Dissertations / Theses on the topic 'Parity equations'

There are two main threads of this thesis, namely, an unknown (unmeasurable) input reconstruction and fault detection and diagnosis. The developed methods are in the form of parity equations, i.e. finite impulse response filters of the available input and output measurements. In the first thread the design of parity equations for the purpose of an unknown input reconstruction of linear, time-invariant, discrete-time, stochastic systems is taken into consideration. An underlying assumption is that both measurable system inputs as well as the outputs can be subjected to noise, which leads to an errors-in-variables framework. The main contribution of the scheme is accommodation of the Lagrange multiplier method in order to minimise the influence of the noise on the unknown input estimate. Two potential applications of the novel input reconstruction method are proposed, which are a control enhancement of a hot strip steel rolling mill and an estimation of a pollutant level in a river. Furthermore, initial research is conducted in the field of the unknown input recon- struction for a class of nonlinear systems, namely, Hammerstein-Wiener systems, where a linear dynamic block is preceded and followed by a static nonlinear function. Many man-made as well as naturally occurring systems can be accurately described using Hammerstein-Wiener models. However, it is considered that not much attention has been paid to Hammerstein-Wiener systems in the errors-in-variables framework and in this thesis it is aimed to narrow this gap. The second thread considers a problem of robust (disturbance decoupled) fault de- tection as well as fault isolation and identification. Unmeasurable external stimuli, parameter variations or discrepancies between the system and the model act as distur- bances, which can obstruct the fault detection process and lead to false alarms. Thus, a fault detection filter needs to be decoupled from the disturbances. In this thesis the right eigenstructure assignment method used for the robust fault detection filter design is extended to systems with unstable invariant zeros. Another contribution re- gards the design of robust parity equations of any arbitrary order using both left and right eigenstructure assignment. Furthermore, a parity equation-based fault isolation and identification filter is designed which provides an estimate of the fault. A simple method for the calculation of thresholds whose violation indicates a fault occurrence is also proposed for the errors-in-variables framework.

We consider transmission over a binary erasure wiretap channel using the code construction method introduced by Rathi et al. based on two edge type Low-Density Parity-Check (LDPC) codes and the coset encoding scheme. By generalizing the method of computing conditional entropy for standard LDPC ensembles introduced by Méasson, Montanari, and Urbanke to two edge type LDPC ensembles, we show how the equivocation for the wiretapper can be computed. We find that relatively simple constructions give very good secrecy performance and are close to the secrecy capacity.

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The safety and functionality of engineering systems can be affected adversely by faults or wear in system components. Therefore, methods for detecting such faults/wear and ameliorating their effects to avoid system failure are important. Designing schemes for the detection and diagnosis of faults is becoming increasingly important in engineering due to the complexity of modern industrial systems and growing demands for quality, cost efficiency, reliability, and the safety issue. In safety/mission critical applications, fault detection can be combined with accommodation/reconfiguration (after a fault) to achieve fault tolerance allowing the system to complete or abort its function in a way that is sub-optimal but does achieve the design objective. This thesis discusses research carried-out on the development and validation of a model-based fault detection and isolation (FDI) system for a pneumatically actuated Stewart platform. The Stewart-Gough platform provides six degrees of freedom consisting of three translational and three rotational degrees of freedom (x, y, z, pitch, roll, & yaw). As these platforms can be fast acting (rapid motion) and can handle reasonable loads, they can become dangerous, especially when fault(s) in the platform mechanism, drivetrain or control system occur. Therefore, as a safety critical application it is imperative that fault tolerant schemes are applied in order to provide a safe working environment. The design concept of the FDI scheme for the full Stewart-Gough platform is first designed using a single cylinder set-up. This modular concept is adopted so that a robust fault tolerant control scheme can be designed basically off-line (i.e. not attached to the Stewart Gough platform). This approach is adopted as requirements are easier to understand using a single cylinder set-up. The modular design approach subdivides the whole system into smaller sections (modules) that can be independently created and then used in the complete Stewart-Gough platform. The main contributions of the work are that a pneumatically actuated Stewart-Gough platform has been designed, built, and commissioned. A mathematical model has been developed and has been validated against experimental results. Two control approaches have been designed and compared. A fundamental comparative study of parity equations and Kalman filter observer banks for fault detection in pneumatic actuators has been conducted. The parity equations and Kalman filter approaches have been extended to provide a combined fault detection scheme. The FDI and control schemes have been combined in a modular Fault Tolerant Control (FTC) scheme for a pneumatic cylinder. The resulting FTC scheme has been validated by experimentation and demonstrated on the single cylinder test rig. The FTC scheme has been extended to all 6 cylinders (and including fault management at top level) of Stewart-Gough platform. The FTC scheme has been validated by experimentation and demonstrated on the Stewart-Gough platform test rig.

A diagnosis procedure is an algorithm to detect and locate (isolate) faulty components in a dynamic process. In 1994 the California Air Resource Board released a regulation, called OBD II, demanding a thorough diagnosis system on board automotive vehicles. These legislative demands indicate that diagnosis will become increasingly important for automotive engines in the next few years. To achieve diagnosis, redundancy has to be included in the system. This redundancy can be either hardware redundancy or analytical redundancy. Hardware redundancy, e.g. an extra sensor or extra actuator, can be space consuming or expensive. Methods based on analytical redundancy need no extra hardware, the redundancy here is generated from a process model instead. In this thesis, approaches utilizing analytical redundancy is examined. A literature study is made, surveying a number of approaches to the diagnosis problem. Three approaches, based on both linear and non-linear models, are selected and further analyzed and complete design examples are performed. A mathematical model of an SI-engine is derived to enable simulations of the designed methods.

It has long been argued that underlying values should hold a central role in political analysis. This would seem particularly relevant in an era of de-alignment and catch-all parties in which political actors often make direct values orientated appeals to the electorate. With the expansion in appropriate data and measures available to empirical researchers, the last two decades have seen a substantial increase in the number of studies directly addressing the values-voting relationship. Values based explanations of vote choice have contributed to a more nuanced understanding of the processes underlying voter preferences and the structure of public opinion within democratic electorates. This existing empirical literature has generally focused on analysing the role of values on voting in single electoral contexts. While this approach has generated many useful findings that establish the role of values in differentiating political choice, it has only partially explored the contextual mechanisms through which values influence vote choice. This is necessary in order to understand under what political conditions values are likely to become more relevant to vote choice decisions. This thesis is an attempt to address three aspects of this gap in the cross-national research literature on values and voting using analyses of data from the 1990 and 2008 waves of the European Values Survey. Firstly it provides a cross-national analysis of core political values that enables a comparison of the role of values in structuring electoral competition across 15 West European countries. Secondly, it estimates the role that left-right political identity has in mediating the influence of values on vote choice using a structural path model. This provides a cross-national test of this mechanism and therefore assesses variation in the values-voting relationship across different national contexts. Thirdly, the thesis provides a systematic empirical analysis of the influence of political context on the values-voting relationship by testing the effect that macro level system factors, such as polarisation and the number of parties, have on the influence of values. The headline findings of the thesis are that political values are dynamic constructs that can demonstrate subtle variations in the preferences of voters across different electoral contexts. Political values have a multi-dimensional influence on electoral choice; with variation in voter preferences being highlighted by both value differentiation (having opposite preferences for the same value dimension) and emphasis (having a preference for different values). Left-right identity can act as both a mediator and a confounder of political values influence on vote choice. Political context is primarily relevant to the influence of values on voting through the content of supply side party competition as opposed to the structure of that competition. Overall, the study argues the findings show that supply side political context plays a crucial role in defining the parameters and strength of the values-voting relationship in each specific electoral arena.

For the last ten years, DVD-R media have played an important role in the storage of large amounts of digital data throughout the world. During this time it was assumed that the DVD-R was as long-lasting and stable as its predecessor, the CD-R. Several reports have surfaced over the last few years questioning the DVD-R's ability to maintain many of its claims regarding archival quality life spans. These reports have shown a wide range of longevity between the different brands. While some DVD-Rs may last a while, others may result in an early and unexpected failure. Compounding this problem is the lack of information available for consumers to know the quality of the media they own. While the industry works on devising a standard for labeling the quality of future media, it is currently up to the consumer to pay close attention to their own DVD-R archives and work diligently to prevent data loss. This research shows that through accelerated aging and the use of logistic regression analysis on data collected through periodic monitoring of disc read-back errors it is possible to accurately predict unrecoverable failures in the test discs. This study analyzed various measurements of PIE errors, PIE8 Sum errors, POF errors and jitter data from three areas of the disc: the whole disc, the region of the disc where it first failed as well as the last half of the disc. From this data five unique predictive equations were produced, each with the ability to predict disc failure. In conclusion, the relative value of these equations for end-of-life predictions is discussed.

Cette thèse étudie le contrôle optimal de la dynamique de type McKean-Vlasov et ses applications en mathématiques financières. La thèse contient deux parties. Dans la première partie, nous développons la méthode de la programmation dynamique pour résoudre les problèmes de contrôle stochastique de type McKean-Vlasov. En utilisant les contrôles admissibles appropriés, nous pouvons reformuler la fonction valeur en fonction de la loi (resp. la loi conditionnelle) du processus comme seule variable d’état et obtenir la propriété du flot de la loi (resp. la loi conditionnelle) du processus, qui permettent d’obtenir en toute généralité le principe de la programmation dynamique. Ensuite nous obtenons l’équation de Bellman correspondante, en s’appuyant sur la notion de différentiabilité par rapport aux mesures de probabilité introduite par P.L. Lions [Lio12] et la formule d’Itô pour le flot de probabilité. Enfin nous montrons la propriété de viscosité et l’unicité de la fonction valeur de l’équation de Bellman. Dans le premier chapitre, nous résumons quelques résultats utiles du calcul différentiel et de l’analyse stochastique sur l’espace de Wasserstein. Dans le deuxième chapitre, nous considérons le contrôle optimal stochastique de système à champ moyen non linéaire en temps discret. Le troisième chapitre étudie le problème de contrôle optimal stochastique d’EDS de type McKean-Vlasov sans bruit commun en temps continu où les coefficients peuvent dépendre de la loi joint de l’état et du contrôle, et enfin dans le dernier chapitre de cette partie nous nous intéressons au contrôle optimal de la dynamique stochastique de type McKean-Vlasov en présence de bruit commun en temps continu. Dans la deuxième partie, nous proposons un modèle d’allocation de portefeuille robuste permettant l’incertitude sur la rentabilité espérée et la matrice de corrélation des actifs multiples, dans un cadre de moyenne-variance en temps continu. Ce problème est formulé comme un jeu différentiel à champ moyen. Nous montrons ensuite un principe de séparation pour le problème associé. Nos résultats explicites permettent de justifier quantitativement la sous-diversification, comme le montrent les études empiriques
This thesis deals with the study of optimal control of McKean-Vlasov dynamics and its applications in mathematical finance. This thesis contains two parts. In the first part, we develop the dynamic programming (DP) method for solving McKean-Vlasov control problem. Using suitable admissible controls, we propose to reformulate the value function of the problem with the law (resp. conditional law) of the controlled state process as sole state variable and get the flow property of the law (resp. conditional law) of the process, which allow us to derive in its general form the Bellman programming principle. Then by relying on the notion of differentiability with respect to probability measures introduced by P.L. Lions [Lio12], and Itô’s formula along measure-valued processes, we obtain the corresponding Bellman equation. At last we show the viscosity property and uniqueness of the value function to the Bellman equation. In the first chapter, we summarize some useful results of differential calculus and stochastic analysis on the Wasserstein space. In the second chapter, we consider the optimal control of nonlinear stochastic dynamical systems in discrete time of McKean-Vlasov type. The third chapter focuses on the stochastic optimal control problem of McKean-Vlasov SDEs without common noise in continuous time where the coefficients may depend upon the joint law of the state and control. In the last chapter, we are interested in the optimal control of stochastic McKean-Vlasov dynamics in the presence of common noise in continuous time.In the second part, we propose a robust portfolio selection model, which takes into account ambiguity about both expected rate of return and correlation matrix of multiply assets, in a continuous-time mean-variance setting. This problem is formulated as a mean-field type differential game. Then we derive a separation principle for the associated problem. Our explicit results provide an explanation to under-diversification, as documented in empirical studies

Tato diplomová práce je zaměřena na vývoj simulačního prostředí v Matlab/Simulink zvoleného letadla ve známém letovém režimu. Pozice a orientace letadla pohybujícího se ve vzduchu je popsána pohybovými rovnicemi se šesti stup\v{n}i volnosti. Soustava translačních, rotačních a kinematických rovnic tvoří soustavu devíti nelineárních diferenciálních rovnic prvního řádu. Tyto rovnice lze linearizovat okolo nějakého rovnovážného stavu, který budeme nazývat letovým režimem. Součástí simulačního prostředí je řídící systém letadla založený na PID regulaci. Základem je návrh autopilota, který řídí úhel podélného sklonu a úhel příčného náklonu. Součástí návrhu jsou takzvané „flight director\textquotedblright \phantom{s}m\'dy jako udržení výšky, volba kursu, regulace vertikální rychlosti, změna výšky, zachycení požadované výšky a navigační m\'{o}d založený na nelineárním navigačním zákonu. Optimalizace regulátorů za použití PSO algoritmu a Pareto optimalitě je využita pro nastavení parametrů PID regulátoru. Simulační prostředí je vizualizováno v softwaru FlightGear.

The overall security posture of information systems (IS) depends on the behaviors of the IS users. Several studies have shown that users are the greatest vulnerability to IS security. The proliferation of smartphones is introducing an entirely new set of risks, threats, and vulnerabilities. Smartphone devices amplify this data exposure problem by enabling instantaneous transmission and storage of personally identifiable information (PII) by smartphone users, which is becoming a major security risk. Moreover, companies are also capitalizing on the availability and powerful computing capabilities of these smartphone devices and developing a bring-your-own-device (BYOD) program, which makes companies susceptible to divulgence of organizational proprietary information and sensitive customer information. In addition to users being the greatest risk to IS security, several studies have shown that many people do not implement even the most basic security countermeasures on their smartphones. The lack of security countermeasures implementation, risky user behavior, and the amount of sensitive information stored and transmitted on smartphones is becoming an ever-increasing problem. A literature review revealed a significant gap in literature pertaining to smartphone security. This study identified six socio-cognitive factors from the domain of traditional computer security which have shown to have an impact on user security behaviors and practices. The six factors this study identified and analyzed are mobile information security self-efficacy, institutional trust, party trust, and awareness of smartphone risks, threats, and vulnerabilities and their influence on smartphone security practices and behaviors. The analysis done in this research was confirmatory factor analysis (CFA) – structural equation modeling (SEM). The goal of this study was to cross-validate previously validated factors within the context of traditional computer security and assess their applicability in the context of smartphone security. Additionally, this study assessed the influential significance of these factors on the security behaviors and practices of smartphone users. This study used a Web-based survey and was distributed to approximately 539 users through Facebook® and LinkedIn® social media outlets which resulted in 275 responses for a 51% response rate. After pre-analysis data screening was completed, there were a total of 19 responses that had to be eliminated due to unengaged responses and outliers leaving 256 responses left to analyze. The results of the analysis found that vulnerability awareness, threat awareness, and risk awareness are interrelated to one another which all in turn had significance in predicting self-efficacy, security practices, and behaviors. This intricate relationship revealed in this study indicates that a user has to have an increased awareness in all three categories of awareness before they can fully understand how to protect themselves. Having an increased awareness in one category does not impact the overall security posture of the user and that risk, threat, and vulnerability awareness all work together. Another interesting find was that as risk awareness increased the less the smartphone users protected themselves. This finding warrants additional research to investigate why the user is more averse to risk, and willing to accept the risk, despite their increased awareness. Finally, institutional trust and party trust was found not to have any significance on any of the factors. These findings should give smartphone users and organizations insight into specific areas to focus on in minimizing inappropriate security behaviors and practices of smartphone users. More specifically, users and organizations need to focus on educating users on all three factors of threats, risks, and vulnerabilities in order for there to have any impact on increasing self-efficacy and reducing inappropriate security behaviors and practices.

We introduce a symbolic computational approach to detecting all permutation and parity symmetries in any general evolution equation, and to generating associated invariant polynomials, from given monomials, under the action of these symmetries. Traditionally, discrete point symmetries of differential equations are systemically found by solving complicated nonlinear systems of partial differential equations; in the presence of Lie symmetries, the process can be simplified further. Here, we show how to find parity- and permutation-type discrete symmetries purely based on algebraic calculations. Furthermore, we show that such symmetries always form groups, thereby allowing for the generation of new group-invariant conserved quantities from known conserved quantities. This work also contains an implementation of the said results in Mathematica. In addition, it includes, as a motivation for this work, an investigation of the connection between variational symmetries, described by local Lie groups, and conserved quantities in Hamiltonian systems.

With over 830,000 km of operating pipeline in Canada alone, their safe and continued functioning underpins much of daily life. A key type of risk associated with pipelines is third-party damage, damage caused by actions not associated with the pipelines normal operation. The question of whether the pressurized structure like pipeline or pressure vessel would undergo “unzipping” due to the third-party impact is crucial for the safety of pipelines or pressure vessels in service needs to be answered. Thus, we endeavour to develop a methodology for assessment of design solutions effectiveness to prevent a pipeline or pressure vessel failure in an abrupt explosion-like fashion due to third-party damage. Model of crack propagation determining whether the “unzipping” rupture will occur is viewed as a key element in the safety-driven design procedure providing significant effect on the safety of operation. The crack propagation modeling is achieved through the use of nonlinear fracture mechanics technique. The method of singular integral equations is used to calculate the critical stress required for the catastrophic failure of pipeline or pressure vessel damaged due to third-party interference. The model was implemented as a FORTRAN program. Testing of the developed numerical tool was performed using experimental data available in the literature, with the results showing promising agreement.
February 2016

Multiple gravity assist (MGA) spacecraft trajectories can be difficult to find, an intractable problem to solve completely. However, these trajectories have enormous benefits for missions to challenging destinations such as outer planets and primitive bodies. Techniques are presented to aid in solving this problem with a global search tool and additional investigation into one particular proximity operations option is discussed. Explore is a global grid-search MGA trajectory pathsolving tool. An efficient sequential tree search eliminates v∞ discontinuities and prunes trajectories. Performance indices may be applied to further prune the search, with multiple objectives handled by allowing these indices to change between trajectory segments and by pruning with a Pareto-optimality ranking. The MGA search is extended to include deep space maneuvers (DSM), v∞ leveraging transfers (VILT) and low-thrust (LT) transfers. In addition, rendezvous or nπ sequences can patch the transfers together, enabling automatic augmentation of the MGA sequence. Details of VILT segments and nπ sequences are presented: A boundaryvalue problem (BVP) VILT formulation using a one-dimensional root-solve enables inclusion of an efficient class of maneuvers with runtime comparable to solving ballistic transfers. Importantly, the BVP VILT also allows the calculation of velocity-aligned apsidal maneuvers (VAM), including inter-body transfers and orbit insertion maneuvers. A method for automated inclusion of nπ transfers such as resonant returns and back-flip trajectories is introduced: a BVP is posed on the v∞ sphere and solved with one or more nπ transfers – which may additionally fulfill specified science objectives. The nπ sequence BVP is implemented within the broader search, combining nπ and other transfers in the same trajectory. To aid proximity operations around small bodies, analytical methods are used to investigate stability regions in the presence of significant solar radiation pressure (SRP) and body oblateness perturbations. The interactions of these perturbations allow for heliotropic orbits, a stable family of low-altitude orbits investigated in detail. A novel constrained double-averaging technique analytically determines inclined heliotropic orbits. This type of knowledge is uniquely valuable for small body missions where SRP and irregular body shape are very important and where target selection is often a part of the mission design.

Lattice Monte Carlo methods are used to investigate far from and out-of-equilibrium systems, including surface growth, spin systems and solid mixtures. Applications range from the determination of universal growth or aging behaviors to palpable systems, where coarsening of nanocomposites or self-organization of functional nanostructures are of interest. Such studies require observations of large systems over long times scales, to allow structures to grow over orders of magnitude, which necessitates massively parallel simulations. This work addresses the problem of parallel processing introducing correlations in Monte Carlo updates and proposes a virtually correlation-free domain decomposition scheme to solve it. The effect of correlations on scaling and dynamical properties of surface growth systems and related lattice gases is investigated further by comparing results obtained by correlation-free and intrinsically correlated but highly efficient simulations using a stochastic cellular automaton (SCA). Efficient massively parallel implementations on graphics processing units (GPUs) were developed, which enable large-scale simulations leading to unprecedented precision in the final results. The primary subject of study is the Kardar–Parisi–Zhang (KPZ) surface growth in (2 + 1) dimensions, which is simulated using a dimer lattice gas and the restricted solid-on-solid model (RSOS) model. Using extensive simulations, conjectures regard- ing growth, autocorrelation and autoresponse properties are tested and new precise numerical predictions for several universal parameters are made.:1. Introduction 1.1. Motivations and Goals 1.2. Overview 2. Methods and Models 2.1. Estimation of Scaling Exponents and Error Margins 2.2. From Continuum- to Atomistic Models 2.3. Models for Phase Ordering and Nanostructure Evolution 2.3.1. The Kinetic Metropolis Lattice Monte-Carlo Method 2.3.2. The Potts Model 2.4. The Kardar–Parisi–Zhang and Edwards–Wilkinson Universality Classes 2.4.0.1. Physical Aging 2.4.1. The Octahedron Model 2.4.2. The Restricted Solid on Solid Model 3. Parallel Implementation: Towards Large-Scale Simulations 3.1. Parallel Architectures and Programming Models 3.1.1. CPU 3.1.2. GPU 3.1.3. Heterogeneous Parallelism and MPI 3.1.4. Bit-Coding of Lattice Sites 3.2. Domain Decomposition for Stochastic Lattice Models 3.2.1. DD for Asynchronous Updates 3.2.1.1. Dead border (DB) 3.2.1.2. Double tiling (DT) 3.2.1.3. DT DD with random origin (DTr) 3.2.1.4. Implementation 3.2.2. Second DD Layer on GPUs 3.2.2.1. Single-Hit DT 3.2.2.2. Single-Hit dead border (DB) 3.2.2.3. DD Parameters for the Octahedron Model 3.2.3. Performance 3.3. Lattice Level DD: Stochastic Cellular Automaton 3.3.1. Local Approach for the Octahedron Model 3.3.2. Non-Local Approach for the Octahedron Model 3.3.2.1. Bit-Vectorized GPU Implementation 3.3.3. Performance of SCA Implementations 3.4. The Multi-Surface Coding Approach 3.4.0.1. Vectorization 3.4.0.2. Scalar Updates 3.4.0.3. Domain Decomposition 3.4.1. Implementation: SkyMC 3.4.1.1. 2d Restricted Solid on Solid Model 3.4.1.2. 2d and 3d Potts Model 3.4.1.3. Sequential CPU Reference 3.4.2. SkyMC Benchmarks 3.5. Measurements 3.5.0.1. Measurement Intervals 3.5.0.2. Measuring using Heterogeneous Resources 4. Monte-Carlo Investigation of the Kardar–Parisi–Zhang Universality Class 4.1. Evolution of Surface Roughness 4.1.1. Comparison of Parallel Implementations of the Octahedron Model 4.1.1.1. The Growth Regime 4.1.1.2. Distribution of Interface Heights in the Growth Regime 4.1.1.3. KPZ Ansatz for the Growth Regime 4.1.1.4. The Steady State 4.1.2. Investigations using RSOS 4.1.2.1. The Growth Regime 4.1.2.2. The Steady State 4.1.2.3. Consistency of Fine-Size Scaling with Respect to DD 4.1.3. Results for Growth Phase and Steady State 4.2. Autocorrelation Functions 4.2.1. Comparison of DD Methods for RS Dynamics 4.2.1.1. Device-Layer DD 4.2.1.2. Block-Layer DD 4.2.2. Autocorrelation Properties under RS Dynamics 4.2.3. Autocorrelation Properties under SCA Dynamics 4.2.3.1. Autocorrelation of Heights 4.2.3.2. Autocorrelation of Slopes 4.2.4. Autocorrelation in the SCA Steady State 4.2.5. Autocorrelation in the EW Case under SCA 4.2.5.1. Autocorrelation of Heights 4.2.5.2. Autocorrelations of Slopes 4.3. Autoresponse Functions 4.3.1. Autoresponse Properties 4.3.1.1. Autoresponse of Heights 4.3.1.2. Autoresponse of Slopes 4.3.1.3. Self-Averaging 4.4. Summary 5. Further Topics 5.1. Investigations of the Potts Model 5.1.1. Testing Results from the Parallel Implementations 5.1.2. Domain Growth in Disordered Potts Models 5.2. Local Scale Invariance in KPZ Surface Growth 6. Conclusions and Outlook Acknowledgements A. Coding Details A.1. Bit-Coding A.2. Packing and Unpacking Signed Integers A.3. Random Number Generation
Gitter-Monte-Carlo-Methoden werden zur Untersuchung von Systemen wie Oberflächenwachstum, Spinsystemen oder gemischten Feststoffen verwendet, welche fern eines Gleichgewichtes bleiben oder zu einem streben. Die Anwendungen reichen von der Bestimmung universellen Wachstums- und Alterungsverhaltens hin zu konkreten Systemen, in denen die Reifung von Nanokompositmaterialien oder die Selbstorganisation von funktionalen Nanostrukturen von Interesse sind. In solchen Studien müssen große Systemen über lange Zeiträume betrachtet werden, um Strukturwachstum über mehrere Größenordnungen zu erlauben. Dies erfordert massivparallele Simulationen. Diese Arbeit adressiert das Problem, dass parallele Verarbeitung Korrelationen in Monte-Carlo-Updates verursachen und entwickelt eine praktisch korrelationsfreie Domänenzerlegungsmethode, um es zu lösen. Der Einfluss von Korrelationen auf Skalierungs- und dynamische Eigenschaften von Oberflächenwachtums- sowie verwandten Gittergassystemen wird weitergehend durch den Vergleich von Ergebnissen aus korrelationsfreien und intrinsisch korrelierten Simulationen mit einem stochastischen zellulären Automaten untersucht. Effiziente massiv parallele Implementationen auf Grafikkarten wurden entwickelt, welche großskalige Simulationen und damit präzedenzlos genaue Ergebnisse ermöglichen. Das primäre Studienobjekt ist das (2 + 1)-dimensionale Kardar–Parisi–Zhang- Oberflächenwachstum, welches durch ein Dimer-Gittergas und das Kim-Kosterlitz-Modell simuliert wird. Durch massive Simulationen werden Thesen über Wachstums-, Autokorrelations- und Antworteigenschaften getestet und neue, präzise numerische Vorhersagen zu einigen universellen Parametern getroffen.:1. Introduction 1.1. Motivations and Goals 1.2. Overview 2. Methods and Models 2.1. Estimation of Scaling Exponents and Error Margins 2.2. From Continuum- to Atomistic Models 2.3. Models for Phase Ordering and Nanostructure Evolution 2.3.1. The Kinetic Metropolis Lattice Monte-Carlo Method 2.3.2. The Potts Model 2.4. The Kardar–Parisi–Zhang and Edwards–Wilkinson Universality Classes 2.4.0.1. Physical Aging 2.4.1. The Octahedron Model 2.4.2. The Restricted Solid on Solid Model 3. Parallel Implementation: Towards Large-Scale Simulations 3.1. Parallel Architectures and Programming Models 3.1.1. CPU 3.1.2. GPU 3.1.3. Heterogeneous Parallelism and MPI 3.1.4. Bit-Coding of Lattice Sites 3.2. Domain Decomposition for Stochastic Lattice Models 3.2.1. DD for Asynchronous Updates 3.2.1.1. Dead border (DB) 3.2.1.2. Double tiling (DT) 3.2.1.3. DT DD with random origin (DTr) 3.2.1.4. Implementation 3.2.2. Second DD Layer on GPUs 3.2.2.1. Single-Hit DT 3.2.2.2. Single-Hit dead border (DB) 3.2.2.3. DD Parameters for the Octahedron Model 3.2.3. Performance 3.3. Lattice Level DD: Stochastic Cellular Automaton 3.3.1. Local Approach for the Octahedron Model 3.3.2. Non-Local Approach for the Octahedron Model 3.3.2.1. Bit-Vectorized GPU Implementation 3.3.3. Performance of SCA Implementations 3.4. The Multi-Surface Coding Approach 3.4.0.1. Vectorization 3.4.0.2. Scalar Updates 3.4.0.3. Domain Decomposition 3.4.1. Implementation: SkyMC 3.4.1.1. 2d Restricted Solid on Solid Model 3.4.1.2. 2d and 3d Potts Model 3.4.1.3. Sequential CPU Reference 3.4.2. SkyMC Benchmarks 3.5. Measurements 3.5.0.1. Measurement Intervals 3.5.0.2. Measuring using Heterogeneous Resources 4. Monte-Carlo Investigation of the Kardar–Parisi–Zhang Universality Class 4.1. Evolution of Surface Roughness 4.1.1. Comparison of Parallel Implementations of the Octahedron Model 4.1.1.1. The Growth Regime 4.1.1.2. Distribution of Interface Heights in the Growth Regime 4.1.1.3. KPZ Ansatz for the Growth Regime 4.1.1.4. The Steady State 4.1.2. Investigations using RSOS 4.1.2.1. The Growth Regime 4.1.2.2. The Steady State 4.1.2.3. Consistency of Fine-Size Scaling with Respect to DD 4.1.3. Results for Growth Phase and Steady State 4.2. Autocorrelation Functions 4.2.1. Comparison of DD Methods for RS Dynamics 4.2.1.1. Device-Layer DD 4.2.1.2. Block-Layer DD 4.2.2. Autocorrelation Properties under RS Dynamics 4.2.3. Autocorrelation Properties under SCA Dynamics 4.2.3.1. Autocorrelation of Heights 4.2.3.2. Autocorrelation of Slopes 4.2.4. Autocorrelation in the SCA Steady State 4.2.5. Autocorrelation in the EW Case under SCA 4.2.5.1. Autocorrelation of Heights 4.2.5.2. Autocorrelations of Slopes 4.3. Autoresponse Functions 4.3.1. Autoresponse Properties 4.3.1.1. Autoresponse of Heights 4.3.1.2. Autoresponse of Slopes 4.3.1.3. Self-Averaging 4.4. Summary 5. Further Topics 5.1. Investigations of the Potts Model 5.1.1. Testing Results from the Parallel Implementations 5.1.2. Domain Growth in Disordered Potts Models 5.2. Local Scale Invariance in KPZ Surface Growth 6. Conclusions and Outlook Acknowledgements A. Coding Details A.1. Bit-Coding A.2. Packing and Unpacking Signed Integers A.3. Random Number Generation