose novel reversion formulas for the stock price and volatility within the Black-Scholes and Heston models. The test for this formula within the Black-Scholes model shows that the storage of data is reduced and also the corresponding energy consumption.

In this thesis we extend the worst-case modeling approach as first introduced by Hua and Wilmott (1997) (option pricing in discrete time) and Korn and Wilmott (2002) (portfolio optimization in continuous time) in various directions.
In the continuous-time worst-case portfolio optimization model (as first introduced by Korn and Wilmott (2002)), the financial market is assumed to be under the threat of a crash in the sense that the stock price may crash by an unknown fraction at an unknown time. It is assumed that only an upper bound on the size of the crash is known and that the investor prepares for the worst-possible crash scenario. That is, the investor aims to find the strategy maximizing her objective function in the worst-case crash scenario.
In the first part of this thesis, we consider the model of Korn and Wilmott (2002) in the presence of proportional transaction costs. First, we treat the problem without crashes and show that the value function is the unique viscosity solution of a dynamic programming equation (DPE) and then construct the optimal strategies. We then consider the problem in the presence of crash threats, derive the corresponding DPE and characterize the value function as the unique viscosity solution of this DPE.
In the last part, we consider the worst-case problem with a random number of crashes by proposing a regime switching model in which each state corresponds to a different crash regime. We interpret each of the crash-threatened regimes of the market as states in which a financial bubble has formed which may lead to a crash. In this model, we prove that the value function is a classical solution of a system of DPEs and derive the optimal strategies.

In this work we focus on the regression models with asymmetrical error distribution,
more precisely, with extreme value error distributions. This thesis arises in the framework
of the project "Robust Risk Estimation". Starting from July 2011, this project won
three years funding by the Volkswagen foundation in the call "Extreme Events: Modelling,
Analysis, and Prediction" within the initiative "New Conceptual Approaches to
Modelling and Simulation of Complex Systems". The project involves applications in
Financial Mathematics (Operational and Liquidity Risk), Medicine (length of stay and
cost), and Hydrology (river discharge data). These applications are bridged by the
common use of robustness and extreme value statistics.
Within the project, in each of these applications arise issues, which can be dealt with by
means of Extreme Value Theory adding extra information in the form of the regression
models. The particular challenge in this context concerns asymmetric error distributions,
which significantly complicate the computations and make desired robustification
extremely difficult. To this end, this thesis makes a contribution.
This work consists of three main parts. The first part is focused on the basic notions
and it gives an overview of the existing results in the Robust Statistics and Extreme
Value Theory. We also provide some diagnostics, which is an important achievement of
our project work. The second part of the thesis presents deeper analysis of the basic
models and tools, used to achieve the main results of the research.
The second part is the most important part of the thesis, which contains our personal
contributions. First, in Chapter 5, we develop robust procedures for the risk management
of complex systems in the presence of extreme events. Mentioned applications use time
structure (e.g. hydrology), therefore we provide extreme value theory methods with time
dynamics. To this end, in the framework of the project we considered two strategies. In
the first one, we capture dynamic with the state-space model and apply extreme value
theory to the residuals, and in the second one, we integrate the dynamics by means of
autoregressive models, where the regressors are described by generalized linear models.
More precisely, since the classical procedures are not appropriate to the case of outlier
presence, for the first strategy we rework classical Kalman smoother and extended
Kalman procedures in a robust way for different types of outliers and illustrate the performance
of the new procedures in a GPS application and a stylized outlier situation.
To apply approach to shrinking neighborhoods we need some smoothness, therefore for
the second strategy, we derive smoothness of the generalized linear model in terms of
L2 differentiability and create sufficient conditions for it in the cases of stochastic and
deterministic regressors. Moreover, we set the time dependence in these models by
linking the distribution parameters to the own past observations. The advantage of
our approach is its applicability to the error distributions with the higher dimensional
parameter and case of regressors of possibly different length for each parameter. Further,
we apply our results to the models with generalized Pareto and generalized extreme value
error distributions.
Finally, we create the exemplary implementation of the fixed point iteration algorithm
for the computation of the optimally robust in
uence curve in R. Here we do not aim to
provide the most
exible implementation, but rather sketch how it should be done and
retain points of particular importance. In the third part of the thesis we discuss three applications,
operational risk, hospitalization times and hydrological river discharge data,
and apply our code to the real data set taken from Jena university hospital ICU and
provide reader with the various illustrations and detailed conclusions.

The work consists of two parts.
In the first part an optimization problem of structures of linear elastic material with contact modeled by Robin-type boundary conditions is considered. The structures model textile-like materials and possess certain quasiperiodicity properties. The homogenization method is used to represent the structures by homogeneous elastic bodies and is essential for formulations of the effective stress and Poisson's ratio optimization problems. At the micro-level, the classical one-dimensional Euler-Bernoulli beam model extended with jump conditions at contact interfaces is used. The stress optimization problem is of a PDE-constrained optimization type, and the adjoint approach is exploited. Several numerical results are provided.
In the second part a non-linear model for simulation of textiles is proposed. The yarns are modeled by hyperelastic law and have no bending stiffness. The friction is modeled by the Capstan equation. The model is formulated as a problem with the rate-independent dissipation, and the basic continuity and convexity properties are investigated. The part ends with numerical experiments and a comparison of the results to a real measurement.

In this thesis, we investigate several upcoming issues occurring in the context of conceiving and building a decision support system. We elaborate new algorithms for computing representative systems with special quality guarantees, provide concepts for supporting the decision makers after a representative system was computed, and consider a methodology of combining two optimization problems.
We review the original Box-Algorithm for two objectives by Hamacher et al. (2007) and discuss several extensions regarding coverage, uniformity, the enumeration of the whole nondominated set, and necessary modifications if the underlying scalarization problem cannot be solved to optimality. In a next step, the original Box-Algorithm is extended to the case of three objective functions to compute a representative system with desired coverage error. Besides the investigation of several theoretical properties, we prove the correctness of the algorithm, derive a bound on the number of iterations needed by the algorithm to meet the desired coverage error, and propose some ideas for possible extensions.
Furthermore, we investigate the problem of selecting a subset with desired cardinality from the computed representative system, the Hypervolume Subset Selection Problem (HSSP). We provide two new formulations for the bicriteria HSSP, a linear programming formulation and a \(k\)-link shortest path formulation. For the latter formulation, we propose an algorithm for which we obtain the currently best known complexity bound for solving the bicriteria HSSP. For the tricriteria HSSP, we propose an integer programming formulation with a corresponding branch-and-bound scheme.
Moreover, we address the issue of how to present the whole set of computed representative points to the decision makers. Based on common illustration methods, we elaborate an algorithm guiding the decision makers in choosing their preferred solution.
Finally, we step back and look from a meta-level on the issue of how to combine two given optimization problems and how the resulting combinations can be related to each other. We come up with several different combined formulations and give some ideas for the practical approach.

The central topic of this thesis is Alperin's weight conjecture, a problem concerning the representation theory of finite groups.
This conjecture, which was first proposed by J. L. Alperin in 1986, asserts that for any finite group the number of its irreducible Brauer characters coincides with the number of conjugacy classes of its weights. The blockwise version of Alperin's conjecture partitions this problem into a question concerning the number of irreducible Brauer characters and weights belonging to the blocks of finite groups.
A proof for this conjecture has not (yet) been found. However, the problem has been reduced to a question on non-abelian finite (quasi-) simple groups in the sense that there is a set of conditions, the so-called inductive blockwise Alperin weight condition, whose verification for all non-abelian finite simple groups implies the blockwise Alperin weight conjecture. Now the objective is to prove this condition for all non-abelian finite simple groups, all of which are known via the classification of finite simple groups.
In this thesis we establish the inductive blockwise Alperin weight condition for three infinite series of finite groups of Lie type: the special linear groups \(SL_3(q)\) in the case \(q>2\) and \(q \not\equiv 1 \bmod 3\), the Chevalley groups \(G_2(q)\) for \(q \geqslant 5\), and Steinberg's triality groups \(^3D_4(q)\).

In this thesis we develop a shape optimization framework for isogeometric analysis in the optimize first–discretize then setting. For the discretization we use
isogeometric analysis (iga) to solve the state equation, and search optimal designs in a space of admissible b-spline or nurbs combinations. Thus a quite
general class of functions for representing optimal shapes is available. For the
gradient-descent method, the shape derivatives indicate both stopping criteria and search directions and are determined isogeometrically. The numerical treatment requires solvers for partial differential equations and optimization methods, which introduces numerical errors. The tight connection between iga and geometry representation offers new ways of refining the geometry and analysis discretization by the same means. Therefore, our main concern is to develop the optimize first framework for isogeometric shape optimization as ground work for both implementation and an error analysis. Numerical examples show that this ansatz is practical and case studies indicate that it allows local refinement.

The Wilkie model is a stochastic asset model, developed by A.D. Wilkie in 1984 with a purpose to explore the behaviour of investment factors of insurers within the United Kingdom. Even so, there is still no analysis that studies the Wilkie model in a portfolio optimization framework thus far. Originally, the Wilkie model is considering a discrete-time horizon and we apply the concept of Wilkie model to develop a suitable ARIMA model for Malaysian data by using Box-Jenkins methodology. We obtained the estimated parameters for each sub model within the Wilkie model that suits the case of Malaysia, and permits us to analyse the result based on statistics and economics view. We then tend to review the continuous time case which was initially introduced by Terence Chan in 1998. The continuous-time Wilkie model inspired is then being employed to develop the wealth equation of a portfolio that consists of a bond and a stock. We are interested in building portfolios based on three well-known trading strategies, a self-financing strategy, a constant growth optimal strategy as well as a buy-and-hold strategy. In dealing with the portfolio optimization problems, we use the stochastic control technique consisting of the maximization problem itself, the Hamilton-Jacobi-equation, the solution to the Hamilton-Jacobi-equation and finally the verification theorem. In finding the optimal portfolio, we obtained the specific solution of the Hamilton-Jacobi-equation and proved the solution via the verification theorem. For a simple buy-and-hold strategy, we use the mean-variance analysis to solve the portfolio optimization problem.

Motivated by the results of infinite dimensional Gaussian analysis and especially white noise analysis, we construct a Mittag-Leffler analysis. This is an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which we call Mittag-Leffler measures. Our results indicate that the Wick ordered polynomials, which play a key role in Gaussian analysis, cannot be generalized to this non-Gaussian case. We provide evidence that a system of biorthogonal polynomials, called generalized Appell system, is applicable to the Mittag-Leffler measures, instead of using Wick ordered polynomials. With the help of an Appell system, we introduce a test function and a distribution space. Furthermore we give characterizations of the distribution space and we characterize the weak integrable functions and the convergent sequences within the distribution space. We construct Donsker's delta in a non-Gaussian setting as an application.
In the second part, we develop a grey noise analysis. This is a special application of the Mittag-Leffler analysis. In this framework, we introduce generalized grey Brownian motion and prove differentiability in a distributional sense and the existence of generalized grey Brownian motion local times. Grey noise analysis is then applied to the time-fractional heat equation and the time-fractional Schrödinger equation. We prove a generalization of the fractional Feynman-Kac formula for distributional initial values. In this way, we find a Green's function for the time-fractional heat equation which coincides with the solutions given in the literature.

Lithium-ion batteries are increasingly becoming an ubiquitous part of our everyday life - they are present in mobile phones, laptops, tools, cars, etc. However, there are still many concerns about their longevity and their safety. In this work we focus on the simulation of several degradation mechanisms on the microscopic scale, where one can resolve the active materials inside the electrodes of the lithium-ion batteries as porous structures. We mainly study two aspects - heat generation and mechanical stress. For the former we consider an electrochemical non-isothermal model on the spatially resolved porous scale to observe the temperature increase inside a battery cell, as well as to observe the individual heat sources to assess their contributions to the total heat generation. As a result from our experiments, we determined that the temperature has very small spatial variance for our test cases and thus allows for an ODE formulation of the heat equation.
The second aspect that we consider is the generation of mechanical stress as a result of the insertion of lithium ions in the electrode materials. We study two approaches - using small strain models and finite strain models. For the small strain models, the initial geometry and the current geometry coincide. The model considers a diffusion equation for the lithium ions and equilibrium equation for the mechanical stress. First, we test a single perforated cylindrical particle using different boundary conditions for the displacement and with Neumann boundary conditions for the diffusion equation. We also test for cylindrical particles, but with boundary conditions for the diffusion equation in the electrodes coming from an isothermal electrochemical model for the whole battery cell. For the finite strain models we take in consideration the deformation of the initial geometry as a result of the intercalation and the mechanical stress. We compare two elastic models to study the sensitivity of the predicted elastic behavior on the specific model used. We also consider a softening of the active material dependent on the concentration of the lithium ions and using data for silicon electrodes. We recover the general behavior of the stress from known physical experiments.
Some models, like the mechanical models we use, depend on the local values of the concentration to predict the mechanical stress. In that sense we perform a short comparative study between the Finite Element Method with tetrahedral elements and the Finite Volume Method with voxel volumes for an isothermal electrochemical model.
The spatial discretizations of the PDEs are done using the Finite Element Method. For some models we have discontinuous quantities where we adapt the FEM accordingly. The time derivatives are discretized using the implicit Backward Euler method. The nonlinear systems are linearized using the Newton method. All of the discretized models are implemented in a C++ framework developed during the thesis.

Lithium-ion batteries are broadly used nowadays in all kinds of portable electronics, such as laptops, cell phones, tablets, e-book readers, digital cameras, etc. They are preferred to other types of rechargeable batteries due to their superior characteristics, such as light weight and high energy density, no memory effect, and a big number of charge/discharge cycles. The high demand and applicability of Li-ion batteries naturally give rise to the unceasing necessity of developing better batteries in terms of performance and lifetime. The aim of the mathematical modelling of Li-ion batteries is to help engineers test different battery configurations and electrode materials faster and cheaper. Lithium-ion batteries are multiscale systems. A typical Li-ion battery consists of multiple connected electrochemical battery cells. Each cell has two electrodes - anode and cathode, as well as a separator between them that prevents a short circuit.
Both electrodes have porous structure composed of two phases - solid and electrolyte. We call macroscale the lengthscale of the whole electrode and microscale - the lengthscale at which we can distinguish the complex porous structure of the electrodes. We start from a Li-ion battery model derived on the microscale. The model is based on nonlinear diffusion type of equations for the transport of Lithium ions and charges in the electrolyte and in the active material. Electrochemical reactions on the solid-electrolyte interface couple the two phases. The interface kinetics is modelled by the highly nonlinear Butler-Volmer interface conditions. Direct numerical simulations with standard methods, such as the Finite Element Method or Finite Volume Method, lead to ill-conditioned problems with a huge number of degrees of freedom which are difficult to solve. Therefore, the aim of this work is to derive upscaled models on the lengthscale of the whole electrode so that we do not have to resolve all the small-scale features of the porous microstructure thus reducing the computational time and cost. We do this by applying two different upscaling techniques - the Asymptotic Homogenization Method and the Multiscale Finite Element Method (MsFEM). We consider the electrolyte and the solid as two self-complementary perforated domains and we exploit this idea with both upscaling methods. The first method is restricted only to periodic media and periodically oscillating solutions while the second method can be applied to randomly oscillating solutions and is based on the Finite Element Method framework. We apply the Asymptotic Homogenization Method to derive a coupled macro-micro upscaled model under the assumption of periodic electrode microstructure. A crucial step in the homogenization procedure is the upscaling of the Butler-Volmer interface conditions. We rigorously determine the asymptotic order of the interface exchange current densities and we perform a comprehensive numerical study in order to validate the derived homogenized Li-ion battery model. In order to upscale the microscale battery problem in the case of random electrode microstructure we apply the MsFEM, extended to problems in perforated domains with Neumann boundary conditions on the holes. We conduct a detailed numerical investigation of the proposed algorithm and we show numerical convergence of the method that we design. We also apply the developed technique to a simplified two-dimensional Li-ion battery problem and we show numerical convergence of the solution obtained with the MsFEM to the reference microscale one.

Many tasks in image processing can be tackled by modeling an appropriate data fidelity term \(\Phi: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}\) and then solve one of the regularized minimization problems \begin{align*}
&{}(P_{1,\tau}) \qquad \mathop{\rm argmin}_{x \in \mathbb R^n} \big\{ \Phi(x) \;{\rm s.t.}\; \Psi(x) \leq \tau \big\} \\ &{}(P_{2,\lambda}) \qquad \mathop{\rm argmin}_{x \in \mathbb R^n} \{ \Phi(x) + \lambda \Psi(x) \}, \; \lambda > 0 \end{align*} with some function \(\Psi: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}\) and a good choice of the parameter(s). Two tasks arise naturally here: \begin{align*} {}& \text{1. Study the solver sets \({\rm SOL}(P_{1,\tau})\) and
\({\rm SOL}(P_{2,\lambda})\) of the minimization problems.} \\ {}& \text{2. Ensure that the minimization problems have solutions.} \end{align*} This thesis provides contributions to both tasks: Regarding the first task for a more special setting we prove that there are intervals \((0,c)\) and \((0,d)\) such that the setvalued curves \begin{align*}
\tau \mapsto {}& {\rm SOL}(P_{1,\tau}), \; \tau \in (0,c) \\ {} \lambda \mapsto {}& {\rm SOL}(P_{2,\lambda}), \; \lambda \in (0,d) \end{align*} are the same, besides an order reversing parameter change \(g: (0,c) \rightarrow (0,d)\). Moreover we show that the solver sets are changing all the time while \(\tau\) runs from \(0\) to \(c\) and \(\lambda\) runs from \(d\) to \(0\).
In the presence of lower semicontinuity the second task is done if we have additionally coercivity. We regard lower semicontinuity and coercivity from a topological point of view and develop a new technique for proving lower semicontinuity plus coercivity.
Dropping any lower semicontinuity assumption we also prove a theorem on the coercivity of a sum of functions.

The overall goal of the work is to simulate rarefied flows inside geometries with moving boundaries. The behavior of a rarefied flow is characterized through the Knudsen number \(Kn\), which can be very small (\(Kn < 0.01\) continuum flow) or larger (\(Kn > 1\) molecular flow). The transition region (\(0.01 < Kn < 1\)) is referred to as the transition flow regime.
Continuum flows are mainly simulated by using commercial CFD methods, which are used to solve the Euler equations. In the case of molecular flows one uses statistical methods, such as the Direct Simulation Monte Carlo (DSMC) method. In the transition region Euler equations are not adequate to model gas flows. Because of the rapid increase of particle collisions the DSMC method tends to fail, as well
Therefore, we develop a deterministic method, which is suitable to simulate problems of rarefied gases for any Knudsen number and is appropriate to simulate flows inside geometries with moving boundaries. Thus, the method we use is the Finite Pointset Method (FPM), which is a mesh-free numerical method developed at the ITWM Kaiserslautern and is mainly used to solve fluid dynamical problems.
More precisely, we develop a method in the FPM framework to solve the BGK model equation, which is a simplification of the Boltzmann equation. This equation is mainly used to describe rarefied flows.
The FPM based method is implemented for one and two dimensional physical and velocity space and different ranges of the Knudsen number. Numerical examples are shown for problems with moving boundaries. It is seen, that our method is superior to regular grid methods with respect to the implementation of boundary conditions. Furthermore, our results are comparable to reference solutions gained through CFD- and DSMC methods, respectevly.

We present a numerical scheme to simulate a moving rigid body with arbitrary shape suspended in a rarefied gas micro flows, in view of applications to complex computations of moving structures in micro or vacuum systems. The rarefied gas is simulated by solving the Boltzmann equation using a DSMC particle method. The motion of the rigid body is governed by the Newton-Euler equations, where the force and the torque on the rigid body is computed from the momentum transfer of the gas molecules colliding with the body. The resulting motion of the rigid body affects in turn again the gas flow in the surroundings. This means that a two-way coupling has been modeled. We validate the scheme by performing various numerical experiments in 1-, 2- and 3-dimensional computational domains. We have presented 1-dimensional actuator problem, 2-dimensional cavity driven flow problem, Brownian diffusion of a spherical particle both with translational and rotational motions, and finally thermophoresis on a spherical particles. We compare the numerical results obtained from the numerical simulations with the existing theories in each test examples.

Since the early days of representation theory of finite groups in the 19th century, it was known that complex linear representations of finite groups live over number fields, that is, over finite extensions of the field of rational numbers.
While the related question of integrality of representations was answered negatively by the work of Cliff, Ritter and Weiss as well as by Serre and Feit, it was not known how to decide integrality of a given representation.
In this thesis we show that there exists an algorithm that given a representation of a finite group over a number field decides whether this representation can be made integral.
Moreover, we provide theoretical and numerical evidence for a conjecture, which predicts the existence of splitting fields of irreducible characters with integrality properties.
In the first part, we describe two algorithms for the pseudo-Hermite normal form, which is crucial when handling modules over ring of integers.
Using a newly developed computational model for ideal and element arithmetic in number fields, we show that our pseudo-Hermite normal form algorithms have polynomial running time.
Furthermore, we address a range of algorithmic questions related to orders and lattices over Dedekind domains, including computation of genera, testing local isomorphism, computation of various homomorphism rings and computation of Solomon zeta functions.
In the second part we turn to the integrality of representations of finite groups and show that an important ingredient is a thorough understanding of the reduction of lattices at almost all prime ideals.
By employing class field theory and tools from representation theory we solve this problem and eventually describe an algorithm for testing integrality.
After running the algorithm on a large set of examples we are led to a conjecture on the existence of integral and nonintegral splitting fields of characters.
By extending techniques of Serre we prove the conjecture for characters with rational character field and Schur index two.

Functional data analysis is a branch of statistics that deals with observations \(X_1,..., X_n\) which are curves. We are interested in particular in time series of dependent curves and, specifically, consider the functional autoregressive process of order one (FAR(1)), which is defined as \(X_{n+1}=\Psi(X_{n})+\epsilon_{n+1}\) with independent innovations \(\epsilon_t\). Estimates \(\hat{\Psi}\) for the autoregressive operator \(\Psi\) have been investigated a lot during the last two decades, and their asymptotic properties are well understood. Particularly difficult and different from scalar- or vector-valued autoregressions are the weak convergence properties which also form the basis of the bootstrap theory.
Although the asymptotics for \(\hat{\Psi}{(X_{n})}\) are still tractable, they are only useful for large enough samples. In applications, however, frequently only small samples of data are available such that an alternative method for approximating the distribution of \(\hat{\Psi}{(X_{n})}\) is welcome. As a motivation, we discuss a real-data example where we investigate a changepoint detection problem for a stimulus response dataset obtained from the animal physiology group at the Technical University of Kaiserslautern.
To get an alternative for asymptotic approximations, we employ the naive or residual-based bootstrap procedure. In this thesis, we prove theoretically and show via simulations that the bootstrap provides asymptotically valid and practically useful approximations of the distributions of certain functions of the data. Such results may be used to calculate approximate confidence bands or critical bounds for tests.

This thesis is concerned with interest rate modeling by means of the potential approach. The contribution of this work is twofold. First, by making use of the potential approach and the theory of affine Markov processes, we develop a general class of rational models to the term structure of interest rates which we refer to as "the affine rational potential model". These models feature positive interest rates and analytical pricing formulae for zero-coupon bonds, caps, swaptions, and European currency options. We present some concrete models to illustrate the scope of the affine rational potential model and calibrate a model specification to real-world market data. Second, we develop a general family of "multi-curve potential models" for post-crisis interest rates. Our models feature positive stochastic basis spreads, positive term structures, and analytic pricing formulae for interest rate derivatives. This modeling framework is also flexible enough to accommodate negative interest rates and positive basis spreads.

Gröbner bases are one of the most powerful tools in computer algebra and commutative algebra, with applications in algebraic geometry and singularity theory. From the theoretical point of view, these bases can be computed over any field using Buchberger's algorithm. In practice, however, the computational efficiency depends on the arithmetic of the coefficient field.
In this thesis, we consider Gröbner bases computations over two types of coefficient fields. First, consider a simple extension \(K=\mathbb{Q}(\alpha)\) of \(\mathbb{Q}\), where \(\alpha\) is an algebraic number, and let \(f\in \mathbb{Q}[t]\) be the minimal polynomial of \(\alpha\). Second, let \(K'\) be the algebraic function field over \(\mathbb{Q}\) with transcendental parameters \(t_1,\ldots,t_m\), that is, \(K' = \mathbb{Q}(t_1,\ldots,t_m)\). In particular, we present efficient algorithms for computing Gröbner bases over \(K\) and \(K'\). Moreover, we present an efficient method for computing syzygy modules over \(K\).
To compute Gröbner bases over \(K\), starting from the ideas of Noro [35], we proceed by joining \(f\) to the ideal to be considered, adding \(t\) as an extra variable. But instead of avoiding superfluous S-pair reductions by inverting algebraic numbers, we achieve the same goal by applying modular methods as in [2,4,27], that is, by inferring information in characteristic zero from information in characteristic \(p > 0\). For suitable primes \(p\), the minimal polynomial \(f\) is reducible over \(\mathbb{F}_p\). This allows us to apply modular methods once again, on a second level, with respect to the
modular factors of \(f\). The algorithm thus resembles a divide and conquer strategy and
is in particular easily parallelizable. Moreover, using a similar approach, we present an algorithm for computing syzygy modules over \(K\).
On the other hand, to compute Gröbner bases over \(K'\), our new algorithm first specializes the parameters \(t_1,\ldots,t_m\) to reduce the problem from \(K'[x_1,\ldots,x_n]\) to \(\mathbb{Q}[x_1,\ldots,x_n]\). The algorithm then computes a set of Gröbner bases of specialized ideals. From this set of Gröbner bases with coefficients in \(\mathbb{Q}\), it obtains a Gröbner basis of the input ideal using sparse multivariate rational interpolation.
At current state, these algorithms are probabilistic in the sense that, as for other modular Gröbner basis computations, an effective final verification test is only known for homogeneous ideals or for local monomial orderings. The presented timings show that for most examples, our algorithms, which have been implemented in SINGULAR [17], are considerably faster than other known methods.

This thesis deals with risk measures based on utility functions and time consistency of dynamic risk measures. It is therefore aimed at readers interested in both, the theory of static and dynamic financial risk measures in the sense of Artzner, Delbaen, Eber and Heath [7], [8] and the theory of preferences in the tradition of von Neumann and Morgenstern [134].
A main contribution of this thesis is the introduction of optimal expected utility (OEU) risk measures as a new class of utility-based risk measures. We introduce OEU, investigate its main properties, and its applicability to risk measurement and put it in perspective to alternative risk measures and notions of certainty equivalents. To the best of our knowledge, OEU is the only existing utility-based risk measure that is (non-trivial and) coherent if the utility function u has constant relative risk aversion. We present several different risk measures that can be derived with special choices of u and illustrate that OEU reacts in a more sensitive way to slight changes of the probability of a financial loss than value at risk (V@R) and average value at risk.
Further, we propose implied risk aversion as a coherent rating methodology for retail structured products (RSPs). Implied risk aversion is based on optimal expected utility risk measures and, in contrast to standard V@R-based ratings, takes into account both the upside potential and the downside risks of such products. In addition, implied risk aversion is easily interpreted in terms of an individual investor's risk aversion: A product is attractive (unattractive) for an investor if its implied risk aversion is higher (lower) than his individual risk aversion. We illustrate this approach in a case study with more than 15,000 warrants on DAX ® and find that implied risk aversion is able to identify favorable products; in particular, implied risk aversion is not necessarily increasing with respect to the strikes of call warrants.
Another main focus of this thesis is on consistency of dynamic risk measures. To this end, we study risk measures on the space of distributions, discuss concavity on the level of distributions and slightly generalize Weber's [137] findings on the relation of time consistent dynamic risk measures to static risk measures to the case of dynamic risk measures with time-dependent parameters. Finally, this thesis investigates how recursively composed dynamic risk measures in discrete time, which are time consistent by construction, can be related to corresponding dynamic risk measures in continuous time. We present different approaches to establish this link and outline the theoretical basis and the practical benefits of this relation. The thesis concludes with a numerical implementation of this theory.

In this thesis, mathematical research questions related to recursive utility and stochastic differential utility (SDU) are explored.
First, a class of backward equations under nonlinear expectations is investigated: Existence and uniqueness of solutions are established, and the issues of stability and discrete-time approximation are addressed. It is then shown that backward equations of this class naturally appear as a continuous-time limit in the context of recursive utility with nonlinear expectations.
Then, the Epstein-Zin parametrization of SDU is studied. The focus is on specifications with both relative risk aversion and elasitcity of intertemporal substitution greater that one. A concave utility functional is constructed and a utility gradient inequality is established.
Finally, consumption-portfolio problems with recursive preferences and unspanned risk are investigated. The investor's optimal strategies are characterized by a specific semilinear partial differential equation. The solution of this equation is constructed by a fixed point argument, and a corresponding efficient and accurate method to calculate optimal strategies numerically is given.

Inflation modeling is a very important tool for conducting an efficient monetary policy. This doctoral thesis reviewed inflation models, in particular the Phillips curve models of inflation dynamics. We focused on a well known and widely used model, the so-called three equation new Keynesian model which is a system of equations consisting of a new Keynesian Phillips curve (NKPC), an investment and saving (IS) curve and an interest rate rule.
We gave a detailed derivation of these equations. The interest rate rule used in this model is normally determined by using a Lagrangian method to solve an optimal control problem constrained by a standard discrete time NKPC which describes the inflation dynamics and an IS curve that represents the output gaps dynamics. In contrast to the real world, this method assumes that the policy makers intervene continuously. This means that the costs resulting from the change in the interest rates are ignored. We showed also that there are approximation errors made, when one log-linearizes non linear equations, by doing the derivation of the standard discrete time NKPC.
We agreed with other researchers as mentioned in this thesis, that errors which result from ignoring such log-linear approximation errors and the costs of altering interest rates by determining interest rate rule, can lead to a suboptimal interest rate rule and hence to non-optimal paths of output gaps and inflation rate.
To overcome such a problem, we proposed a stochastic optimal impulse control method. We formulated the problem as a stochastic optimal impulse control problem by considering the costs of change in interest rates and the approximation error terms. In order to formulate this problem, we first transform the standard discrete time NKPC and the IS curve into their high-frequency versions and hence into their continuous time versions where error terms are described by a zero mean Gaussian white noise with a finite and constant variance. After formulating this problem, we use the quasi-variational inequality approach to solve analytically a special case of the central bank problem, where an inflation rate is supposed to be on target and a central bank has to optimally control output gap dynamics. This method gives an optimal control band in which output gap process has to be maintained and an optimal control strategy, which includes the optimal size of intervention and optimal intervention time, that can be used to keep the process into the optimal control band.
Finally, using a numerical example, we examined the impact of some model parameters on optimal control strategy. The results show that an increase in the output gap volatility as well as in the fixed and proportional costs of the change in interest rate lead to an increase in the width of the optimal control band. In this case, the optimal intervention requires the central bank to wait longer before undertaking another control action.

By using Gröbner bases of ideals of polynomial algebras over a field, many implemented algorithms manage to give exciting examples and counter examples in Commutative Algebra and Algebraic Geometry. Part A of this thesis will focus on extending the concept of Gröbner bases and Standard bases for polynomial algebras over the ring of integers and its factors \(\mathbb{Z}_m[x]\). Moreover we implemented two algorithms for this case in Singular which use different approaches in detecting useless computations, the classical Buchberger algorithm and a F5 signature based algorithm. Part B includes two algorithms that compute the graded Hilbert depth of a graded module over a polynomial algebra \(R\) over a field, as well as the depth and the multigraded Stanley depth of a factor of monomial ideals of \(R\). The two algorithms provide faster computations and examples that lead B. Ichim and A. Zarojanu to a counter example of a question of J. Herzog. A. Duval, B. Goeckner, C. Klivans and J. Martin have recently discovered a counter example for the Stanley Conjecture. We prove in this thesis that the Stanley Conjecture holds in some special cases. Part D explores the General Neron Desingularization in the frame of Noetherian local domains of dimension 1. We have constructed and implemented in Singular and algorithm that computes a strong Artin Approximation for Cohen-Macaulay local rings of dimension 1.

A vehicles fatigue damage is a highly relevant figure in the complete vehicle design process.
Long term observations and statistical experiments help to determine the influence of differnt parts of the vehicle, the driver and the surrounding environment.
This work is focussing on modeling one of the most important influence factors of the environment: road roughness. The quality of the road is highly dependant on several surrounding factors which can be used to create mathematical models.
Such models can be used for the extrapolation of information and an estimation of the environment for statistical studies.
The target quantity we focus on in this work ist the discrete International Roughness Index or discrete IRI. The class of models we use and evaluate is a discriminative classification model called Conditional Random Field.
We develop a suitable model specification and show new variants of stochastic optimizations to train the model efficiently.
The model is also applied to simulated and real world data to show the strengths of our approach.

Advantage of Filtering for Portfolio Optimization in Financial Markets with Partial Information
(2016)

In a financial market we consider three types of investors trading with a finite
time horizon with access to a bank account as well as multliple stocks: the
fully informed investor, the partially informed investor whose only source of
information are the stock prices and an investor who does not use this infor-
mation. The drift is modeled either as following linear Gaussian dynamics
or as being a continuous time Markov chain with finite state space. The
optimization problem is to maximize expected utility of terminal wealth.
The case of partial information is based on the use of filtering techniques.
Conditions to ensure boundedness of the expected value of the filters are
developed, in the Markov case also for positivity. For the Markov modulated
drift, boundedness of the expected value of the filter relates strongly to port-
folio optimization: effects are studied and quantified. The derivation of an
equivalent, less dimensional market is presented next. It is a type of Mutual
Fund Theorem that is shown here.
Gains and losses eminating from the use of filtering are then discussed in
detail for different market parameters: For infrequent trading we find that
both filters need to comply with the boundedness conditions to be an advan-
tage for the investor. Losses are minimal in case the filters are advantageous.
At an increasing number of stocks, again boundedness conditions need to be
met. Losses in this case depend strongly on the added stocks. The relation
of boundedness and portfolio optimization in the Markov model leads here to
increasing losses for the investor if the boundedness condition is to hold for
all numbers of stocks. In the Markov case, the losses for different numbers
of states are negligible in case more states are assumed then were originally
present. Assuming less states leads to high losses. Again for the Markov
model, a simplification of the complex optimal trading strategy for power
utility in the partial information setting is shown to cause only minor losses.
If the market parameters are such that shortselling and borrowing constraints
are in effect, these constraints may lead to big losses depending on how much
effect the constraints have. They can though also be an advantage for the
investor in case the expected value of the filters does not meet the conditions
for boundedness.
All results are implemented and illustrated with the corresponding numerical
findings.

The thesis consists of two parts. In the first part we consider the stable Auslander--Reiten quiver of a block \(B\) of a Hecke algebra of the symmetric group at a root of unity in characteristic zero. The main theorem states that if the ground field is algebraically closed and \(B\) is of wild representation type, then the tree class of every connected component of the stable Auslander--Reiten quiver \(\Gamma_{s}(B)\) of \(B\) is \(A_{\infty}\). The main ingredient of the proof is a skew group algebra construction over a quantum complete intersection. Also, for these algebras the stable Auslander--Reiten quiver is computed in the case where the defining parameters are roots of unity. As a result, the tree class of every connected component of the stable Auslander--Reiten quiver is \(A_{\infty}\).\[\]
In the second part of the thesis we are concerned with branching rules for Hecke algebras of the symmetric group at a root of unity. We give a detailed survey of the theory initiated by I. Grojnowski and A. Kleshchev, describing the Lie-theoretic structure that the Grothendieck group of finite-dimensional modules over a cyclotomic Hecke algebra carries. A decisive role in this approach is played by various functors that give branching rules for cyclotomic Hecke algebras that are independent of the underlying field. We give a thorough definition of divided power functors that will enable us to reformulate the Scopes equivalence of a Scopes pair of blocks of Hecke algebras of the symmetric group. As a consequence we prove that two indecomposable modules that correspond under this equivalence have a common vertex. In particular, we verify the Dipper--Du Conjecture in the case where the blocks under consideration have finite representation type.