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- D-1250-2010 (1)
The investigation of multivariate generalized Pareto distributions (GPDs) in the framework of extreme value theory has begun only lately. Recent results show that they can, as in the univariate case, be used in Peaks over Threshold approaches. In this manuscript we investigate the definition of GPDs from Section 5.1 of Falk et al. (2004), which does not differ in the area of interest from those of other authors. We first show some theoretical properties and introduce important examples of GPDs. For the further investigation of these distributions simulation methods are an important part. We describe several methods of simulating GPDs, beginning with an efficient method for the logistic GPD. This algorithm is based on the Shi transformation, which was introduced by Shi (1995) and was used in Stephenson (2003) for the simulation of multivariate extreme value distributions of logistic type. We also present nonparametric and parametric estimation methods in GPD models. We estimate the angular density nonparametrically in arbitrary dimension, where the bivariate case turns out to be a special case. The asymptotic normality of the corresponding estimators is shown. Also in the parametric estimations, which are mainly based on maximum likelihood methods, the asymptotic normality of the estimators is shown under certain regularity conditions. Finally the methods are applied to a real hydrological data set containing water discharges of the rivers Altmühl and Danube in southern Bavaria.
In this PhD thesis, we develop models for the numerical simulation of epitaxial crystal growth, as realized, e.g., in molecular beam epitaxy (MBE). The basic idea is to use a discrete lattice gas representation of the crystal structure, and to apply kinetic Monte Carlo (KMC) simulations for the description of the growth dynamics. The main advantage of the KMC approach is the possibility to account for atomistic details and at the same time cover MBE relevant time scales in the simulation. In chapter 1, we describe the principles of MBE, pointing out relevant physical processes and the influence of experimental control parameters. We discuss various methods used in the theoretical description of epitaxial growth. Subsequently, the underlying concepts of the KMC method and the lattice gas approach are presented. Important aspects concerning the design of a lattice gas model are considered, e.g. the solid-on-solid approximation or the choice of an appropriate lattice topology. A key element of any KMC simulation is the selection of allowed events and the evaluation of Arrhenius rates for thermally activated processes. We discuss simplifying schemes that are used to approximate the corresponding energy barriers if detailed knowledge about the barriers is not available. Finally, the efficient implementation of the MC kinetics using a rejection-free algorithm is described. In chapter 2, we present a solid-on-solid lattice gas model which aims at the description of II-VI(001) semiconductor surfaces like CdTe(001). The model accounts for the zincblende structure and the relevant surface reconstructions of Cd- and Te-terminated surfaces. Particles at the surface interact via anisotropic nearest and next nearest neighbor interactions, whereas interactions in the bulk are isotropic. The anisotropic surface interactions reflect known properties of CdTe(001) like the small energy difference between the c(2x2) and (2x1) vacancy structures of Cd-terminated surfaces. A key element of the model is the presence of additional Te atoms in a weakly bound Te* state, which is motivated by experimental observations of Te coverages exceeding one monolayer at low temperatures and high Te fluxes. The true mechanism of binding excess Te to the surface is still unclear. Here, we use a mean-field approach assuming a Te* reservoir with limited occupation. In chapter 3, we perform KMC simulations of atomic layer epitaxy (ALE) of CdTe(001). We study the self-regulation of the ALE growth rate and demonstrate how the interplay of the Te* reservoir occupation with the surface kinetics results in two different regimes: at high temperatures the growth rate is limited to one half layer of CdTe per ALE cycle, whereas at low enough temperatures each cycle adds a complete layer. The temperature where the transition between the two regimes occurs depends mainly on the particle fluxes. The temperature dependence of the growth rate and the flux dependence of the transition temperature are in good qualitative agreement with experimental results. Comparing the macroscopic activation energy for Te* desorption in our model with experimental values we find semiquantitative agreement. In chapter 4, we study the formation of nanostructures with alternating stripes during submonolayer heteroepitaxy of two different adsorbate species on a given substrate. We evaluate the influence of two mechanisms: kinetic segregation due to chemically induced diffusion barriers, and strain relaxation by alternating arrangement of the adsorbate species. KMC simulations of a simple cubic lattice gas with weak inter-species binding energy show that kinetic effects are sufficient to account for stripe formation during growth. The dependence of the stripe width on control parameters is investigated. We find an Arrhenius temperature dependence, in agreement with experimental investigations of phase separation in binary or ternary material systems. Canonical MC simulations show that the observed stripes are not stable under equilibrium conditions: the adsorbate species separate into very large domains. Off-lattice simulations which account for the lattice misfit of the involved particle species show that, under equilibrium conditions, the competition between binding and strain energy results in regular stripe patterns with a well-defined width depending on both misfit and binding energies. In KMC simulations, the stripe-formation and the experimentally reported ramification of adsorbate islands are reproduced. To clarify the origin of the island ramification, we investigate an enhanced lattice gas model whose parameters are fitted to match characteristic off-lattice diffusion barriers. The simulation results show that a satisfactory explanation of experimental observations within the lattice gas framework requires a detailed incorporation of long-range elastic interactions. In the appendix we discuss supplementary topics related to the lattice gas simulations in chapter 4.
This thesis contains two major parts: The first part introduces the reader into three independent concepts of treating strongly correlated many body physics. These are, on the analytical side the SO(5)-theory (Chap.3), which poses the general frame. On the numerical side these are the Stochastic Series Expansion (SSE) (Chap.1) and the Contractor Renormalization Group (CORE) approach (Chap. 2}). The central idea of this thesis was to combine these above concepts, in order to achieve a better understanding of the high-T_c superconductors (HTSC). The results obtained by this combination can be found in the second major part of this thesis (chapters 4 and 5). The main idea of this thesis, i.e., to combine the SO(5)-theory with the capabilities of bosonic Quantum-Monte Carlo simulations and those of the CORE approach, has been proven to be a very successful Ansatz. Two different approaches, one based on symmetry and one on renormalization-group arguments, motivate an effective bosonic Hamiltonian. In a subsequent step the effective Hamiltonian has been simulated efficiently using the SSE. The results reproduce salient experiments on high-T_c superconductors. In addition, it has been shown that the model can be extended to capture also charge ordering. These results also form a profound basis for further studies, for example one could address the open question of SO(5)-symmetry restoration at a multicritical point in the extended pSO(5) model, where longer ranged interactions are included.
In this thesis a new and powerful approach for modeling laser cavity eigenmodes is presented. This approach is based on an eigenvalue problem for singularly perturbed partial differential operators with complex coefficients; such operators have not been investigated in detail until now. The eigenvalue problem is discretized by finite elements, and convergence of the approximate solution is proved by using an abstract convergence theory also developed in this dissertation. This theory for the convergence of an approximate solution of a (quadratic) eigenvalue problem, which particularly can be applied to a finite element discretization, is interesting on its own, since the ideas can conceivably be used to handle equations with a more complex nonlinearity. The discretized eigenvalue problem essentially is solved by preconditioned GMRES, where the preconditioner is constructed according to the underlying physics of the problem. The power and correctness of the new approach for computing laser cavity eigenmodes is clearly demonstrated by successfully simulating a variety of different cavity configurations. The thesis is organized as follows: Chapter 1 contains a short overview on solving the so-called Helmholtz equation with the help of finite elements. The main part of Chapter 2 is dedicated to the analysis of a one-dimensional model problem containing the main idea of a new model for laser cavity eigenmodes which is derived in detail in Chapter 3. Chapter 4 comprises a convergence theory for the approximate solution of quadratic eigenvalue problems. In Chapter 5, a stabilized finite element discretization of the new model is described and its convergence is proved by applying the theory of Chapter 4. Chapter 6 contains computational aspects of solving the resulting system of equations and, finally, Chapter 7 presents numerical results for various configurations, demonstrating the practical relevance of our new approach.
In der Arbeit wird ein neues Konzept für Fahrsimulator-Datenbasen vorgestellt. Der Anwender entwirft eine auf seine Fragestellung zugeschnittene Datenbasis mithilfe einer einfachen Skriptsprache. Das Straßennetzwerk wird auf einer topologischen Ebene repäsentiert. In jedem Simulationsschritt wird hieraus im Sichtbarkeitsbereich des Fahrers die geometrische Repäsentation berechnet. Die für den Fahrer unsichtbaren Teile des Straßenetzwerks können während der Simulation verändert werden. Diese Veränderungen können von der Route des Fahrers oder von den in der Simulation erhobenen Messerten abhängen. Zudem kann der Anwender das Straßennetzwerk interaktiv verändern. Das vorgestellte Konzept bietet zahlreiche Möglichkeiten zur Erzeugung reproduzierbarer Szenarien für Experimente in Fahrsimulatoren.
Aktivitätsbasierte Verhaltensmodellierung und ihre Unterstützung bei Multiagentensimulationen
(2000)
Durch Zusammenführung traditioneller Methoden zur individuenbasierten Simulation und dem Konzept der Multiagentensysteme steht mit der Multiagentensimulation eine Methodik zur Verfügung, die es ermöglicht, sowohl technisch als auch konzeptionell eine neue Ebene an Detaillierung bei Modellbildung und Simulation zu erreichen. Ein Modell beruht dabei auf dem Konzept einer Gesellschaft: Es besteht aus einer Menge interagierender, aber in ihren Entscheidungen autonomen Einheiten, den Agenten. Diese ändern durch ihre Aktionen ihre Umwelt und reagieren ebenso auf die für sie wahrnehmbaren Änderungen in der Umwelt. Durch die Simulation jedes Agenten zusammen mit der Umwelt, in der er "lebt", wird die Dynamik im Gesamtsystem beobachtbar. In der vorliegenden Dissertation wurde ein Repräsentationsschema für Multiagentensimulationen entwickelt werden, das es Fachexperten, wie zum Beispiel Biologen, ermöglicht, selbständig ohne traditionelles Programmieren Multiagentenmodelle zu implementieren und mit diesen Experimente durchzuführen. Dieses deklarative Schema beruht auf zwei Basiskonzepten: Der Körper eines Agenten besteht aus Zustandsvariablen. Das Verhalten des Agenten kann mit Regeln beschrieben werden. Ausgehend davon werden verschiedene Strukturierungsansätze behandelt. Das wichtigste Konzept ist das der "Aktivität", einer Art "Verhaltenszustand": Während der Agent in einer Aktivität A verweilt, führt er die zugehörigen Aktionen aus und dies solange, bis eine Regel feuert, die diese Aktivität beendet und eine neue Aktivität auswählt. Durch Indizierung dieser Regeln bei den zugehörigen Aktivitäten und Einführung von abstrakten Aktivitäten entsteht ein Schema für eine vielfältig strukturierbare Verhaltensbeschreibung. Zu diesem Schema wurde ein Interpreter entwickelt, der ein derartig repräsentiertes Modell ausführt und so Simulationsexperimente mit dem Multiagentenmodell erlaubt. Auf dieser Basis wurde die Modellierungs- und Experimentierumgebung SeSAm ("Shell für Simulierte Agentensysteme") entwickelt. Sie verwendet vorhandene Konzepte aus dem visuellen Programmieren. Mit dieser Umgebung wurden Anwendungsmodelle aus verschiedenen Domänen realisiert: Neben abstrakten Spielbeispielen waren dies vor allem Fragestellungen zu sozialen Insekten, z.B. zum Verhalten von Ameisen, Bienen oder der Interaktion zwischen Bienenvölkern und Milbenpopulationen.