Kaiserslautern - Fachbereich Mathematik
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Faculty / Organisational entity
Understanding human crowd behaviour has been an intriguing topic of interdisciplinary research in recent decades. Modelling of crowd dynamics using differential equations is an indispensable approach to unraveling the various complex dynamics involved in such interacting particle systems. Numerical simulation of pedestrian crowd via these mathematical models allows us to study different realistic scenarios beyond the limitations of studies via controlled experiments.
In this thesis, the main objective is to understand and analyse the dynamics in a domain shared by both pedestrians and moving obstacles. We model pedestrian motion by combining the social force concept with the idea of optimal path computation. This leads to a system of ordinary differential equations governing the dynamics of individual pedestrians via the interaction forces (social forces) between them. Additionally, a non-local force term involving the optimal path and desired velocity governs the pedestrian trajectory. The optimal path computation involves solving a time-independent Eikonal equation, which is coupled to the system of ODEs. A hydrodynamic model is developed from this microscopic model via the mean-field limit.
To consider the interaction with moving obstacles in the domain, we model a set of kinematic equations for the obstacle motion. Two kinds of obstacles are considered - "passive", which move in their predefined trajectories and have only a one-way interaction with pedestrians, and "dynamic", which have a feedback interaction with pedestrians and have their trajectories changing dynamically. The coupled model of pedestrians and obstacles is used to discern pedestrian collision avoidance behaviour in different computational scenarios in a long rectangular domain. We observe that pedestrians avoid collisions through route choice strategies that involve changes in speed and path. We extend this model to consider the interaction between pedestrians and vehicular traffic. We appropriately model the interactions of vehicles, following lane traffic, based on the car-following approach. We observe how the deceleration and braking mechanism of vehicles is executed at pedestrian crossings depending on the right of way on the roads.
As a second objective, we study the disease contagion in moving crowds. We consider the influence of the crowd motion in a complex dynamical environment on the course of infection of pedestrians. A hydrodynamic model for multi-group pedestrian flow is derived from the kinetic equations based on a social force model. It is coupled along with an Eikonal equation to a non-local SEIS contagion model for disease spread. Here, apart from the description of local contacts, the influence of contact times has also been modelled. We observe that the nature of the flow and the geometry of the domain lead to changes in density which affect the contact time and, consequently, the rate of spread of infection.
Finally, the social force model is compared to a variable speed based rational behaviour pedestrian model. We derive a hierarchy of the heuristics-based model from microscopic to macroscopic scales and numerically investigate these models in different density scenarios. Various numerical test cases are considered, including uni- and bi-directional flows and scenarios with and without obstacles. We observe that in low-density scenarios, collision avoidance forces arising from the behavioural heuristics give valid results. Whereas in high-density scenarios, repulsive force terms are essential.
The numerical simulations of all the models are carried out using a mesh-free particle method based on least square approximations. The meshfree numerical framework provides an efficient and elegant way to handle complex geometric situations involving boundaries and stationary or moving obstacles.
The focus of this work has been to develop two families of wavelet solvers for the inner displacement boundary-value problem of elastostatics. Our methods are particularly suitable for the deformation analysis corresponding to geoscientifically relevant (regular) boundaries like sphere, ellipsoid or the actual Earth's surface. The first method, a spatial approach to wavelets on a regular (boundary) surface, is established for the classical (inner) displacement problem. Starting from the limit and jump relations of elastostatics we formulate scaling functions and wavelets within the framework of the Cauchy-Navier equation. Based on numerical integration rules a tree algorithm is constructed for fast wavelet computation. This method can be viewed as a first attempt to "short-wavelength modelling", i.e. high resolution of the fine structure of displacement fields. The second technique aims at a suitable wavelet approximation associated to Green's integral representation for the displacement boundary-value problem of elastostatics. The starting points are tensor product kernels defined on Cauchy-Navier vector fields. We come to scaling functions and a spectral approach to wavelets for the boundary-value problems of elastostatics associated to spherical boundaries. Again a tree algorithm which uses a numerical integration rule on bandlimited functions is established to reduce the computational effort. For numerical realization for both methods, multiscale deformation analysis is investigated for the geoscientifically relevant case of a spherical boundary using test examples. Finally, the applicability of our wavelet concepts is shown by considering the deformation analysis of a particular region of the Earth, viz. Nevada, using surface displacements provided by satellite observations. This represents the first step towards practical applications.
In this thesis, we have dealt with two modeling approaches of the credit risk, namely the structural (firm value) and the reduced form. In the former one, the firm value is modeled by a stochastic process and the first hitting time of this stochastic process to a given boundary defines the default time of the firm. In the existing literature, the stochastic process, triggering the firm value, has been generally chosen as a diffusion process. Therefore, on one hand it is possible to obtain closed form solutions for the pricing problems of credit derivatives and on the other hand the optimal capital structure of a firm can be analysed by obtaining closed form solutions of firm's corporate securities such as; equity value, debt value and total firm value, see Leland(1994). We have extended this approach by modeling the firm value as a jump-diffusion process. The choice of the jump-diffusion process was a crucial step to obtain closed form solutions for corporate securities. As a result, we have chosen a jump-diffusion process with double exponentially distributed jump heights, which enabled us to analyse the effects of jump on the optimal capital structure of a firm. In the second part of the thesis, by following the reduced form models, we have assumed that the default is triggered by the first jump of a Cox process. Further, by following Schönbucher(2005), we have modeled the forward default intensity of a firm as a geometric Brownian motion and derived pricing formulas for credit default swap options in a more general setup than the ones in Schönbucher(2005).
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.
Abstract
The main theme of this thesis is about Graph Coloring Applications and Defining Sets in Graph Theory.
As in the case of block designs, finding defining sets seems to be difficult problem, and there is not a general conclusion. Hence we confine us here to some special types of graphs like bipartite graphs, complete graphs, etc.
In this work, four new concepts of defining sets are introduced:
• Defining sets for perfect (maximum) matchings
• Defining sets for independent sets
• Defining sets for edge colorings
• Defining set for maximal (maximum) clique
Furthermore, some algorithms to find and construct the defining sets are introduced. A review on some known kinds of defining sets in graph theory is also incorporated, in chapter 2 the basic definitions and some relevant notations used in this work are introduced.
chapter 3 discusses the maximum and perfect matchings and a new concept for a defining set for perfect matching.
Different kinds of graph colorings and their applications are the subject of chapter 4.
Chapter 5 deals with defining sets in graph coloring. New results are discussed along with already existing research results, an algorithm is introduced, which enables to determine a defining set of a graph coloring.
In chapter 6, cliques are discussed. An algorithm for the determination of cliques using their defining sets. Several examples are included.
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.
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.
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.
The main aim of this work was to obtain an approximate solution of the seismic traveltime tomography problems with the help of splines based on reproducing kernel Sobolev spaces. In order to be able to apply the spline approximation concept to surface wave as well as to body wave tomography problems, the spherical spline approximation concept was extended for the case where the domain of the function to be approximated is an arbitrary compact set in R^n and a finite number of discontinuity points is allowed. We present applications of such spline method to seismic surface wave as well as body wave tomography, and discuss the theoretical and numerical aspects of such applications. Moreover, we run numerous numerical tests that justify the theoretical considerations.
We investigate the long-term behaviour of diffusions on the non-negative real numbers under killing at some random time. Killing can occur at zero as well as in the interior of the state space. The diffusion follows a stochastic differential equation driven by a Brownian motion. The diffusions we are working with will almost surely be killed. In large parts of this thesis we only assume the drift coefficient to be continuous. Further, we suppose that zero is regular and that infinity is natural. We condition the diffusion on survival up to time t and let t tend to infinity looking for a limiting behaviour.
An autoregressive-ARCH model with possible exogeneous variables is treated. We estimate the conditional volatility of the model by applying feedforward networks to the residuals and prove consistency and asymptotic normality for the estimates under the rate of feedforward networks complexity. Recurrent neural networks estimates of GARCH and value-at-risk is studied. We prove consistency and asymptotic normality for the recurrent neural networks ARMA estimator under the rate of recurrent networks complexity. We also overcome the estimation problem in stochastic variance models in discrete time by feedforward networks and the introduction of a new distributions on the innovations. We use the method to calculate market risk such as expected shortfall and Value-at risk. We tested this distribution together with other new distributions on the GARCH family models against other common distributions on the financial market such as Normal Inverse Gaussian, normal and the Student's t- distributions. As an application of the models, some German stocks are studied and the different approaches are compared together with the most common method of GARCH(1,1) fit.
Laser-induced interstitial thermotherapy (LITT) is a minimally invasive procedure to destroy liver
tumors through thermal ablation. Mathematical models are the basis for computer simulations
of LITT, which support the practitioner in planning and monitoring the therapy.
In this thesis, we propose three potential extensions of an established mathematical model of
LITT, which is based on two nonlinearly coupled partial differential equations (PDEs) modeling
the distribution of the temperature and the laser radiation in the liver.
First, we introduce the Cattaneo–LITT model for delayed heat transfer in this context, prove its
well-posedness and study the effect of an inherent delay parameter numerically.
Second, we model the influence of large blood vessels in the heat-transfer model by means
of a spatially varying blood-perfusion rate. This parameter is unknown at the beginning of
each therapy because it depends on the individual patient and the placement of the LITT
applicator relative to the liver. We propose a PDE-constrained optimal-control problem for the
identification of the blood-perfusion rate, prove the existence of an optimal control and prove
necessary first-order optimality conditions. Furthermore, we introduce a numerical example
based on which we demonstrate the algorithmic solution of this problem.
Third, we propose a reformulation of the well-known PN model hierarchy with Marshak
boundary conditions as a coupled system of second-order PDEs to approximate the radiative-transfer
equation. The new model hierarchy is derived in a general context and is applicable
to a wide range of applications other than LITT. It can be generated in an automated way by
means of algebraic transformations and allows the solution with standard finite-element tools.
We validate our formulation in a general context by means of various numerical experiments.
Finally, we investigate the coupling of this new model hierarchy with the LITT model numerically.
Mixed Isogeometric Methods for Hodge–Laplace Problems induced by Second-Order Hilbert Complexes
(2024)
Partial differential equations (PDEs) play a crucial role in mathematics and physics to describe numerous physical processes. In numerical computations within the scope of PDE problems, the transition from classical to weak solutions is often meaningful. The latter may not precisely satisfy the original PDE, but they fulfill a weak variational formulation, which, in turn, is suitable for the discretization concept of Finite Elements (FE). A central concept in this context is the
well-posed problem. A class of PDE problems for which not only well-posedness statements but also suitable weak formulations are known are the so-called abstract Hodge–Laplace problems. These can be derived from Hilbert complexes and constitute a central aspect of the Finite Element Exterior Calculus (FEEC).
This thesis addresses the discretization of mixed formulations of Hodge-Laplace problems, focusing on two key aspects. Firstly, we utilize Isogeometric Analysis (IGA) as a specific paradigm for discretization, combining geometric representations with Non-Uniform Rational B-Splines (NURBS) and Finite Element discretizations.
Secondly, we primarily concentrate on mixed formulations exhibiting a saddle-point structure and generated from Hilbert complexes with second-order derivative operators. We go beyond the well-known case of the classical de Rham
complex, considering complexes such as the Hessian or elasticity complex. The BGG (Bernstein–Gelfand–Gelfand) method is employed to define and examine these second-order complexes. The main results include proofs of discrete well-posedness and a priori error estimates for two different discretization approaches. One approach demonstrates, through the introduction of a Lagrange multiplier, how the so-called isogeometric discrete differential forms can be reused.
A second method addresses the question of how standard NURBS basis functions, through a modification of the mixed formulation, can also lead to convergent procedures. Numerical tests and examples, conducted using MATLAB and the open-source software GeoPDEs, illustrate the theoretical findings. Our primary application extends to linear elasticity theory, extensively
discussing mixed methods with and without strong symmetry of the stress tensor.
The work demonstrates the potential of IGA in numerical computations, particularly in the challenging scenario of second-order Hilbert complexes. It also provides insights into how IGA and FEEC can be meaningfully combined, even for non-de Rham complexes.
In this thesis, we present the basic concepts of isogeometric analysis (IGA) and we consider Poisson's equation as model problem. Since in IGA the physical domain is parametrized via a geometry function that goes from a parameter domain, e.g. the unit square or unit cube, to the physical one, we present a class of parametrizations that can be viewed as a generalization of polar coordinates, known as the scaled boundary parametrizations (SB-parametrizations). These are easy to construct and are particularly attractive when only the boundary of a domain is available. We then present an IGA approach based on these parametrizations, that we call scaled boundary isogeometric analysis (SB-IGA). The SB-IGA derives the weak form of partial differential equations in a different way from the standard IGA. For the discretization projection
on a finite-dimensional space, we choose in both cases Galerkin's method. Thanks to this technique, we state an equivalence theorem for linear elliptic boundary value problems between the standard IGA, when it makes use of an SB-parametrization,
and the SB-IGA. We solve Poisson's equation with Dirichlet boundary conditions on different geometries and with different SB-parametrizations.
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.
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.
For the last decade, optimization of beam orientations in intensity-modulated radiation therapy (IMRT) has been shown to be successful in improving the treatment plan. Unfortunately, the quality of a set of beam orientations depends heavily on its corresponding beam intensity profiles. Usually, a stochastic selector is used for optimizing beam orientation, and then a single objective inverse treatment planning algorithm is used for the optimization of beam intensity profiles. The overall time needed to solve the inverse planning for every random selection of beam orientations becomes excessive. Recently, considerable improvement has been made in optimizing beam intensity profiles by using multiple objective inverse treatment planning. Such an approach results in a variety of beam intensity profiles for every selection of beam orientations, making the dependence between beam orientations and its intensity profiles less important. This thesis takes advantage of this property to accelerate the optimization process through an approximation of the intensity profiles that are used for multiple selections of beam orientations, saving a considerable amount of calculation time. A dynamic algorithm (DA) and evolutionary algorithm (EA), for beam orientations in IMRT planning will be presented. The DA mimics, automatically, the methods of beam's eye view and observer's view which are recognized in conventional conformal radiation therapy. The EA is based on a dose-volume histogram evaluation function introduced as an attempt to minimize the deviation between the mathematical and clinical optima. To illustrate the efficiency of the algorithms they have been applied to different clinical examples. In comparison to the standard equally spaced beams plans, improvements are reported for both algorithms in all the clinical examples even when, for some cases, fewer beams are used. A smaller number of beams is always desirable without compromising the quality of the treatment plan. It results in a shorter treatment delivery time, which reduces potential errors in terms of patient movements and decreases discomfort.
In this thesis we have discussed the problem of decomposing an integer matrix \(A\) into a weighted sum \(A=\sum_{k \in {\mathcal K}} \alpha_k Y^k\) of 0-1 matrices with the strict consecutive ones property. We have developed algorithms to find decompositions which minimize the decomposition time \(\sum_{k \in {\mathcal K}} \alpha_k\) and the decomposition cardinality \(|\{ k \in {\mathcal K}: \alpha_k > 0\}|\). In the absence of additional constraints on the 0-1 matrices \(Y^k\) we have given an algorithm that finds the minimal decomposition time in \({\mathcal O}(NM)\) time. For the case that the matrices \(Y^k\) are restricted to shape matrices -- a restriction which is important in the application of our results in radiotherapy -- we have given an \({\mathcal O}(NM^2)\) algorithm. This is achieved by solving an integer programming formulation of the problem by a very efficient combinatorial algorithm. In addition, we have shown that the problem of minimizing decomposition cardinality is strongly NP-hard, even for matrices with one row (and thus for the unconstrained as well as the shape matrix decomposition). Our greedy heuristics are based on the results for the decomposition time problem and produce better results than previously published algorithms.
Many open problems in graph theory aim to verify that a specific class of graphs has a certain property.
One example, which we study extensively in this thesis, is the 3-decomposition conjecture.
It states that every cubic graph can be decomposed into a spanning tree, cycles, and a matching.
Our most noteworthy contributions to this conjecture are a proof that graphs which are star-like satisfy the conjecture and that several small graphs, which we call forbidden subgraphs, cannot be part of minimal counterexamples.
These star-like graphs are a natural generalisation of Hamiltonian graphs in this context and encompass an infinite family of graphs for which the conjecture was not known previously.
Moreover, we use the forbidden subgraphs we determined to deduce that 3-connected cubic graphs of path-width at most 4 satisfy the 3-decomposition conjecture:
we do this by showing that the path-width restriction causes one of these forbidden subgraphs to appear.
In the second part of this thesis, we delve deeper into two steps of the proof that 3-connected cubic graphs of path-width 4 satisfy the conjecture.
These steps involve a significant amount of case distinctions and, as such, are impractical to extend to larger path-width values.
We show how to formalise the techniques used in such a way that they can be implemented and solved algorithmically.
As a result, only the work that is "interesting" to do remains and the many "straightforward" parts can now be done by a computer.
While one step is specific to the 3-decomposition conjecture, we derive a general algorithm for the other.
This algorithm takes a class of graphs \(\mathcal G\) as an input, together with a set of graphs \(\mathcal U\), and a path-width bound \(k\).
It then attempts to answer the following question:
does any graph in \(\mathcal G\) that has path-width at most \(k\) contain a subgraph in \(\mathcal U\)?
We show that this problem is undecidable in general, so our algorithm does not always terminate, but we also provide a general criterion that guarantees termination.
In the final part of this thesis we investigate two connectivity problems on directed graphs.
We prove that verifying the existence of an \(st\)-path in a local certification setting, cannot be achieved with a constant number of bits.
More precisely, we show that a proof labelling scheme needs \(\Theta(\log \Delta)\) many bits, where \(\Delta\) denotes the maximum degree.
Furthermore, we investigate the complexity of the separating by forbidden pairs problem, which asks for the smallest number of arc pairs that are needed such that any \(st\)-path completely contains at least one such pair.
We show that the corresponding decision problem in \(\mathsf{\Sigma_2P}\)-complete.
In the first part of this work, called Simple node singularity, are computed matrix factorizations of all isomorphism classes, up to shiftings, of rank one and two, graded, indecomposable maximal Cohen--Macaulay (shortly MCM) modules over the affine cone of the simple node singularity. The subsection 2.2 contains a description of all rank two graded MCM R-modules with stable sheafification on the projective cone of R, by their matrix factorizations. It is given also a general description of such modules, of any rank, over a projective curve of arithmetic genus 1, using their matrix factorizations. The non-locally free rank two MCM modules are computed using an alghorithm presented in the Introduction of this work, that gives a matrix factorization of any extension of two MCM modules over a hypersurface. In the second part, called Fermat surface, are classified all graded, rank two, MCM modules over the affine cone of the Fermat surface. For the classification of the orientable rank two graded MCM R-modules, is used a description of the orientable modules (over normal rings) with the help of codimension two Gorenstein ideals, realized by Herzog and Kühl. It is proven (in section 4), that they have skew symmetric matrix factorizations (over any normal hypersurface ring). For the classification of the non-orientable rank two MCM R-modules, we use a similar idea as in the case of the orientable ones, only that the ideal is not any more Gorenstein.
Methods for scale and orientation invariant analysis of lower dimensional structures in 3d images
(2023)
This thesis is motivated by two groups of scientific disciplines: engineering sciences and mathematics. On the one hand, engineering sciences such as civil engineering want to design sustainable and cost-effective materials with desirable mechanical properties. The material behaviour depends on physical properties and production parameters. Therefore, physical properties are measured experimentally from real samples. In our case, computed tomography (CT) is used to non-destructively gain insight into the materials’ microstructure. This results in large 3d images which yield information on geometric microstructure characteristics. On the other hand, mathematical sciences are interested in designing methods with suitable and guaranteed properties. For example, a natural assumption of human vision is to analyse images regardless of object position, orientation, or scale. This assumption is formalized through the concepts of equivariance and invariance.
In Part I, we deal with oriented structures in materials such as concrete or fiber-reinforced composites. In image processing, knowledge of the local structure orientation can be used for various tasks, e.g. structure enhancement. The idea of using banks of directed filters parameterized in the orientation space is effective in 2d. However, this class of methods is prohibitive in 3d due to the high computational burden of filtering when using a fine discretization of the unit sphere. Hence, we introduce a method for 3d pixel-wise orientation estimation and directional filtering inspired by the idea of adaptive refinement in discretized settings. Furthermore, an operator for distinction between isotropic and anisotropic structures is defined based on our method. Finally, usefulness of the method is shown on 3d CT images in three different tasks on a fiber-reinforced polymer, concrete with cracks, and partially closed foams. Additionally, our method is extended to construct line granulometry and characterize fiber length and orientation distributions in fiber-reinforced polymers produced by either 3d printing or by injection moulding.
In Part II, we investigate how to introduce scale invariance for neural networks by using the Riesz transform. In classical convolutional neural networks, scale invariance is typically achieved by data augmentation. However, when presented with a scale far outside the range covered by the training set, the network may fail to generalize. Here, we introduce the Riesz network, a novel scale invariant neural network. Instead of standard 2d or 3d convolutions for combining spatial information, the Riesz network is based on the Riesz transform, a scale equivariant operator. As a consequence, this network naturally generalizes to unseen or even arbitrary scales in a single forward pass. As an application example, we consider segmenting cracks in CT images of concrete. In this context, 'scale' refers to the crack thickness which may vary strongly even within the same sample. To prove its scale invariance, the Riesz network is trained on one fixed crack width. We then validate its performance in segmenting simulated and real CT images featuring a wide range of crack widths. As an alternative to deep learning models, the Riesz transform is utilized to construct a scale equivariant scattering network, which does not require a lengthy training procedure and works with very few training examples. Mathematical foundations behind this representation are laid out and analyzed. We show that this representation with 4 times less features than the original scattering networks from Mallat performs comparably well on texture classification and gives superior performance when dealing with scales outside the training set distribution.
The nowadays increasing number of fields where large quantities of data are collected generates an emergent demand for methods for extracting relevant information from huge databases. Amongst the various existing data mining models, decision trees are widely used since they represent a good trade-off between accuracy and interpretability. However, one of their main problems is that they are very instable, which complicates the process of the knowledge discovery because the users are disturbed by the different decision trees generated from almost the same input learning samples. In the current work, binary tree classifiers are analyzed and partially improved. The analysis of tree classifiers goes from their topology from the graph theory point of view to the creation of a new tree classification model by means of combining decision trees and soft comparison operators (Mlynski, 2003) with the purpose to not only overcome the well known instability problem of decision trees, but also in order to confer the ability of dealing with uncertainty. In order to study and compare the structural stability of tree classifiers, we propose an instability coefficient which is based on the notion of Lipschitz continuity and offer a metric to measure the proximity between decision trees. This thesis converges towards its main part with the presentation of our model ``Soft Operators Decision Tree\'\' (SODT). Mainly, we describe its construction, application and the consistency of the mathematical formulation behind this. Finally we show the results of the implementation of SODT and compare numerically the stability and accuracy of a SODT and a crisp DT. The numerical simulations support the stability hypothesis and a smaller tendency to overfitting the training data with SODT than with crisp DT is observed. A further aspect of this inclusion of soft operators is that we choose them in a way so that the resulting goodness function (used by this method) is differentiable and thus allows to calculate the best split points by means of gradient descent methods. The main drawback of SODT is the incorporation of the unpreciseness factor, which increases the complexity of the algorithm.
Nowadays one of the major objectives in geosciences is the determination of the gravitational field of our planet, the Earth. A precise knowledge of this quantity is not just interesting on its own but it is indeed a key point for a vast number of applications. The important question is how to obtain a good model for the gravitational field on a global scale. The only applicable solution - both in costs and data coverage - is the usage of satellite data. We concentrate on highly precise measurements which will be obtained by GOCE (Gravity Field and Steady State Ocean Circulation Explorer, launch expected 2006). This satellite has a gradiometer onboard which returns the second derivatives of the gravitational potential. Mathematically seen we have to deal with several obstacles. The first one is that the noise in the different components of these second derivatives differs over several orders of magnitude, i.e. a straightforward solution of this outer boundary value problem will not work properly. Furthermore we are not interested in the data at satellite height but we want to know the field at the Earth's surface, thus we need a regularization (downward-continuation) of the data. These two problems are tackled in the thesis and are now described briefly. Split Operators: We have to solve an outer boundary value problem at the height of the satellite track. Classically one can handle first order side conditions which are not tangential to the surface and second derivatives pointing in the radial direction employing integral and pseudo differential equation methods. We present a different approach: We classify all first and purely second order operators which fulfill that a harmonic function stays harmonic under their application. This task is done by using modern algebraic methods for solving systems of partial differential equations symbolically. Now we can look at the problem with oblique side conditions as if we had ordinary i.e. non-derived side conditions. The only additional work which has to be done is an inversion of the differential operator, i.e. integration. In particular we are capable to deal with derivatives which are tangential to the boundary. Auto-Regularization: The second obstacle is finding a proper regularization procedure. This is complicated by the fact that we are facing stochastic rather than deterministic noise. The main question is how to find an optimal regularization parameter which is impossible without any additional knowledge. However we could show that with a very limited number of additional information, which are obtainable also in practice, we can regularize in an asymptotically optimal way. In particular we showed that the knowledge of two input data sets allows an order optimal regularization procedure even under the hard conditions of Gaussian white noise and an exponentially ill-posed problem. A last but rather simple task is combining data from different derivatives which can be done by a weighted least squares approach using the information we obtained out of the regularization procedure. A practical application to the downward-continuation problem for simulated gravitational data is shown.
This thesis covers two important fields in financial mathematics, namely the continuous time portfolio optimisation and credit risk modelling. We analyse optimisation problems of portfolios of Call and Put options on the stock and/or the zero coupon bond issued by a firm with default risk. We use the martingale approach for dynamic optimisation problems. Our findings show that the riskier the option gets, the less proportion of his wealth the investor allocates to the risky asset. Further, we analyse the Credit Default Swap (CDS) market quotes on the Eurobonds issued by Turkish sovereign for building the term structure of the sovereign credit risk. Two methods are introduced and compared for bootstrapping the risk-neutral probabilities of default (PD) in an intensity based (or reduced form) credit risk modelling approach. We compare the market-implied PDs with the actual PDs reported by credit rating agencies based on historical experience. Our results highlight the market price of the sovereign credit risk depending on the assigned rating category in the sampling period. Finally, we find an optimal leverage strategy for delivering the payments promised by a Constant Proportion Debt Obligation (CPDO). The problem is solved via the introduction and explicit solution of a stochastic control problem by transforming the related Hamilton-Jacobi-Bellman Equation into its dual. Contrary to the industry practise, the optimal leverage function we derive is a non-linear function of the CPDO asset value. The simulations show promising behaviour of the optimal leverage function compared with the one popular among practitioners.
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.
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 thesis diverse problems concerning inflation-linked products are dealt with. To start with, two models for inflation are presented, including a geometric Brownian motion for consumer price index itself and an extended Vasicek model for inflation rate. For both suggested models the pricing formulas of inflation-linked products are derived using the risk-neutral valuation techniques. As a result Black and Scholes type closed form solutions for a call option on inflation index for a Brownian motion model and inflation evolution for an extended Vasicek model as well as for an inflation-linked bond are calculated. These results have been already presented in Korn and Kruse (2004) [17]. In addition to these inflation-linked products, for the both inflation models the pricing formulas of a European put option on inflation, an inflation cap and floor, an inflation swap and an inflation swaption are derived. Consequently, basing on the derived pricing formulas and assuming the geometric Brownian motion process for an inflation index, different continuous-time portfolio problems as well as hedging problems are studied using the martingale techniques as well as stochastic optimal control methods. These utility optimization problems are continuous-time portfolio problems in different financial market setups and in addition with a positive lower bound constraint on the final wealth of the investor. When one summarizes all the optimization problems studied in this work, one will have the complete picture of the inflation-linked market and both counterparts of market-participants, sellers as well as buyers of inflation-linked financial products. One of the interesting results worth mentioning here is naturally the fact that a regular risk-averse investor would like to sell and not buy inflation-linked products due to the high price of inflation-linked bonds for example and an underperformance of inflation-linked bonds compared to the conventional risk-free bonds. The relevance of this observation is proved by investigating a simple optimization problem for the extended Vasicek process, where as a result we still have an underperforming inflation-linked bond compared to the conventional bond. This situation does not change, when one switches to an optimization of expected utility from the purchasing power, because in its nature it is only a change of measure, where we have a different deflator. The negativity of the optimal portfolio process for a normal investor is in itself an interesting aspect, but it does not affect the optimality of handling inflation-linked products compared to the situation not including these products into investment portfolio. In the following, hedging problems are considered as a modeling of the other half of inflation market that is inflation-linked products buyers. Natural buyers of these inflation-linked products are obviously institutions that have payment obligations in the future that are inflation connected. That is why we consider problems of hedging inflation-indexed payment obligations with different financial assets. The role of inflation-linked products in the hedging portfolio is shown to be very important by analyzing two alternative optimal hedging strategies, where in the first one an investor is allowed to trade as inflation-linked bond and in the second one he is not allowed to include an inflation-linked bond into his hedging portfolio. Technically this is done by restricting our original financial market, which is made of a conventional bond, inflation index and a stock correlated with inflation index, to the one, where an inflation index is excluded. As a whole, this thesis presents a wide view on inflation-linked products: inflation modeling, pricing aspects of inflation-linked products, various continuous-time portfolio problems with inflation-linked products as well as hedging of inflation-related payment obligations.
This thesis is primarily motivated by a project with Deutsche Bahn about offer preparation in rail freight transport. At its core, a customer should be offered three train paths to choose from in response to a freight train request. As part of this cooperation with DB Netz AG, we investigated how to compute these train paths efficiently. They should be all "good" but also "as different as possible". We solved this practical problem using combinatorial optimization techniques.
At the beginning of this thesis, we describe the practical aspects of our research collaboration. The more theoretical problems, which we consider afterwards, are divided into two parts.
In Part I, we deal with a dual pair of problems on directed graphs with two designated end-vertices. The Almost Disjoint Paths (ADP) problem asks for a maximum number of paths between the end-vertices any two of which have at most one arc in common. In comparison, for the Separating by Forbidden Pairs (SFP) problem we have to select as few arc pairs as possible such that every path between the end-vertices contains both arcs of a chosen pair. The main results of this more theoretical part are the classifications of ADP as an NP-complete and SFP as a Sigma-2-P-complete problem.
In Part II, we address a simplified version of the practical project: the Fastest Path with Time Profiles and Waiting (FPTPW) problem. In a directed acyclic graph with durations on the arcs and time windows at the vertices, we search for a fastest path from a source to a target vertex. We are only allowed to be at a vertex within its time windows, and we are only allowed to wait at specified vertices. After introducing departure-duration functions we develop solution algorithms based on these. We consider special cases that significantly reduce the complexity or are of practical relevance. Furthermore, we show that already this simplified problem is in general NP-hard and investigate the complexity status more closely.
Solving probabilistic-robust optimization problems using methods from semi-infinite optimization
(2023)
Optimization under uncertainty is one field of mathematics which is strongly inspired by real world problems. To handle uncertainties several models have arisen. One of these is the probust model where a combination of probabilistic and worst-case uncertainty is considered. So far, just problem instances with a special structure can be dealt with. In this thesis, we introduce solving techniques applicable for any probust optimization problem. On the one hand, we create upper bounds for the solution value by solving a sequence of chance constrained optimization problems. These bounds are based on discretization schemes which are inspired by semi-infinite optimization. On the other hand, we create lower bounds by solving a sequence of set-approximation problems. Here, we substitute the original event set by an appropriate family of sets. We examine the performance of the corresponding algorithms on simple packing problems where we can provide the probust solution analytically. Afterwards, we solve a water reservoir and a distillation problem and compare the probust solutions with solutions arising from other uncertainty models.
This thesis concerns itself with the long-term behavior of generalized Langevin dynamics with multiplicative noise,
i.e. the solutions to a class of two-component stochastic differential equations in \( \mathbb{R}^{d_1}\times\mathbb{R}^{d_2} \)
subject to outer influence induced by potentials \( \Phi \) and \( \Psi \),
where the stochastic term is only present in the second component, on which it is dependent.
In particular, convergence to an equilibrium defined by an invariant initial distribution \( \mu \) is shown
for weak solutions to the generalized Langevin equation obtained via generalized Dirichlet forms,
and the convergence rate is estimated by applying hypocoercivity methods relying on weak or classical Poincaré inequalities.
As a prerequisite, the space of compactly supported smooth functions is proven to be a domain of essential m-dissipativity
for the associated Kolmogorov backward operator on \(L^2(\mu)\).
In the second part of the thesis, similar Langevin dynamics are considered, however defined on a product of infinite-dimensional separable Hilbert spaces.
The set of finitely based smooth bounded functions is shown to be a domain of essential m-dissipativity for the corresponding Kolmogorov operator \( L \) on \( L^2(\mu) \)
for a Gaussian measure \( \mu \), by applying the previous finite-dimensional result to appropriate restrictions of \( L \).
Under further bounding conditions on the diffusion coefficient relative to the covariance operators of \( \mu \),
hypocoercivity of the generated semigroup is proved, as well as the existence of an associated weakly continuous Markov process
which analytically weakly provides a weak solution to the considered Langevin equation.
Many loads acting on a vehicle depend on the condition and quality of roads
traveled as well as on the driving style of the motorist. Thus, during vehicle development,
good knowledge on these further operations conditions is advantageous.
For that purpose, usage models for different kinds of vehicles are considered. Based
on these mathematical descriptions, representative routes for multiple user
types can be simulated in a predefined geographical region. The obtained individual
driving schedules consist of coordinates of starting and target points and can
thus be routed on the true road network. Additionally, different factors, like the
topography, can be evaluated along the track.
Available statistics resulting from travel survey are integrated to guarantee reasonable
trip length. Population figures are used to estimate the number of vehicles in
contained administrative units. The creation of thousands of those geo-referenced
trips then allows the determination of realistic measures of the durability loads.
Private as well as commercial use of vehicles is modeled. For the former, commuters
are modeled as the main user group conducting daily drives to work and
additional leisure time a shopping trip during workweek. For the latter, taxis as
example for users of passenger cars are considered. The model of light-duty commercial
vehicles is split into two types of driving patterns, stars and tours, and in
the common traffic classes of long-distance, local and city traffic.
Algorithms to simulate reasonable target points based on geographical and statistical
data are presented in detail. Examples for the evaluation of routes based
on topographical factors and speed profiles comparing the influence of the driving
style are included.
The high complexity of civil engineering structures makes it difficult to satisfactorily evaluate their reliability. However, a good risk assessment of such structures is incredibly important to avert dangers and possible disasters for public life. For this purpose, we need algorithms that reliably deliver estimates for their failure probabilities with high efficiency and whose results enable a better understanding of their reliability. This is a major challenge, especially when dynamics, for example due to uncertainties or time-dependent states, must be included in the model.
The contributions are centered around Subset Simulation, a very popular adaptive Monte Carlo method for reliability analysis in the engineering sciences. It particularly well estimates small failure probabilities in high dimensions and is therefore tailored to the demands of many complex problems. We modify Subset Simulation and couple it with interpolation methods in order to keep its remarkable properties and receive all conditional failure probabilities with respect to one variable of the structural reliability model. This covers many sorts of model dynamics with several model constellations, such as time-dependent modeling, sensitivity and uncertainty, in an efficient way, requiring similar computational demands as a static reliability analysis for one model constellation by Subset Simulation. The algorithm offers many new opportunities for reliability evaluation and can even be used to verify results of Subset Simulation by artificially manipulating the geometry of the underlying limit state in numerous ways, allowing to provide correct results where Subset Simulation systematically fails. To improve understanding and further account for model uncertainties, we present a new visualization technique that matches the extensive information on reliability we get as a result from the novel algorithm.
In addition to these extensions, we are also dedicated to the fundamental analysis of Subset Simulation, partially bridging the gap between theory and results by simulation where inconsistencies exist. Based on these findings, we also extend practical recommendations on selection of the intermediate probability with respect to the implementation of the algorithm and derive a formula for correction of the bias. For a better understanding, we also provide another stochastic interpretation of the algorithm and offer alternative implementations which stick to the theoretical assumptions, typically made in analysis.
Adjoint-Based Shape Optimization and Optimal Control with Applications to Microchannel Systems
(2021)
Optimization problems constrained by partial differential equations (PDEs) play an important role in many areas of science and engineering. They often arise in the optimization of technological applications, where the underlying physical effects are modeled by PDEs. This thesis investigates such problems in the context of shape optimization and optimal control with microchannel systems as novel applications. Such systems are used, e.g., as cooling systems, heat exchangers, or chemical reactors as their high surface-to-volume ratio, which results in beneficial heat and mass transfer characteristics, allows them to excel in these settings. Additionally, this thesis considers general PDE constrained optimization problems with particular regard to their efficient solution.
As our first application, we study a shape optimization problem for a microchannel cooling system: We rigorously analyze this problem, prove its shape differentiability, and calculate the corresponding shape derivative. Afterwards, we consider the numerical optimization of the cooling system for which we employ a hierarchy of reduced models derived via porous medium modeling and a dimension reduction technique. A comparison of the models in this context shows that the reduced models approximate the original one very accurately while requiring substantially less computational resources.
Our second application is the optimization of a chemical microchannel reactor for the Sabatier process using techniques from PDE constrained optimal control. To treat this problem, we introduce two models for the reactor and solve a parameter identification problem to determine the necessary kinetic reaction parameters for our models. Thereafter, we consider the optimization of the reactor's operating conditions with the objective of improving its product yield, which shows considerable potential for enhancing the design of the reactor.
To provide efficient solution techniques for general shape optimization problems, we introduce novel nonlinear conjugate gradient methods for PDE constrained shape optimization and analyze their performance on several well-established benchmark problems. Our results show that the proposed methods perform very well, making them efficient and appealing gradient-based shape optimization algorithms.
Finally, we continue recent software-based developments for PDE constrained optimization and present our novel open-source software package cashocs. Our software implements and automates the adjoint approach and, thus, facilitates the solution of general PDE constrained shape optimization and optimal control problems. Particularly, we highlight our software's user-friendly interface, straightforward applicability, and mesh independent behavior.
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 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.
In this thesis we classify simple coherent sheaves on Kodaira fibers of types II, III and IV (cuspidal and tacnode cubic curves and a plane configuration of three concurrent lines). Indecomposable vector bundles on smooth elliptic curves were classified in 1957 by Atiyah. In works of Burban, Drozd and Greuel it was shown that the categories of vector bundles and coherent sheaves on cycles of projective lines are tame. It turns out, that all other degenerations of elliptic curves are vector-bundle-wild. Nevertheless, we prove that the category of coherent sheaves of an arbitrary reduced plane cubic curve, (including the mentioned Kodaira fibers) is brick-tame. The main technical tool of our approach is the representation theory of bocses. Although, this technique was mainly used for purely theoretical purposes, we illustrate its computational potential for investigating tame behavior in wild categories. In particular, it allows to prove that a simple vector bundle on a reduced cubic curve is determined by its rank, multidegree and determinant, generalizing Atiyah's classification. Our approach leads to an interesting class of bocses, which can be wild but are brick-tame.
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 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.
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.
An increasing number of nowadays tasks, such as speech recognition, image generation,
translation, classification or prediction are performed with the help of machine learning.
Especially artificial neural networks (ANNs) provide convincing results for these tasks.
The reasons for this success story are the drastic increase of available data sources in
our more and more digitalized world as well as the development of remarkable ANN
architectures. This development has led to an increasing number of model parameters
together with more and more complex models. Unfortunately, this yields a loss in the
interpretability of deployed models. However, there exists a natural desire to explain the
deployed models, not just by empirical observations but also by analytical calculations.
In this thesis, we focus on variational autoencoders (VAEs) and foster the understanding
of these models. As the name suggests, VAEs are based on standard autoencoders (AEs)
and therefore used to perform dimensionality reduction of data. This is achieved by a
bottleneck structure within the hidden layers of the ANN. From a data input the encoder,
that is the part up to the bottleneck, produces a low dimensional representation. The
decoder, the part from the bottleneck to the output, uses this representation to reconstruct
the input. The model is learned by minimizing the error from the reconstruction.
In our point of view, the most remarkable property and, hence, also a central topic
in this thesis is the auto-pruning property of VAEs. Simply speaking, the auto-pruning
is preventing the VAE with thousands of parameters from overfitting. However, such a
desirable property comes with the risk for the model of learning nothing at all. In this
thesis, we look at VAEs and the auto-pruning from two different angles and our main
contributions to research are the following:
(i) We find an analytic explanation of the auto-pruning. We do so, by leveraging the
framework of generalized linear models (GLMs). As a result, we are able to explain
training results of VAEs before conducting the actual training.
(ii) We construct a time dependent VAE and show the effects of the auto-pruning in
this model. As a result, we are able to model financial data sequences and estimate
the value-at-risk (VaR) of associated portfolios. Our results show that we surpass
the standard benchmarks for VaR estimation.
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.
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 is devoted to applying symbolic methods to the problems of decoding linear codes and of algebraic cryptanalysis. The paradigm we employ here is as follows. We reformulate the initial problem in terms of systems of polynomial equations over a finite field. The solution(s) of such systems should yield a way to solve the initial problem. Our main tools for handling polynomials and polynomial systems in such a paradigm is the technique of Gröbner bases and normal form reductions. The first part of the thesis is devoted to formulating and solving specific polynomial systems that reduce the problem of decoding linear codes to the problem of polynomial system solving. We analyze the existing methods (mainly for the cyclic codes) and propose an original method for arbitrary linear codes that in some sense generalizes the Newton identities method widely known for cyclic codes. We investigate the structure of the underlying ideals and show how one can solve the decoding problem - both the so-called bounded decoding and more general nearest codeword decoding - by finding reduced Gröbner bases of these ideals. The main feature of the method is that unlike usual methods based on Gröbner bases for "finite field" situations, we do not add the so-called field equations. This tremendously simplifies the underlying ideals, thus making feasible working with quite large parameters of codes. Further we address complexity issues, by giving some insight to the Macaulay matrix of the underlying systems. By making a series of assumptions we are able to provide an upper bound for the complexity coefficient of our method. We address also finding the minimum distance and the weight distribution. We provide solid experimental material and comparisons with some of the existing methods in this area. In the second part we deal with the algebraic cryptanalysis of block iterative ciphers. Namely, we analyze the small-scale variants of the Advanced Encryption Standard (AES), which is a widely used modern block cipher. Here a cryptanalyst composes the polynomial systems which solutions should yield a secret key used by communicating parties in a symmetric cryptosystem. We analyze the systems formulated by researchers for the algebraic cryptanalysis, and identify the problem that conventional systems have many auxiliary variables that are not actually needed for the key recovery. Moreover, having many such auxiliary variables, specific to a given plaintext/ciphertext pair, complicates the use of several pairs which is common in cryptanalysis. We thus provide a new system where the auxiliary variables are eliminated via normal form reductions. The resulting system in key-variables only is then solved. We present experimental evidence that such an approach is quite good for small scaled ciphers. We investigate further our approach and employ the so-called meet-in-the-middle principle to see how far one can go in analyzing just 2-3 rounds of scaled ciphers. Additional "tuning techniques" are discussed together with experimental material. Overall, we believe that the material of this part of the thesis makes a step further in algebraic cryptanalysis of block ciphers.
Diese Arbeit gehört in die algebraische Geometrie und die Darstellungstheorie und stellt eine Beziehung zwischen beiden Gebieten dar. Man beschäftigt sich mit den abgeleiteten Kategorien auf flachen Entartungen projektiver Geraden und elliptischer Kurven. Als Mittel benutzt man die Technik der Matrixprobleme. Das Hauptergebnis dieser Dissertation ist der folgende Satz: SATZ. Sei X ein Zykel projektiver Geraden. Dann gibt es drei Typen unzerlegbarer Objekte in D^-(Coh_X): - Shifts von Wolkenkratzergarben in einem regulären Punkt; - Bänder B(w,m,lambda), - Saiten S(w). Ganz analog beweist man die Zahmheit der abgeleiteten Kategorien vieler assoziativer Algebren.
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.
This study deals with the optimal control problems of the glass tube drawing processes where the aim is to control the cross-sectional area (circular) of the tube by using the adjoint variable approach. The process of tube drawing is modeled by four coupled nonlinear partial differential equations. These equations are derived by the axisymmetric Stokes equations and the energy equation by using the approach based on asymptotic expansions with inverse aspect ratio as small parameter. Existence and uniqueness of the solutions of stationary isothermal model is also proved. By defining the cost functional, we formulated the optimal control problem. Then Lagrange functional associated with minimization problem is introduced and the first and the second order optimality conditions are derived. We also proved the existence and uniqueness of the solutions of the stationary isothermal model. We implemented the optimization algorithms based on the steepest descent, nonlinear conjugate gradient, BFGS, and Newton approaches. In the Newton method, CG iterations are introduced to solve the Newton equation. Numerical results are obtained for two different cases. In the first case, the cross-sectional area for the entire time domain is controlled and in the second case, the area at the final time is controlled. We also compared the performance of the optimization algorithms in terms of the solution iterations, functional evaluations and the computation time.
The lattice Boltzmann method (LBM) is a numerical solver for the Navier-Stokes equations, based on an underlying molecular dynamic model. Recently, it has been extended towardsthe simulation of complex fluids. We use the asymptotic expansion technique to investigate the standard scheme, the initialization problem and possible developments towards moving boundary and fluid-structure interaction problems. At the same time, it will be shown how the mathematical analysis can be used to understand and improve the algorithm. First of all, we elaborate the tool "asymptotic analysis", proposing a general formulation of the technique and explaining the methods and the strategy we use for the investigation. A first standard application to the LBM is described, which leads to the approximation of the Navier-Stokes solution starting from the lattice Boltzmann equation. As next, we extend the analysis to investigate origin and dynamics of initial layers. A class of initialization algorithms to generate accurate initial values within the LB framework is described in detail. Starting from existing routines, we will be able to improve the schemes in term of efficiency and accuracy. Then we study the features of a simple moving boundary LBM. In particular, we concentrate on the initialization of new fluid nodes created by the variations of the computational fluid domain. An overview of existing possible choices is presented. Performing a careful analysis of the problem we propose a modified algorithm, which produces satisfactory results. Finally, to set up an LBM for fluid structure interaction, efficient routines to evaluate forces are required. We describe the Momentum Exchange algorithm (MEA). Precise accuracy estimates are derived, and the analysis leads to the construction of an improved method to evaluate the interface stresses. In conclusion, we test the defined code and validate the results of the analysis on several simple benchmarks. From the theoretical point of view, in the thesis we have developed a general formulation of the asymptotic expansion, which is expected to offer a more flexible tool in the investigation of numerical methods. The main practical contribution offered by this work is the detailed analysis of the numerical method. It allows to understand and improve the algorithms, and construct new routines, which can be considered as starting points for future researches.
In this thesis, we deal with the worst-case portfolio optimization problem occuring in discrete-time markets.
First, we consider the discrete-time market model in the presence of crash threats. We construct the discrete worst-case optimal portfolio strategy by the indifference principle in the case of the logarithmic utility. After that we extend this problem to general utility functions and derive the discrete worst-case optimal portfolio processes, which are characterized by a dynamic programming equation. Furthermore, the convergence of the discrete worst-case optimal portfolio processes are investigated when we deal with the explicit utility functions.
In order to further study the relation of the worst-case optimal value function in discrete-time models to continuous-time models we establish the finite-difference approach. By deriving the discrete HJB equation we verify the worst-case optimal value function in discrete-time models, which satisfies a system of dynamic programming inequalities. With increasing degree of fineness of the time discretization, the convergence of the worst-case value function in discrete-time models to that in continuous-time models are proved by using a viscosity solution method.
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.
We study a multi-scale model for growth of malignant gliomas in the human brain.
Interactions of individual glioma cells with their environment determine the gross tumor shape.
We connect models on different time and length scales to derive a practical description of tumor growth that takes these microscopic interactions into account.
From a simple subcellular model for haptotactic interactions of glioma cells with the white matter we derive a microscopic particle system, which leads to a meso-scale model for the distribution of particles, and finally to a macroscopic description of the cell density.
The main body of this work is dedicated to the development and study of numerical methods adequate for the meso-scale transport model and its transition to the macroscopic limit.