Stochastic differential inclusions can be considered as a generalisation of stochastic differential equations. In particular a multivalued mapping describes the set of equations, in which a solution has to be found. This paper presents an existence result for a special parabolic stochastic inclusion. The proof is based on the method of upper and lower solutions. In the deterministic case this method was effectively introduced by S. Carl.

Anomale Diffusion ist nicht an stochastische Kräfte und eine große Zahl von Freiheitsgraden gebunden, sondern ist auch in chaotischen Systemen mit statischer Unordnung in den Bewegungsgleichungen zu beobachten. Einfache Modelle dieser niedrigdimensionalen Systeme, deren Dynamik durch iterierte Abbildungen vermittelt wird, können auf zufällige Irrfahrten in Zufallsumgebungen (random walks in random environments) abgebildet werden. Sinai-Unordnung beschreibt eine spezielle Klasse dieser zufälligen Irrfahrten in Zufallsumgebungen, für die mit dem asymptotischen Verhalten der Entweichrate, der mittleren quadratischen Versetzung, der Zustandsdichte der Relaxationsraten bis hin zu der als Golosov-Phänomen bekannten dynamischen Lokalisierung analytische Resultate für verschiedene anomale Transporteigenschaften bekannt sind. Die vorliegende Arbeit untersucht numerisch eine rekurrente Erweiterung dieses auf Sprünge zu benachbarten Gitterpunkten beschränkten Modells auf die genannten Transporteigenschaften. Als wesentlicher Unterschied stellt sich dabei die Verletzung von detaillierter Balance im stationären Zustand heraus, der Auswirkungen auf das präasymptotische Verhalten der Transportkoeffizienten hat. Asymptotisch zeigt sich hingegen ein Verhalten wie bei der Sinai-Unordnung.

This study is concerned with the computation of seismic first-arrival traveltimes in anisotropic media using finite difference eikonal methods. For this purpose, different numerical schemes that directly solve the eikonal equation are implemented and assessed numerically. Subsequently, they are used for pre-stack depth migration on synthetic and field data. The thesis starts with a detailed examination of different finite difference methods that have gained popularity in scientific literature for computing seismic traveltimes in isotropic media. The most appropriate for an extension towards anisotropic media are found to be the so-called Fast Marching/Sweeping methods. Both schemes rely on different iteration strategies, but incorporate the same upwind finite difference Godunov schemes that are implemented up to the second order. As a result, the derived methods exhibit high numerical accuracy and perform robustly even in highly contrasted velocity models. Subsequently, the methods are adapted for transversely isotropic media with vertical (VTI) and tilted (TTI) symmetry axes, respectively. Therefore, two different formulations for approximating the anisotropic phase velocities are tested, which are the weakly-anisotropic and the pseudo-acoustic approximation. As expected, the pseudo-acoustic formulation shows superior accuracy especially for strongly anisotropic media. Moreover, it turns out that the tested eikonal schemes are generally more accurate than anisotropic ray tracing approaches, since they do not require an approximation of the group velocity. Numerical experiments are carried out on homogeneous models with varying strengths of anisotropy and the industrial BP 2007 benchmark model. They show that the computed eikonal traveltimes are in good agreement with independent results from finite difference modelling of the isotropic and anisotropic elastic wave equations, and traveltimes estimated by ray-based wavefront construction, respectively. The computational performance of the TI eikonal schemes is largely increased compared to their original isotropic implementations, which is due to the algebraic complexity of the anisotropic phase velocity formulations. At this point, the Fast Marching Method is found to be more efficient on models containing up to 50 million grid points. For larger models, the anisotropic Fast Sweeping implementation gradually becomes advantageous. Here, both techniques perform independently well of the structural complexity of the underlying velocity model. The final step of this thesis is the application of the developed eikonal schemes in pre-stack depth migration. A synthetic experiment over a VTI/TTI layer-cake model demonstrates that the traveltime computation leads to accurate imaging results including a tilted, strongly anisotropic shale layer. The experiment shows further that the estimation of anisotropic velocity models solely from surface reflection data is highly ambiguous. In a second example, the eikonal solvers are applied for depth imaging of two-dimensional field data that were acquired for geothermal exploration in southern Tuscany, Italy. The developed methods also produce clear imaging results in this setting, which illustrates their general applicability for pre-stack depth imaging, particularly in challenging environments.

Local studies of magnetic flux line (vortex) distribution in superconducting thin films and their pinning by natural and artificial defects have been performed using low-temperature magnetic force microscopy (LT-MFM). Taken a 100 nm thin NbN film as an example, the depinning of vortices from natural defects under the influence of the force that the MFM tip exerts on the individual vortex was visualized and the local pinning force was estimated. The good agreement of these results with global transport measurements demonstrates that MFM is a powerful and reliable method to probe the local variation of the pinning landscape. Furthermore, it was demonstrated that the presence of an ordered array of 1-μm-sized ferromagnetic permalloy dots being in a magneticvortex state underneath the Nb film significantly influences the natural pinning landscape of the superconductor leading to commensurate pinning effects. This strong pinning exceeds the repulsive interaction between the superconducting vortices and allows vortex clusters to be located at each dot. Additionally, for industrially applicable YBa$_2$Cu$_3$O$_{7-\delta} thin films the main question discussed was the possibility of a direct correlation between vortices and artificial defects as well as vortex imaging on rough as-prepared thin films. Since the surface roughness (droplets, precipitates) causes a severe problem to the scanning MFM tip, a nanoscale wedge polishing technique that allows to overcome this problem was developed. Mounting the sample under a defined small angle results in a smooth surface and a monotonic thickness reduction of the film along the length of the sample. It provides a continuous insight from the film surface down to the substrate with surface sensitive scanning techniques.

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Der technologische Fortschritt gestattet die Implementierung zunehmend komplexerer Prozessorarchitekturen auf einem Schaltkreis. Ein Trend der letzten Jahre ist die Implementierung von mehr und mehr Verarbeitungseinheiten auf einem Chip. Daraus ergeben sich neue Herausforderungen für die Abbildung von Algorithmen auf solche Architekturen, denn alle Verarbeitungseinheiten sollen effizient bei der Ausführung des Algorithmus genutzt werden. Der Schwerpunkt der eingereichten Dissertation ist die Ausnutzung der Parallelität von Rechenfeldern mit Teilwortparallelität. Solche Architekturen erlauben Parallelverarbeitung auf mehreren Ebenen. Daher wurde eine Abbildungsstrategie, mit besonderem Schwerpunkt auf Teilwortparallelität entwickelt. Diese Abbildungsstrategie basiert auf den Methoden des Rechenfeldentwurfs. Rechenfelder sind regelmäßig angeordnete Prozessorelemente, die nur mit ihren Nachbarelementen kommunizieren. Die Datenein- und -ausgabe wird durch die Prozessorelemente am Rand des Rechenfeldes realisiert. Jedes Prozessorelement kann mehrere Funktionseinheiten besitzen, welche die Rechenoperationen des Algorithmus ausführen. Die Teilwortparallelität bezeichnet die Fähigkeit zur Teilung des Datenpfads der Funktionseinheit in mehrere schmale Datenpfade für die parallele Ausführung von Daten mit geringer Wortbreite. Die entwickelte Abbildungsstrategie unterteilt sich in zwei Schritte, die \"Vorverarbeitung\" und die \"Mehrstufige Modifizierte Copartitionierung\" (kurz: MMC). Die \"Vorverarbeitung\" verändert den Algorithmus in einer solchen Art, dass der veränderte Algorithmus schnell und effizient auf die Zielarchitektur abgebildet werden kann. Hierfür wurde ein Optimierungsproblem entwickelt, welches schrittweise die Parameter für die Transformation des Algorithmus bestimmt. Die \"Mehrstufige Modifizierte Copartitionierung\" wird für die schrittweise Anpassung des Algorithmus an die Zielarchitektur eingesetzt. Darüber hinaus ermöglicht die Abbildungsmethode die Ausnutzung der lokalen Register in den Prozessorelementen und die Anpassung des Algorithmus an die Speicherarchitektur, an die das Rechenfeld angebunden ist. Die erste Stufe der MMC dient der Transformation eines Algorithmus mit Einzeldatenoperationen in einen Algorithmus mit teilwortparallelen Operationen. Mit der zweiten Copartitionierungsstufe wird der Algorithmus an die lokalen Register und an das Rechenfeld angepasst. Weitere Copartitionierungsstufen können zur Anpassung des Algorithmus an die Speicherarchitektur verwendet werden. / The technological progress allows the implementation of complex processor architectures on a chip. One trend of the last years is the implemenation of more and more execution units on one chip. That implies new challenges for the mapping of algorithms on such architectures, because the execution units should be used efficiently during the execution of the algorithm. The focus of the submitted dissertation thesis is the utilization of the parallelism of processor arrays with subword parallelism. Such architectures allow parallel executions on different levels. Therefore an algorithm mapping strategy was developed, where the exploitation of the subword parallelism was in the focus. This algorithm mapping strategy is based on the methods of the processor array design. Processor arrays are regular arranged processor elements, which communicate with their neighbors elements only. The data in- and output will be realized by the processor elements on the border of the array. Each processor element can have several functional units, which execute the computational operations. Subword parallelism means the capability for splitting the data path of the functional units in several smaller chunks for the parallel execution of data with lower word width. The developed mapping strategy is subdivided in two steps, the \"Preprocessing\" and the \"Multi-Level Modified Copartitioning\" (kurz: MMC), whereat the MMC means the method of the step simultaneously. The \"Preprocessing\" alter the algorithm in such a kind, that the altered algorithm can be fast and efficient mapped on the target architecture. Therefore an optimization problem was developed, which determines gradual the parameter for the transformation of the algorithm. The \"Multi-Level Modified Copartitioning\" is used for mapping the algorithm gradual on the target architecture. Furthermore the mapping methodology allows the exploitation of the local registers in the processing elements and the adaptation of the algorithm on the memory architecture, where the processing array is connected on. The first level of the MMC is used for the transformation of an algorithm with operation based on single data to an algorithm with subword parallel operations. With the second level, the algorithm will be adapted to the local registers in the processing elements and to the processor array. Further copartition levels can be used for matching the algorithm to the memory architecture.

Of key importance for metamodelling are appropriate modelling formalisms. Most metamodelling languages permit the development of metamodels that specify tree-structured models enriched with semantics like constraints, references and operations, which extend the models to graphs. However, often the semantics of these semantic constructs is not part of the metamodel, i.e., it is unspeci ed. Therefore, we propose to reuse well-known compiler construction techniques to specify metamodel semantics. To be more precise, we present the application of reference attribute grammars (RAGs) for metamodel semantics and analyse commonalities and differences. Our focus is to pave the way for such a combination, by exemplifying why and how the metamodelling and attribute grammar (AG) world can be combined and by investigating a concrete example - the combination of the Eclipse Modelling Framework (EMF) and JastAdd, an AG evaluator generator.

The discourse presented here is aimed at examining the justification of applications of current analysis to real world problems.

This work focusses on analysing the stability of the MSFR – a molten salt reactor with a fast neutron spectrum. The investigations are based on a model, which was published and studied by the Politecnico di Milano using a linear approach. Since linear methods can only provide stability information to a limited extent, this work continues the conducted investigations by applying nonlinear methods. In order to examine the specified reactor model, the system equations were implemented, adjusted and verified using MATLAB code. With the help of the computational tool MatCont, a so-called fixed-point solution was tracked and its stability monitored during the variation of selected control parameters. It was found that the considered fixed point does not change its stability state and remains stable. Coexisting fixed points or periodic solutions could not be detected. Therefore, the analysed MSFR model is considered to be a stable system, in which the solutions always tend towards a steady state.:1. Introduction 2. Molten Salt Reactor Technology 2.1. Introduction 2.2. Historical Development 2.3. Working Principle of Molten Salt Reactors 2.4. Molten Salt Coolants 2.5. Advantages and Drawbacks 2.6. Classification 2.7. Molten Salt Fast Reactor Design 3. Stability Characteristics of Dynamical Systems 3.1. Introduction 3.2. Dynamical Systems 3.3. Stability Concepts 3.3.1. Introduction 3.3.2. Lagrange Stability (Bounded Stability) 3.3.3. Lyapunov Stability 3.3.4. Poincaré Stability (Orbital Stability) 3.4. Fixed-Point Solutions 3.4.1. Stability Analysis of Fixed-Point Solutions 3.4.2. Bifurcations of Fixed-Point Solutions 3.5. Periodic Solutions 3.5.1. Stability Analysis of Periodic Solutions 3.5.2. Bifurcations of Periodic Solutions 4. Analysed Reactor System 4.1. Introduction 4.2. Specified Reactor Model 4.3. Implementation and Verification of the Linearised System of Equations 4.3.1. Linearised System of Delayed Differential Equations 4.3.2. Comparison with Reference Plots 4.3.3. Adaptation of Parameter Values 4.4. Implementation and Verification of the Nonlinear System of Equations 4.4.1. Nonlinear System of Delayed Differential Equations 4.4.2. Delayed Neutron Precursor Equation Adjustments 4.4.3. Salt Temperature Equation Adjustments 4.4.4. Nonlinear System of Ordinary Differential Equations 4.4.5. Verification of the Nonlinear System of Ordinary Differential Equations 5. Conducted Stability Analyses 5.1. Introduction 5.2. Nonlinear Stability Analysis 5.2.1. Implementation 5.2.2. Results 5.2.3. Interpretation 5.3. Linear Stability Analysis 5.3.1. Comparison Between the Linearised and Nonlinearised MSFR System of Equations 5.3.2. Stability Investigations Using a Linear Criterion 5.4. MatCont Reliability Test Using an MSBR Model 6. Conclusions and Recommendations for Future Studies / Im Fokus dieser Arbeit steht die Stabilitätsanalyse des MSFR – eines Flüssigsalzreaktors mit schnellem Neutronenspektrum. Als Grundlage wurde ein Modell verwendet, das am Politecnico di Milano erstellt und dort mittels linearer Methoden untersucht wurde. Da lineare Betrachtungen nur eingeschränkte Stabilitätsaussagen treffen können, erweitert diese Arbeit die Untersuchungen um die nichtlineare Stabilitätsanalyse. Zur Untersuchung des vorgegebenen Reaktormodells wurden die Systemgleichungen in MATLAB übertragen und verifiziert. Mithilfe der Rechensoftware MatCont wurde eine sogenannten Fixpunkt-Lösung des Modells unter der Variation ausgewählter Parameter verfolgt und deren Stabilität überprüft. Es hat sich gezeigt, dass der betrachtete Fixpunkt seinen Stabilitätszustand dabei nicht verändert und stabil bleibt. Koexistierende Fixpunkte oder periodische Lösungen konnten nicht nachgewiesen werden. Daher gilt das betrachtete MSFR-Modell als ein stabiles System, dessen Lösungen immer auf einen stationären Zustand zulaufen.:1. Introduction 2. Molten Salt Reactor Technology 2.1. Introduction 2.2. Historical Development 2.3. Working Principle of Molten Salt Reactors 2.4. Molten Salt Coolants 2.5. Advantages and Drawbacks 2.6. Classification 2.7. Molten Salt Fast Reactor Design 3. Stability Characteristics of Dynamical Systems 3.1. Introduction 3.2. Dynamical Systems 3.3. Stability Concepts 3.3.1. Introduction 3.3.2. Lagrange Stability (Bounded Stability) 3.3.3. Lyapunov Stability 3.3.4. Poincaré Stability (Orbital Stability) 3.4. Fixed-Point Solutions 3.4.1. Stability Analysis of Fixed-Point Solutions 3.4.2. Bifurcations of Fixed-Point Solutions 3.5. Periodic Solutions 3.5.1. Stability Analysis of Periodic Solutions 3.5.2. Bifurcations of Periodic Solutions 4. Analysed Reactor System 4.1. Introduction 4.2. Specified Reactor Model 4.3. Implementation and Verification of the Linearised System of Equations 4.3.1. Linearised System of Delayed Differential Equations 4.3.2. Comparison with Reference Plots 4.3.3. Adaptation of Parameter Values 4.4. Implementation and Verification of the Nonlinear System of Equations 4.4.1. Nonlinear System of Delayed Differential Equations 4.4.2. Delayed Neutron Precursor Equation Adjustments 4.4.3. Salt Temperature Equation Adjustments 4.4.4. Nonlinear System of Ordinary Differential Equations 4.4.5. Verification of the Nonlinear System of Ordinary Differential Equations 5. Conducted Stability Analyses 5.1. Introduction 5.2. Nonlinear Stability Analysis 5.2.1. Implementation 5.2.2. Results 5.2.3. Interpretation 5.3. Linear Stability Analysis 5.3.1. Comparison Between the Linearised and Nonlinearised MSFR System of Equations 5.3.2. Stability Investigations Using a Linear Criterion 5.4. MatCont Reliability Test Using an MSBR Model 6. Conclusions and Recommendations for Future Studies

Systeme mit Zeitverzögerung sind dadurch charakterisiert, dass deren zukünftige Entwicklung durch den Zustand zum aktuellen Zeitpunkt nicht eindeutig festgelegt ist. Die Historie des Zustands muss in einem Zeitraum bekannt sein, dessen Länge Totzeit genannt wird und die Gedächtnislänge festlegt. In dieser Arbeit werden fundamentale Effekte untersucht, die sich ergeben, wenn die Totzeit zeitlich variiert wird. Im ersten Teil werden zwei Klassen periodischer Totzeitvariationen eingeführt. Da diese von den dynamischen Eigenschaften einer eindimensionalen iterierten Abbildung abgeleitet werden, die über die Totzeit definiert wird, werden die Klassen entsprechend der zugehörigen Dynamik konservativ oder dissipativ genannt. Systeme mit konservativer Totzeit können in Systeme mit konstanter Totzeit transformiert werden und besitzen gleiche charakteristische Eigenschaften. Dagegen weisen Systeme mit dissipativer Totzeit fundamentale Unterschiede z.B. in der Tangentialraumdynamik auf. Im zweiten Teil werden diese Ergebnisse auf Systeme angewendet, deren Totzeit im Vergleich zur internen Relaxationszeit des Systems groß ist. Es zeigt sich, dass ein durch dissipative Totzeitvariationen induzierter Mechanismus, genannt resonanter Dopplereffekt, unter anderem zu neuen Arten chaotischer Dynamik führt. Diese sind im Vergleich zur bekannten chaotischen Dynamik in Systemen mit konstanter Totzeit sehr niedrig-dimensional. Als Spezialfall wird das so genannte laminare Chaos betrachtet, dessen Zeitreihen durch nahezu konstante Phasen periodischer Dauer gekennzeichnet sind, deren Amplitude chaotisch variiert. Im dritten Teil dieser Arbeit wird auf der Basis experimenteller Daten und durch die Analyse einer nichtlinearen retardierten Langevin-Gleichung gezeigt, dass laminares Chaos robust gegenüber Störungen wie zum Beispiel Rauschen ist und experimentell realisiert werden kann. Es werden Methoden zur Zeitreihenanalyse entwickelt, um laminares Chaos in experimentellen Daten ohne Kenntnis des erzeugenden Systems zu detektieren. Mit diesen Methoden ist selbst dann eine Detektion möglich, wenn das Rauschen so stark ist, dass laminares Chaos mit bloßem Auge nur schwer erkennbar ist.:1. Introduction 2. Dissipative and conservative delays in systems with time-varying delay 3. Laminar Chaos and the resonant Doppler effect 4. Laminar Chaos: a robust phenomenon 5. Summary and concluding remarks A. Appendix / In systems with time-delay, the evolution of a system is not uniquely determined by the state at the current time. The history of the state must be known for a time period of finite duration, where the duration is called delay and determines the memory length of the system. In this work, fundamental effects arising from a temporal variation of the time-delay are investigated. In the first part, two classes of periodically time-varying delays are introduced. They are related to a specific dynamics of a one-dimensional iterated map that is defined by the time-varying delay. Referring to the related map dynamics the classes are called conservative or dissipative. Systems with conservative delay can be transformed into systems with constant delay, and thus have the same characteristic properties as constant delay systems. In contrast, there are fundamental differences, for instance, in the tangent space dynamics, between systems with dissipative delay and systems with constant delay. In the second part, these results are applied to systems with a delay that is considered large compared to the internal relaxation time of the system. It is shown that a mechanism induced by dissipative delays leads to new kinds of regular and chaotic dynamics. The dynamics caused by the so-called resonant Doppler effect is fundamentally different from the behavior known from systems with constant delay. For instance, the chaotic attractors in systems with dissipative delay are very low-dimensional compared to typical ones arising in systems with constant delay. An example of this new kind of low-dimensional dynamics is given by the so-called Laminar Chaos. It is characterized by nearly constant laminar phases of periodic duration, where the amplitude varies chaotically. In the third part of this work, it is shown that Laminar Chaos is a robust phenomenon, which survives perturbations such as noise and can be observed experimentally. Therefore experimental data is provided and a nonlinear delayed Langevin equation is analyzed. Using the robust features that characterize Laminar Chaos, methods for time series analysis are developed, which enable us to detect Laminar Chaos without the knowledge of the specific system that has generated the time series. By these methods Laminar Chaos can be detected even for comparably large noise strengths, where the characteristic properties are nearly invisible to the eye.:1. Introduction 2. Dissipative and conservative delays in systems with time-varying delay 3. Laminar Chaos and the resonant Doppler effect 4. Laminar Chaos: a robust phenomenon 5. Summary and concluding remarks A. Appendix