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

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.

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.

Model uncertainty is a challenge that is inherent in many applications of mathematical models in various areas, for instance in mathematical finance and stochastic control. Optimization procedures in general take place under a particular model. This model, however, might be misspecified due to statistical estimation errors and incomplete information. In that sense, any specified model must be understood as an approximation of the unknown "true" model. Difficulties arise since a strategy which is optimal under the approximating model might perform rather bad in the true model. A natural way to deal with model uncertainty is to consider worst-case optimization.
The optimization problems that we are interested in are utility maximization problems in continuous-time financial markets. It is well known that drift parameters in such markets are notoriously difficult to estimate. To obtain strategies that are robust with respect to a possible misspecification of the drift we consider a worst-case utility maximization problem with ellipsoidal uncertainty sets for the drift parameter and with a constraint on the strategies that prevents a pure bond investment.
By a dual approach we derive an explicit representation of the optimal strategy and prove a minimax theorem. This enables us to show that the optimal strategy converges to a generalized uniform diversification strategy as uncertainty increases.
To come up with a reasonable uncertainty set, investors can use filtering techniques to estimate the drift of asset returns based on return observations as well as external sources of information, so-called expert opinions. In a Black-Scholes type financial market with a Gaussian drift process we investigate the asymptotic behavior of the filter as the frequency of expert opinions tends to infinity. We derive limit theorems stating that the information obtained from observing the discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process which can be interpreted as a continuous-time expert. Our convergence results carry over to convergence of the value function in a portfolio optimization problem with logarithmic utility.
Lastly, we use our observations about how expert opinions improve drift estimates for our robust utility maximization problem. We show that our duality approach carries over to a financial market with non-constant drift and time-dependence in the uncertainty set. A time-dependent uncertainty set can then be defined based on a generic filter. We apply this to various investor filtrations and investigate which effect expert opinions have on the robust strategies.

In this dissertation we consider complex, projective hypersurfaces with many isolated singularities. The leading questions concern the maximal number of prescribed singularities of such hypersurfaces in a given linear system, and geometric properties of the equisingular stratum. In the first part a systematic introduction to the theory of equianalytic families of hypersurfaces is given. Furthermore, the patchworking method for constructing hypersurfaces with singularities of prescribed types is described. In the second part we present new existence results for hypersurfaces with many singularities. Using the patchworking method, we show asymptotically proper results for hypersurfaces in P^n with singularities of corank less than two. In the case of simple singularities, the results are even asymptotically optimal. These statements improve all previous general existence results for hypersurfaces with these singularities. Moreover, the results are also transferred to hypersurfaces defined over the real numbers. The last part of the dissertation deals with the Castelnuovo function for studying the cohomology of ideal sheaves of zero-dimensional schemes. Parts of the theory of this function for schemes in P^2 are generalized to the case of schemes on general surfaces in P^3. As an application we show an H^1-vanishing theorem for such schemes.

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 this text we survey some large deviation results for diffusion processes. The first chapters present results from the literature such as the Freidlin-Wentzell theorem for diffusions with small noise. We use these results to prove a new large deviation theorem about diffusion processes with strong drift. This is the main result of the thesis. In the later chapters we give another application of large deviation results, namely to determine the exponential decay rate for the Bayes risk when separating two different processes. The final chapter presents techniques which help to experiment with rare events for diffusion processes by means of computer simulations.

In this thesis we explicitly solve several portfolio optimization problems in a very realistic setting. The fundamental assumptions on the market setting are motivated by practical experience and the resulting optimal strategies are challenged in numerical simulations.
We consider an investor who wants to maximize expected utility of terminal wealth by trading in a high-dimensional financial market with one riskless asset and several stocks.
The stock returns are driven by a Brownian motion and their drift is modelled by a Gaussian random variable. We consider a partial information setting, where the drift is unknown to the investor and has to be estimated from the observable stock prices in addition to some analyst’s opinion as proposed in [CLMZ06]. The best estimate given these observations is the well known Kalman-Bucy-Filter. We then consider an innovations process to transform the partial information setting into a market with complete information and an observable Gaussian drift process.
The investor is restricted to portfolio strategies satisfying several convex constraints.
These constraints can be due to legal restrictions, due to fund design or due to client's specifications. We cover in particular no-short-selling and no-borrowing constraints.
One popular approach to constrained portfolio optimization is the convex duality approach of Cvitanic and Karatzas. In [CK92] they introduce auxiliary stock markets with shifted market parameters and obtain a dual problem to the original portfolio optimization problem that can be better solvable than the primal problem.
Hence we consider this duality approach and using stochastic control methods we first solve the dual problems in the cases of logarithmic and power utility.
Here we apply a reverse separation approach in order to obtain areas where the corresponding Hamilton-Jacobi-Bellman differential equation can be solved. It turns out that these areas have a straightforward interpretation in terms of the resulting portfolio strategy. The areas differ between active and passive stocks, where active stocks are invested in, while passive stocks are not.
Afterwards we solve the auxiliary market given the optimal dual processes in a more general setting, allowing for various market settings and various dual processes.
We obtain explicit analytical formulas for the optimal portfolio policies and provide an algorithm that determines the correct formula for the optimal strategy in any case.
We also show optimality of our resulting portfolio strategies in different verification theorems.
Subsequently we challenge our theoretical results in a historical and an artificial simulation that are even closer to the real world market than the setting we used to derive our theoretical results. However, we still obtain compelling results indicating that our optimal strategies can outperform any benchmark in a real market in general.

Cell migration is essential for embryogenesis, wound healing, immune surveillance, and
progression of diseases, such as cancer metastasis. For the migration to occur, cellular
structures such as actomyosin cables and cell-substrate adhesion clusters must interact.
As cell trajectories exhibit a random character, so must such interactions. Furthermore,
migration often occurs in a crowded environment, where the collision outcome is deter-
mined by altered regulation of the aforementioned structures. In this work, guided by a
few fundamental attributes of cell motility, we construct a minimal stochastic cell migration
model from ground-up. The resulting model couples a deterministic actomyosin contrac-
tility mechanism with stochastic cell-substrate adhesion kinetics, and yields a well-defined
piecewise deterministic process. The signaling pathways regulating the contractility and
adhesion are considered as well. The model is extended to include cell collectives. Numer-
ical simulations of single cell migration reproduce several experimentally observed results,
including anomalous diffusion, tactic migration, and contact guidance. The simulations
of colliding cells explain the observed outcomes in terms of contact induced modification
of contractility and adhesion dynamics. These explained outcomes include modulation
of collision response and group behavior in the presence of an external signal, as well as
invasive and dispersive migration. Moreover, from the single cell model we deduce a pop-
ulation scale formulation for the migration of non-interacting cells. In this formulation,
the relationships concerning actomyosin contractility and adhesion clusters are maintained.
Thus, we construct a multiscale description of cell migration, whereby single, collective,
and population scale formulations are deduced from the relationships on the subcellular
level in a mathematically consistent way.

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.

In this thesis we address two instances of duality in commutative algebra.
In the first part, we consider value semigroups of non irreducible singular algebraic curves
and their fractional ideals. These are submonoids of Z^n closed under minima, with a conductor and which fulfill special compatibility properties on their elements. Subsets of Z^n
fulfilling these three conditions are known in the literature as good semigroups and their ideals, and their class strictly contains the class of value semigroup ideals. We examine
good semigroups both independently and in relation with their algebraic counterpart. In the combinatoric setting, we define the concept of good system of generators, and we
show that minimal good systems of generators are unique. In relation with the algebra side, we give an intrinsic definition of canonical semigroup ideals, which yields a duality
on good semigroup ideals. We prove that this semigroup duality is compatible with the Cohen-Macaulay duality under taking values. Finally, using the duality on good semigroup ideals, we show a symmetry of the Poincaré series of good semigroups with special properties.
In the second part, we treat Macaulay’s inverse system, a one-to-one correspondence
which is a particular case of Matlis duality and an effective method to construct Artinian k-algebras with chosen socle type. Recently, Elias and Rossi gave the structure of the inverse system of positive dimensional Gorenstein k-algebras. We extend their result by establishing a one-to-one correspondence between positive dimensional level k-algebras and certain submodules of the divided power ring. We give several examples to illustrate
our result.

The thesis discusses discrete-time dynamic flows over a finite time horizon T. These flows take time, called travel time, to pass an arc of the network. Travel times, as well as other network attributes, such as, costs, arc and node capacities, and supply at the source node, can be constant or time-dependent. Here we review results on discrete-time dynamic flow problems (DTDNFP) with constant attributes and develop new algorithms to solve several DTDNFPs with time-dependent attributes. Several dynamic network flow problems are discussed: maximum dynamic flow, earliest arrival flow, and quickest flow problems. We generalize the hybrid capacity scaling and shortest augmenting path algorithmic of the static network flow problem to consider the time dependency of the network attributes. The result is used to solve the maximum dynamic flow problem with time-dependent travel times and capacities. We also develop a new algorithm to solve earliest arrival flow problems with the same assumptions on the network attributes. The possibility to wait (or park) at a node before departing on outgoing arc is also taken into account. We prove that the complexity of new algorithm is reduced when infinite waiting is considered. We also report the computational analysis of this algorithm. The results are then used to solve quickest flow problems. Additionally, we discuss time-dependent bicriteria shortest path problems. Here we generalize the classical shortest path problems in two ways. We consider two - in general contradicting - objective functions and introduce a time dependency of the cost which is caused by a travel time on each arc. These problems have several interesting practical applications, but have not attained much attention in the literature. Here we develop two new algorithms in which one of them requires weaker assumptions as in previous research on the subject. Numerical tests show the superiority of the new algorithms. We then apply dynamic network flow models and their associated solution algorithms to determine lower bounds of the evacuation time, evacuation routes, and maximum capacities of inhabited areas with respect to safety requirements. As a macroscopic approach, our dynamic network flow models are mainly used to produce good lower bounds for the evacuation time and do not consider any individual behavior during the emergency situation. These bounds can be used to analyze existing buildings or help in the design phase of planning a building.

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.

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

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.

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.

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.

Competing Neural Networks as Models for Non Stationary Financial Time Series -Changepoint Analysis-
(2005)

The problem of structural changes (variations) play a central role in many scientific fields. One of the most current debates is about climatic changes. Further, politicians, environmentalists, scientists, etc. are involved in this debate and almost everyone is concerned with the consequences of climatic changes. However, in this thesis we will not move into the latter direction, i.e. the study of climatic changes. Instead, we consider models for analyzing changes in the dynamics of observed time series assuming these changes are driven by a non-observable stochastic process. To this end, we consider a first order stationary Markov Chain as hidden process and define the Generalized Mixture of AR-ARCH model(GMAR-ARCH) which is an extension of the classical ARCH model to suit to model with dynamical changes. For this model we provide sufficient conditions that ensure its geometric ergodic property. Further, we define a conditional likelihood given the hidden process and a pseudo conditional likelihood in turn. For the pseudo conditional likelihood we assume that at each time instant the autoregressive and volatility functions can be suitably approximated by given Feedfoward Networks. Under this setting the consistency of the parameter estimates is derived and versions of the well-known Expectation Maximization algorithm and Viterbi Algorithm are designed to solve the problem numerically. Moreover, considering the volatility functions to be constants, we establish the consistency of the autoregressive functions estimates given some parametric classes of functions in general and some classes of single layer Feedfoward Networks in particular. Beside this hidden Markov Driven model, we define as alternative a Weighted Least Squares for estimating the time of change and the autoregressive functions. For the latter formulation, we consider a mixture of independent nonlinear autoregressive processes and assume once more that the autoregressive functions can be approximated by given single layer Feedfoward Networks. We derive the consistency and asymptotic normality of the parameter estimates. Further, we prove the convergence of Backpropagation for this setting under some regularity assumptions. Last but not least, we consider a Mixture of Nonlinear autoregressive processes with only one abrupt unknown changepoint and design a statistical test that can validate such changes.

The topic of this thesis is the coupling of an atomistic and a coarse scale region in molecular dynamics simulations with the focus on the reflection of waves at the interface between the two scales and the velocity of waves in the coarse scale region for a non-equilibrium process. First, two models from the literature for such a coupling, the concurrent coupling of length scales and the bridging scales method are investigated for a one dimensional system with harmonic interaction. It turns out that the concurrent coupling of length scales method leads to the reflection of fine scale waves at the interface, while the bridging scales method gives an approximated system that is not energy conserving. The velocity of waves in the coarse scale region is in both models not correct. To circumvent this problems, we present a coupling based on the displacement splitting of the bridging scales method together with choosing appropriate variables in orthogonal subspaces. This coupling allows the derivation of evolution equations of fine and coarse scale degrees of freedom together with a reflectionless boundary condition at the interface directly from the Lagrangian of the system. This leads to an energy conserving approximated system with a clear separation between modeling errors an errors due to the numerical solution. Possible approximations in the Lagrangian and the numerical computation of the memory integral and other numerical errors are discussed. We further present a method to choose the interpolation from coarse to atomistic scale in such a way, that the fine scale degrees of freedom in the coarse scale region can be neglected. The interpolation weights are computed by comparing the dispersion relations of the coarse scale equations and the fully atomistic system. With this new interpolation weights, the number of degrees of freedom can be drastically reduced without creating an error in the velocity of the waves in the coarse scale region. We give an alternative derivation of the new coupling with the Mori-Zwanzig projection operator formalism, and explain how the method can be extended to non-zero temperature simulations. For the comparison of the results of the approximated with the fully atomistic system, we use a local stress tensor and the energy in the atomistic region. Examples for the numerical solution of the approximated system for harmonic potentials are given in one and two dimensions.

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

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.

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.

Numerical Godeaux surfaces are minimal surfaces of general type with the smallest possible numerical invariants. It is known that the torsion group of a numerical Godeaux surface is cyclic of order \(m\leq 5\). A full classification has been given for the cases \(m=3,4,5\) by the work of Reid and Miyaoka. In each case, the corresponding moduli space is 8-dimensional and irreducible.
There exist explicit examples of numerical Godeaux surfaces for the orders \(m=1,2\), but a complete classification for these surfaces is still missing.
In this thesis we present a construction method for numerical Godeaux surfaces which is based on homological algebra and computer algebra and which arises from an experimental approach by Schreyer. The main idea is to consider the canonical ring \(R(X)\) of a numerical Godeaux surface \(X\) as a module over some graded polynomial ring \(S\). The ring \(S\) is chosen so that \(R(X)\) is finitely generated as an \(S\)-module and a Gorenstein \(S\)-algebra of codimension 3. We prove that the canonical ring of any numerical Godeaux surface, considered as an \(S\)-module, admits a minimal free resolution whose middle map is alternating. Moreover, we show that a partial converse of this statement is true under some additional conditions.
Afterwards we use these results to construct (canonical rings of) numerical Godeaux surfaces. Hereby, we restrict our study to surfaces whose bicanonical system has no fixed component but 4 distinct base points, in the following referred to as marked numerical Godeaux surfaces.
The particular interest of this thesis lies on marked numerical Godeaux surfaces whose torsion group is trivial. For these surfaces we study the fibration of genus 4 over \(\mathbb{P}^1\) induced by the bicanonical system. Catanese and Pignatelli showed that the general fibre is non-hyperelliptic and that the number \(\tilde{h}\) of hyperelliptic fibres is bounded by 3. The two explicit constructions of numerical Godeaux surfaces with a trivial torsion group due to Barlow and Craighero-Gattazzo, respectively, satisfy \(\tilde{h} = 2\).
With the method from this thesis, we construct an 8-dimensional family of numerical Godeaux surfaces with a trivial torsion group and whose general element satisfy \(\tilde{h}=0\).
Furthermore, we establish a criterion for the existence of hyperelliptic fibres in terms of a minimal free resolution of \(R(X)\). Using this criterion, we verify experimentally the
existence of a numerical Godeaux surface with \(\tilde{h}=1\).

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.

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.

In this thesis we consider the directional analysis of stationary point processes. We focus on three non-parametric methods based on second order analysis which we have defined as Integral method, Ellipsoid method, and Projection method. We present the methods in a general setting and then focus on their application in the 2D and 3D case of a particular type of anisotropy mechanism called geometric anisotropy. We mainly consider regular point patterns motivated by our application to real 3D data coming from glaciology. Note that directional analysis of 3D data is not so prominent in the literature.
We compare the performance of the methods, which depends on the relative parameters, in a simulation study both in 2D and 3D. Based on the results we give recommendations on how to choose the methods´ parameters in practice.
We apply the directional analysis to the 3D data coming from glaciology, which consist in the locations of air-bubbles in polar ice cores. The aim of this study is to provide information about the deformation rate in the ice and the corresponding thinning of ice layers at different depths. This information is substantial for the glaciologists in order to build ice dating models and consequently to give a correct interpretation of the climate information which can be found by analyzing ice cores. In this thesis we consider data coming from three different ice cores: the Talos Dome core, the EDML core and the Renland core.
Motivated by the ice application, we study how isotropic and stationary noise influences the directional analysis. In fact, due to the relaxation of the ice after drilling, noise bubbles can form within the ice samples. In this context we take two classification algorithms into consideration, which aim to classify points in a superposition of a regular isotropic and stationary point process with Poisson noise.
We introduce two methods to visualize anisotropy, which are particularly useful in 3D and apply them to the ice data. Finally, we consider the problem of testing anisotropy and the limiting behavior of the geometric anisotropy transform.

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.

Traffic flow on road networks has been a continuous source of challenging mathematical problems. Mathematical modelling can provide an understanding of dynamics of traffic flow and hence helpful in organizing the flow through the network. In this dissertation macroscopic models for the traffic flow in road networks are presented. The primary interest is the extension of the existing macroscopic road network models based on partial differential equations (PDE model). In order to overcome the difficulty of high computational costs of PDE model an ODE model has been introduced. In addition, steady state traffic flow model named as RSA model on road networks has been dicsussed. To obtain the optimal flow through the network cost functionals and corresponding optimal control problems are defined. The solution of these optimization problems provides an information of shortest path through the network subject to road conditions. The resulting constrained optimization problem is solved approximately by solving unconstrained problem invovling exact penalty functions and the penalty parameter. A good estimate of the threshold of the penalty parameter is defined. A well defined algorithm for solving a nonlinear, nonconvex equality and bound constrained optimization problem is introduced. The numerical results on the convergence history of the algorithm support the theoretical results. In addition to this, bottleneck situations in the traffic flow have been treated using a domain decomposition method (DDM). In particular this method could be used to solve the scalar conservation laws with the discontinuous flux functions corresponding to other physical problems too. This method is effective even when the flux function presents more than one discontinuity within the same spatial domain. It is found in the numerical results that the DDM is superior to other schemes and demonstrates good shock resolution.

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.

We construct and study two surface measures on the space C([0,1],M) of paths in a compact Riemannian manifold M embedded into the Euclidean space R^n. The first one is induced by conditioning the usual Wiener measure on C([0,T],R^n) to the event that the Brownian particle does not leave the tubular epsilon-neighborhood of M up to time T, and passing to the limit. The second one is defined as the limit of the laws of reflected Brownian motions with reflection on the boundaries of the tubular epsilon-neighborhoods of M. We prove that the both surface measures exist and compare them with the Wiener measure W_M on C([0,T],M). We show that the first one is equivalent to W_M and compute the corresponding density explicitly in terms of the scalar curvature and the mean curvature vector of M. Further, we show that the second surface measure coincides with W_M. Finally, we study the limit behavior of the both surface measures as T tends to infinity.

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.

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.

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.

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 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.

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.

We introduce and investigate a product pricing model in social networks where the value a possible buyer assigns to a product is influenced by the previous buyers. The selling proceeds in discrete, synchronous rounds for some set price and the individual values are additively altered. Whereas computing the revenue for a given price can be done in polynomial time, we show that the basic problem PPAI, i.e., is there a price generating a requested revenue, is weakly NP-complete. With algorithm Frag we provide a pseudo-polynomial time algorithm checking the range of prices in intervals of common buying behavior we call fragments. In some special cases, e.g., solely positive influences, graphs with bounded in-degree, or graphs with bounded path length, the amount of fragments is polynomial. Since the run-time of Frag is polynomial in the amount of fragments, the algorithm itself is polynomial for these special cases. For graphs with positive influence we show that every buyer does also buy for lower prices, a property that is not inherent for arbitrary graphs. Algorithm FixHighest improves the run-time on these graphs by using the above property.
Furthermore, we introduce variations on this basic model. The version of delaying the propagation of influences and the awareness of the product can be implemented in our basic model by substituting nodes and arcs with simple gadgets. In the chapter on Dynamic Product Pricing we allow price changes, thereby raising the complexity even for graphs with solely positive or negative influences. Concerning Perishable Product Pricing, i.e., the selling of products that are usable for some time and can be rebought afterward, the principal problem is computing the revenue that a given price can generate in some time horizon. In general, the problem is #P-hard and algorithm Break runs in pseudo-polynomial time. For polynomially computable revenue, we investigate once more the complexity to find the best price.
We conclude the thesis with short results in topics of Cooperative Pricing, Initial Value as Parameter, Two Product Pricing, and Bounded Additive Influence.

In change-point analysis the point of interest is to decide if the observations follow one model
or if there is at least one time-point, where the model has changed. This results in two sub-
fields, the testing of a change and the estimation of the time of change. This thesis considers
both parts but with the restriction of testing and estimating for at most one change-point.
A well known example is based on independent observations having one change in the mean.
Based on the likelihood ratio test a test statistic with an asymptotic Gumbel distribution was
derived for this model. As it is a well-known fact that the corresponding convergence rate is
very slow, modifications of the test using a weight function were considered. Those tests have
a better performance. We focus on this class of test statistics.
The first part gives a detailed introduction to the techniques for analysing test statistics and
estimators. Therefore we consider the multivariate mean change model and focus on the effects
of the weight function. In the case of change-point estimators we can distinguish between
the assumption of a fixed size of change (fixed alternative) and the assumption that the size
of the change is converging to 0 (local alternative). Especially, the fixed case in rarely analysed
in the literature. We show how to come from the proof for the fixed alternative to the
proof of the local alternative. Finally, we give a simulation study for heavy tailed multivariate
observations.
The main part of this thesis focuses on two points. First, analysing test statistics and, secondly,
analysing the corresponding change-point estimators. In both cases, we first consider a
change in the mean for independent observations but relaxing the moment condition. Based on
a robust estimator for the mean, we derive a new type of change-point test having a randomized
weight function. Secondly, we analyse non-linear autoregressive models with unknown
regression function. Based on neural networks, test statistics and estimators are derived for
correctly specified as well as for misspecified situations. This part extends the literature as
we analyse test statistics and estimators not only based on the sample residuals. In both
sections, the section on tests and the one on the change-point estimator, we end with giving
regularity conditions on the model as well as the parameter estimator.
Finally, a simulation study for the case of the neural network based test and estimator is
given. We discuss the behaviour under correct and mis-specification and apply the neural
network based test and estimator on two data sets.

This thesis builds a bridge between singularity theory and computer algebra. To an isolated hypersurface singularity one can associate a regular meromorphic connection, the Gauß-Manin connection, containing a lattice, the Brieskorn lattice. The leading terms of the Brieskorn lattice with respect to the weight and V-filtration of the Gauß-Manin connection define the spectral pairs. They correspond to the Hodge numbers of the mixed Hodge structure on the cohomology of the Milnor fibre and belong to the finest known invariants of isolated hypersurface singularities. The differential structure of the Brieskorn lattice can be described by two complex endomorphisms A0 and A1 containing even more information than the spectral pairs. In this thesis, an algorithmic approach to the Brieskorn lattice in the Gauß-Manin connection is presented. It leads to algorithms to compute the complex monodromy, the spectral pairs, and the differential structure of the Brieskorn lattice. These algorithms are implemented in the computer algebra system Singular.

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 \(^3D_4(q)\).

The goal of this work is the development and investigation of an interdisciplinary and in itself closed hydrodynamic approach to the simulation of dilute and dense granular flow. The definition of “granular flow” is a nontrivial task in itself. We say that it is either the flow of grains in a vacuum or in a fluid. A grain is an observable piece of a certain material, for example stone when we mean the flow of sand. Choosing a hydrodynamic view on granular flow, we treat the granular material as a fluid. A hydrodynamic model is developed, that describes the process of flowing granular material. This is done through a system of partial differential equations and algebraic relations. This system is derived by the kinetic theory of granular gases which is characterized by inelastic collisions extended with approaches from soil mechanics. Solutions to the system have to be obtained to understand the process. The equations are so difficult to solve that an analytical solution is out of reach. So approximate solutions must be obtained. Hence the next step is the choice or development of a numerical algorithm to obtain approximate solutions of the model. Common to every problem in numerical simulation, these two steps do not lead to a result without implementation of the algorithm. Hence the author attempts to present this work in the following frame, to participate in and contribute to the three areas Physics, Mathematics and Software implementation and approach the simulation of granular flow in a combined and interdisciplinary way. This work is structured as follows. A continuum model for granular flow which covers the regime of fast dilute flow as well as slow dense flow up to vanishing velocity is presented in the first chapter. This model is strongly nonlinear in the dependence of viscosity and other coefficients on the hydrodynamic variables and it is singular because some coefficients diverge towards the maximum packing fraction of grains. Hence the second difficulty, the challenging task of numerically obtaining approximate solutions for this model is faced in the second chapter. In the third chapter we aim at the validation of both the model and the numerical algorithm through numerical experiments and investigations and show their application to industrial problems. There we focus intensively on the shear flow experiment from the experimental and analytical work of Bocquet et al. which serves well to demonstrate the algorithm, all boundary conditions involved and provides a setting for analytical studies to compare our results. The fourth chapter rounds up the work with the implementation of both the model and the numerical algorithm in a software framework for the solution of complex rheology problems developed as part of this thesis.

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.

Matrix Compression Methods for the Numerical Solution of Radiative Transfer in Scattering Media
(2002)

Radiative transfer in scattering media is usually described by the radiative transfer equation, an integro-differential equation which describes the propagation of the radiative intensity along a ray. The high dimensionality of the equation leads to a very large number of unknowns when discretizing the equation. This is the major difficulty in its numerical solution. In case of isotropic scattering and diffuse boundaries, the radiative transfer equation can be reformulated into a system of integral equations of the second kind, where the position is the only independent variable. By employing the so-called momentum equation, we derive an integral equation, which is also valid in case of linear anisotropic scattering. This equation is very similar to the equation for the isotropic case: no additional unknowns are introduced and the integral operators involved have very similar mapping properties. The discretization of an integral operator leads to a full matrix. Therefore, due to the large dimension of the matrix in practical applcation, it is not feasible to assemble and store the entire matrix. The so-called matrix compression methods circumvent the assembly of the matrix. Instead, the matrix-vector multiplications needed by iterative solvers are performed only approximately, thus, reducing, the computational complexity tremendously. The kernels of the integral equation describing the radiative transfer are very similar to the kernels of the integral equations occuring in the boundary element method. Therefore, with only slight modifications, the matrix compression methods, developed for the latter are readily applicable to the former. As apposed to the boundary element method, the integral kernels for radiative transfer in absorbing and scattering media involve an exponential decay term. We examine how this decay influences the efficiency of the matrix compression methods. Further, a comparison with the discrete ordinate method shows that discretizing the integral equation may lead to reductions in CPU time and to an improved accuracy especially in case of small absorption and scattering coefficients or if local sources are present.

In the filling process of a car tank, the formation of foam plays an unwanted role, as it may prevent the tank from being completely filled or at least delay the filling. Therefore it is of interest to optimize the geometry of the tank using numerical simulation in such a way that the influence of the foam is minimized. In this dissertation, we analyze the behaviour of the foam mathematically on the mezoscopic scale, that is for single lamellae. The most important goals are on the one hand to gain a deeper understanding of the interaction of the relevant physical effects, on the other hand to obtain a model for the simulation of the decay of a lamella which can be integrated in a global foam model. In the first part of this work, we give a short introduction into the physical properties of foam and find that the Marangoni effect is the main cause for its stability. We then develop a mathematical model for the simulation of the dynamical behaviour of a lamella based on an asymptotic analysis using the special geometry of the lamella. The result is a system of nonlinear partial differential equations (PDE) of third order in two spatial and one time dimension. In the second part, we analyze this system mathematically and prove an existence and uniqueness result for a simplified case. For some special parameter domains the system can be further simplified, and in some cases explicit solutions can be derived. In the last part of the dissertation, we solve the system using a finite element approach and discuss the results in detail.

In this work we present and estimate an explanatory model with a predefined system of explanatory equations, a so called lag dependent model. We present a locally optimal, on blocked neural network based lag estimator and theorems about consistensy. We define the change points in context of lag dependent model, and present a powerfull algorithm for change point detection in high dimensional high dynamical systems. We present a special kind of bootstrap for approximating the distribution of statistics of interest in dependent processes.

This thesis discusses methods for the classification of finite projective planes via exhaustive search. In the main part the author classifies all projective planes of order 16 admitting a large quasiregular group of collineations. This is done by a complete search using the computer algebra system GAP. Computational methods for the construction of relative difference sets are discussed. These methods are implemented in a GAP-package, which is available separately. As another result --found in cooperation with U. Dempwolff-- the projective planes defined by planar monomials are classified. Furthermore the full automorphism group of the non-translation planes defined by planar monomials are classified.

In many industrial applications fast and accurate solutions of linear elliptic partial differential equations are needed as one of the building blocks of more complex problems. The domains are often highly complex and meshing turns out to be expensive and difficult to obtain with a sufficient quality. In such cases methods with a regular, not boundary adapted grid offer an attractive alternative. The Explicit Jump Immersed Interface Method is one of these algorithms. The main interest of this work lies in solving the linear elasticity equations. For this purpose the existing EJIIM algorithm has been extended to three dimensions. The Poisson equation is always considered in parallel as the most typical representative of elliptic PDEs. During the work it became clear that EJIIM can have very high computational memory requirements. To overcome this problem an improvement, Reduced EJIIM is proposed. The main theoretical result in this work is the proof of the smoothing property of inverses of elliptic finite difference operators in two and three space dimensions. It is an often observed phenomena that the local truncation error is allowed to be of lower order along some lower dimensional manifold without influencing the global convergence order of the solution.

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