## Doctoral Thesis

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In this dissertation a model of melt spinning (by Doufas, McHugh and Miller) has been investigated. The model (DMM model) which takes into account effects of inertia, air drag, gravity and surface tension in the momentum equation and heat exchange between air and fibre surface, viscous dissipation and crystallization in the energy equation also has a complicated coupling with the microstructure. The model has two parts, before onset of crystallization (BOC) and after onset of crystallization (AOC) with the point of onset of crystallization as the unknown interface. Mathematically the model has been formulated as a Free boundary value problem. Changes have been introduced in the model with respect to the air drag and an interface condition at the free boundary. The mathematical analysis of the nonlinear, coupled free boundary value problem shows that the solution of this problem depends heavily on initial conditions and parameters which renders the global analysis impossible. But by defining a physically acceptable solution, it is shown that for a more restricted set of initial conditions if a unique solution exists for IVP BOC then it is physically acceptable. For this the important property of the positivity of the conformation tensor variables has been proved. Further it is shown that if a physically acceptable solution exists for IVP BOC then under certain conditions it also exists for IVP AOC. This gives an important relation between the initial conditions of IVP BOC and the existence of a physically acceptable solution of IVP AOC. A new investigation has been done for the melt spinning process in the framework of classical mechanics. A Hamiltonian formulation has been done for the melt spinning process for which appropriate Poisson brackets have been derived for the 1-d, elongational flow of a viscoelastic fluid. From the Hamiltonian, cross sectionally averaged balance mass and momentum equations of melt spinning can be derived along with the microstructural equations. These studies show that the complicated problem of melt spinning can also be studied under the framework of classical mechanics. This work provides the basic groundwork on which further investigations on the dynamics of a fibre could be carried out. The Free boundary value problem has been solved numerically using shooting method. Matlab routines have been used to solve the IVPs arising in the problem. Some numerical case studies have been done to study the sensitivity of the ODE systems with respect to the initial guess and parameters. These experiments support the analysis done and throw more light on the stiff nature and ill posedness of the ODE systems. To validate the model, simulations have been performed on sets of data provided by the company. Comparison of numerical results (axial velocity profiles) has been done with the experimental profiles provided by the company. Numerical results have been found to be in excellent agreement with the experimental profiles.

This thesis investigates the constrained form of the spherical Minimax location problem and the spherical Weber location problem. Specifically, we consider the problem of locating a new facility on the surface of the unit sphere in the presence of convex spherical polygonal restricted regions and forbidden regions such that the maximum weighted distance from the new facility on the surface of the unit sphere to m existing facilities is minimized and the sum of the weighted distance from the new facility on the surface of the unit sphere to m existing facilities is minimized. It is assumed that a forbidden region is an area on the surface of the unit sphere where travel and facility location are not permitted and that distance is measured using the great circle arc distance. We represent a polynomial time algorithm for the spherical Minimax location problem for the special case where all the existing facilities are located on the surface of a hemisphere. Further, we have developed algorithms for spherical Weber location problem using barrier distance on a hemisphere as well as on the unit sphere.

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

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

Die vorliegende Arbeit wurde angeregt durch die in A.N. Borodin(2000) [Version of the Feynman-Kac Formula. Journal of Mathematical Sciences, 99(2):1044-1052, 2000] und in B. Simon(2000) [A Feynman-Kac Formula for Unbounded Semigroups. Canadian Math. Soc. Conf. Proc., 28:317-321, 2000] dargestellten Feynman-Kac-Formeln. Sie beschäftigt sich mit dem Problem, den Geltungsbereich der Feynman-Kac-Formel im Hinblick auf die Bedingungen der Potentiale und der Anfangsbedingung der zugehörigen partiellen Differentialgleichung zu erweitern. Es ist bekannt, dass die Feynman-Kac-Formel für beschränkte Potentiale gilt. Ausserdem gilt sie auch für Anfangsbedingungen, die im Raum \(C_{0}(\mathbb{R}^{n})\) oder im Raum \(C_{c}^{2}(\mathbb{R}^{n})\) liegen. Die Darstellung der Feynman-Kac-Formel für die Anfangsbedingung, die im Raum \(C_{c}^{2}(\mathbb{R}^{n})\) liegt, liefert die Lösung der partiellen Differentialgleichung. Wir können sie auch als stark stetige Halbgruppe auf dem Raum \(C_{0}(\mathbb{R}^{n})\) auffassen. Diese zwei verschiedenen Darstellungen sind äquivalent. In dieser Arbeit zeigen wir zunächst, dass die Feynman-Kac-Formel auch für unbeschränkte Potentiale \(V\) gilt, wobei \(|V(x)| \leq \varepsilon ||x||^{2} + C_{\varepsilon} \) für alle \(\varepsilon > 0; C_{\varepsilon} > 0\) und \(x \in \mathbb{R}^{n}\) ist. Ausserdem zeigen wir, dass sie für alle Anfangsbedingungen \(f\) gilt mit \(x \mapsto e^{-\varepsilon |x|^{2}} f(x) \in H^{2,2}(\mathbb{R}^{n})\). Der Beweis ist wahrscheinlichkeitstheoretisch und benutzt keine Spektraltheorie. Der spektraltheoretische Zugang, in dem eine Darstellung des Operators \(e^{-tH}\), wobei \(H = -\frac{1}{2} \Delta + V\) gegeben wird, wurde von B. Simon(2000) auch auf die obige Klasse von Potentialen ausgeweitet. Wir lassen zusätzlich auch Potentiale der Form \(V = V_{1} + V_{2}\) zu, wobei \(V_{1} \in L^{2}(\mathbb{R}^{3})\) ist und für alle \(\varepsilon > 0\) gibt es \(C_{\varepsilon} > 0\), so dass \(|V_{2}(x)| \leq\varepsilon ||x||^{2} + C_{\varepsilon}\) für alle \(x \in \mathbb{R}^{3}\) ist. Im Gegensatz zur klassischen Situation ist \(e^{-tH}\) jetzt ein unbeschränkter Operator. Schließlich wird in dieser Arbeit auch der Zusammenhang zwischen der Feynman-Kac-It\(\hat{o}\)-Formel, der Feynman-Kac-Formel und der Kolmogorov-Rückwärtsgleichung untersucht.

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

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.

It is considered an analytical model of defaultable bond portfolio in terms of its face value process. The face value process dynamically evolves with time and incorporates changes caused by recovery payment on default followed by purchasing of new bonds. The further studies involve properties, distribution and control of the face value process.

This thesis contains the mathematical treatment of a special class of analog microelectronic circuits called translinear circuits. The goal is to provide foundations of a new coherent synthesis approach for this class of circuits. The mathematical methods of the suggested synthesis approach come from graph theory, combinatorics, and from algebraic geometry, in particular symbolic methods from computer algebra. Translinear circuits form a very special class of analog circuits, because they rely on nonlinear device models, but still allow a very structured approach to network analysis and synthesis. Thus, translinear circuits play the role of a bridge between the "unknown space" of nonlinear circuit theory and the very well exploited domain of linear circuit theory. The nonlinear equations describing the behavior of translinear circuits possess a strong algebraic structure that is nonetheless flexible enough for a wide range of nonlinear functionality. Furthermore, translinear circuits offer several technical advantages like high functional density, low supply voltage and insensitivity to temperature. This unique profile is the reason that several authors consider translinear networks as the key to systematic synthesis methods for nonlinear circuits. The thesis proposes the usage of a computer-generated catalog of translinear network topologies as a synthesis tool. The idea to compile such a catalog has grown from the observation that on the one hand, the topology of a translinear network must satisfy strong constraints which severely limit the number of "admissible" topologies, in particular for networks with few transistors, and on the other hand, the topology of a translinear network already fixes its essential behavior, at least for static networks, because the so-called translinear principle requires the continuous parameters of all transistors to be the same. Even though the admissible topologies are heavily restricted, it is a highly nontrivial task to compile such a catalog. Combinatorial techniques have been adapted to undertake this task. In a catalog of translinear network topologies, prototype network equations can be stored along with each topology. When a circuit with a specified behavior is to be designed, one can search the catalog for a network whose equations can be matched with the desired behavior. In this context, two algebraic problems arise: To set up a meaningful equation for a network in the catalog, an elimination of variables must be performed, and to test whether a prototype equation from the catalog and a specified equation of desired behavior can be "matched", a complex system of polynomial equations must be solved, where the solutions are restricted to a finite set of integers. Sophisticated algorithms from computer algebra are applied in both cases to perform the symbolic computations. All mentioned algorithms have been implemented using C++, Singular, and Mathematica, and are successfully applied to actual design problems of humidity sensor circuitry at Analog Microelectronics GmbH, Mainz. As result of the research conducted, an exhaustive catalog of all static formal translinear networks with at most eight transistors is available. The application for the humidity sensor system proves the applicability of the developed synthesis approach. The details and implementations of the algorithms are worked out only for static networks, but can easily be adopted for dynamic networks as well. While the implementation of the combinatorial algorithms is stand-alone software written "from scratch" in C++, the implementation of the algebraic algorithms, namely the symbolic treatment of the network equations and the match finding, heavily rely on the sophisticated Gröbner basis engine of Singular and thus on more than a decade of experience contained in a special-purpose computer algebra system. It should be pointed out that the thesis contains the new observation that the translinear loop equations of a translinear network are precisely represented by the toric ideal of the network's translinear digraph. Altogether, this thesis confirms and strengthenes the key role of translinear circuits as systematically designable nonlinear circuits.

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.

Over the last decades, mathematical modeling has reached nearly all fields of natural science. The abstraction and reduction to a mathematical model has proven to be a powerful tool to gain a deeper insight into physical and technical processes. The increasing computing power has made numerical simulations available for many industrial applications. In recent years, mathematicians and engineers have turned there attention to model solid materials. New challenges have been found in the simulation of solids and fluid-structure interactions. In this context, it is indispensable to study the dynamics of elastic solids. Elasticity is a main feature of solid bodies while demanding a great deal of the numerical treatment. There exists a multitude of commercial tools to simulate the behavior of elastic solids. Anyhow, the majority of these software packages consider quasi-stationary problems. In the present work, we are interested in highly dynamical problems, e.g. the rotation of a solid. The applicability to free-boundary problems is a further emphasis of our considerations. In the last years, meshless or particle methods have attracted more and more attention. In many fields of numerical simulation these methods are on a par with classical methods or superior to them. In this work, we present the Finite Pointset Method (FPM) which uses a moving least squares particle approximation operator. The application of this method to various industrial problems at the Fraunhofer ITWM has shown that FPM is particularly suitable for highly dynamical problems with free surfaces and strongly changing geometries. Thereby, FPM offers exactly the features that we require for the analysis of the dynamics of solid bodies. In the present work, we provide a numerical scheme capable to simulate the behavior of elastic solids. We present the system of partial differential equations describing the dynamics of elastic solids and show its hyperbolic character. In particular, we focus our attention to the constitutive law for the stress tensor and provide evolution equations for the deviatoric part of the stress tensor in order to circumvent limitations of the classical Hooke's law. Furthermore, we present the basic principle of the Finite Pointset Method. In particular, we provide the concept of upwinding in a given direction as a key ingredient for stabilizing hyperbolic systems. The main part of this work describes the design of a numerical scheme based on FPM and an operator splitting to take the different processes within a solid body into account. Each resulting subsystem is treated separately in an adequate way. Hereby, we introduce the notion of system-inherent directions and dimensional upwinding. Finally, a coupling strategy for the subsystems and results are presented. We close this work with some final conclusions and an outlook on future work.