In this dissertation, we consider two different strategies for improving the projection-based reduced order model (ROM) accuracy: (I) adding closure terms to the standard ROM; (II) using Lagrangian data to improve the ROM basis. Following strategy (I), we propose a new data-driven reduced order model (ROM) framework that centers around the hierarchical structure of the variational multiscale (VMS) methodology and utilizes data to increase the ROM accuracy at a modest computational cost. The VMS methodology is a natural fit for the hierarchical structure of the ROM basis: In the first step, we use the ROM projection to separate the scales into three categories: (i) resolved large scales, (ii) resolved small scales, and (iii) unresolved scales. In the second step, we explicitly identify the VMS-ROM closure terms, i.e., the terms representing the interactions among the three types of scales. In the third step, we use available data to model the VMS-ROM closure terms. Thus, instead of phenomenological models used in VMS for standard numerical discretizations (e.g., eddy viscosity models), we utilize available data to construct new structural VMS-ROM closure models. Specifically, we build ROM operators (vectors, matrices, and tensors) that are closest to the true ROM closure terms evaluated with the available data. We test the new data-driven VMS-ROM in the numerical simulation of four test cases: (i) the 1D Burgers equation with viscosity coefficient $nu = 10^{-3}$; (ii) a 2D flow past a circular cylinder at Reynolds numbers $Re=100$, $Re=500$, and $Re=1000$; (iii) the quasi-geostrophic equations at Reynolds number $Re=450$ and Rossby number $Ro=0.0036$; and (iv) a 2D flow over a backward facing step at Reynolds number $Re=1000$. The numerical results show that the data-driven VMS-ROM is significantly more accurate than standard ROMs. Furthermore, we propose a new hybrid ROM framework for the numerical simulation of fluid flows. This hybrid framework incorporates two closure modeling strategies: (i) A structural closure modeling component that involves the recently proposed data-driven variational multiscale ROM approach, and (ii) A functional closure modeling component that introduces an artificial viscosity term. We also utilize physical constraints for the structural ROM operators in order to add robustness to the hybrid ROM. We perform a numerical investigation of the hybrid ROM for the three-dimensional turbulent channel flow at a Reynolds number $Re = 13,750$. In addition, we focus on the mathematical foundations of ROM closures. First, we extend the verifiability concept from large eddy simulation to the ROM setting. Specifically, we call a ROM closure model verifiable if a small ROM closure model error (i.e., a small difference between the true ROM closure and the modeled ROM closure) implies a small ROM error. Second, we prove that a data-driven ROM closure (i.e., the data-driven variational multiscale ROM) is verifiable. For strategy (II), we propose new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct new Lagrangian ROMs. We show that the new Lagrangian ROMs are orders of magnitude more accurate than the standard Eulerian ROMs, i.e., ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs' accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis. / Doctor of Philosophy / Reduced order models (ROMs) are popular in physical and engineering applications: for example, ROMs are widely used in aircraft designing as it can greatly reduce computational cost for the aircraft's aeroelastic predictions while retaining good accuracy. However, for high Reynolds number turbulent flows, such as blood flows in arteries, oil transport in pipelines, and ocean currents, the standard ROMs may yield inaccurate results. In this dissertation, to improve ROM's accuracy for turbulent flows, we investigate three different types of ROMs. In this dissertation, both numerical and theoretical results show that the proposed new ROMs yield more accurate results than the standard ROM and thus can be more useful.

La ecuación de la difusión neutrónica describe la población de neutrones de un reactor nuclear. Este trabajo trata con este modelo para reactores nucleares con geometría hexagonal. En primer lugar se estudia la ecuación de la difusión neutrónica. Este es un problema diferencial de valores propios, llamado problema de los modos Lambda. Para resolver el problema de los modos Lambda se han comparado diferentes métodos en geometrías unidimensionales, resultando como el mejor el método de elementos espectrales. Usando este método discretizamos los operadores en geometrías bidimensiones y tridimensionales, resolviendo el problema algebraica de valores propios resultante con el método de Arnoldi. La distribución de neutrones estado estacionario se utiliza como condición inicial para la integración de la ecuación de la difusión neutrónica dependiente del tiempo. Se utiliza un método de Euler implícito para integrar en el tiempo. Cuando un nodo está parcialmente insertado aparece un comportamiento no físico de la solución, el efecto ``rod cusping'', que se corrige mediante la ponderación de las secciones eficaces con el flujo del paso de tiempo anterior. Cuando la solución de los sistemas algebraicos que surgen en el método hacia atrás, un método de Krylov se utiliza para resolver los sistemas resultantes, y diferentes estrategias de precondicionamiento se evalúan se. La primera consiste en el uso de la estructura de bloque obtenido por los grupos de energía para resolver el sistema por bloques, y diferentes técnicas de aceleración para el esquema iterativo de bloques y un precondicionador utilizando esta estructura de bloque se proponen. Además se estudia un precondicionador espectral, que hace uso de la información en un subespacio de Krylov para precondicionar el siguiente sistema. También se proponen métodos exponenciales de segundo y cuarto orden integrar la ecuación de difusión neutrónica dependiente del tiempo, donde la exponencial de la matriz del sistema tiene qu / González Pintor, S. (2012). Approximation of the Neutron Diffusion Equation on Hexagonal Geometries [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/17829

Over the past century, great strides have been made in the advancement of mine fire knowledge since the 1909 Cherry Mine Fire Disaster, one of the worst in U.S. history. However, fire hazards remain omnipresent in underground coal mines in the U.S. and around the world. A precise fire numerical analysis (simulation) before any fire events can give a broad view of the emergency scenarios, leading to improved emergency response, and better health and safety outcomes. However, the simulation cost of precise large complex dynamical systems such as fire in underground mines makes practical and even theoretical application challenging. This work details a novel methodology to reduce fire and airflow simulation costs in order to make simulation of complex systems around fire and mine ventilation systems viable. This study will examine the development of a Reduced Order Model (ROM) to predict the flow field of an underground mine geometry using proper orthogonal decomposition (POD) to reduce the airflow simulation cost in a nonlinear model. ROM proves to be an effective tool for approximating several possible solutions near a known solution, resulting in significant time savings over calculating full solutions and suitable for ensemble calculations. In addition, a novel iterative methodology was developed based on the physics of the fluid structure, turbulent kinetic energy (TKE) of the dynamical system, and the vortex dynamics to determine the interface boundary in multiscale (3D-1D) fire simulations of underground space environments. The proposed methodology was demonstrated to be a useful technique for the determination of near and far fire fields, and could be applied across a broad range of flow simulations and mine geometries. Moreover, this research develops a methodology to analyze the tenable limits in a methane fire event in an underground coal mine for bare-faced miners, mine rescue teams, and fire brigade teams in order to improve safety and training of personnel trained to fight fires. The outcomes of this research are specific to mining although the methods outlined might have broader impacts on the other fields such as tunneling and underground spaces technology, HVAC, and fire protection engineering industries. / Ph. D. / With the rapid advancement of technology, the mine fire knowledge has progressed significantly. Atmospheric monitoring and early sensing of heating has improved; the numerical analysis has been expedited with the usage of supercomputers, and more regulations and standards have been set to increase health and safety of miners. In spite of advancements in these areas, fire hazards remain a critical hazard in underground mines. Developing an emergency plan for the safe escape and for fighting the fire is one of the most important issues during a fire event in underground space environments such as mines. A precise fire numerical analysis (simulation) before any fire events can give a broad view of the emergency situation that leads to improving the health and safety issues in the mining industry. Unfortunately, the precise simulation of the large complex dynamical system such as a fire in underground spaces is costly. This work details a cutting edge approach to reduce the fire and airflow simulation costs in order to make simulation of complex systems around fire and mine ventilation systems viable. The main focus of this proposal is to develop novel methodologies to decrease the time of the fire and airflow simulations. The developed methodologies prove to be useful techniques for the reduction of fire simulation and airflow simulation costs. In addition, this study will examine the development of a comprehensive methodology to analyze the tenable limits in a fire event in an underground coal mine in order to improve safety and training of personnel trained to fight fires. These simulations, applied to training, will result in more efficient evacuations (e.g., the decision to leave can be made quickly and with less delay), as well as safe and effective firefighting under certain situations. The target of this research is specific to mining industry although the methods outlined might have broader impacts on the other fields such as tunneling and underground spaces technology, HVAC, and fire protection engineering industries. Therefore, this research may have an immense contribution on the improvement of health and safety associated with firefighting.

Dans le premier chapitre, nous définissons la notion d’espaces à partitions pondérées qui généralise la structure d’espaces à murs mesurés et qui fournit un cadre géométrique à l’étude des actions isométriques affines sur des espaces de Banach pour les groupes localement compacts à base dénombrable. Dans un premier temps, nous caractérisons les actions isométriques affines propres sur des espaces de Banach en termes d’actions propres par automorphismes sur des espaces à partitions pondérées. Puis, nous nous intéressons aux structures de partitions pondérées naturelles pour les actions de certaines constructions de groupes : somme directe ; produit semi-directe ; produit en couronne et produit libre. Nous établissons ainsi des résultats de stabilité de la propriété PLp par ces constructions. Notamment, nous généralisons un résultat de Cornulier, Stalder et Valette de la façon suivante : le produit en couronne d’un groupe ayant la propriété PLp par un groupe ayant la propriété de Haagerup possède la propriété PLp. Dans le deuxième chapitre, nous nous intéressons aux espaces métriques quasi-médians - une généralisation des espaces hyperboliques à la Gromov et des espaces médians - et à leurs propriétés. Après l’étude de quelques exemples, nous démontrons qu’un espace δ-médian est δ′-médian pour tout δ′ ≥ δ. Ce résultat nous permet par la suite d’établir la stabilité par produit directe et par produit libre d’espaces métriques - notion que nous développons par la même occasion. Le troisième chapitre est consacré à la définition et l’étude d’une distance propre, invariante à gauche et qui engendre la topologie explicite sur les groupes localement compacts, compactement engendrés. Après avoir montré les propriétés précédentes, nous prouvons que cette distance est quasi-isométrique à la distance des mots sur le groupe et que la croissance du volume des boules est contrôlée exponentiellement. / In the first chapter, we define the notion of spaces with labelled partitions which generalizes the structure of spaces with measured walls : it provides a geometric setting to study isometric affine actions on Banach spaces of second countable locally compact groups. First, we characterise isometric affine actions on Banach spaces in terms of proper actions by automorphisms on spaces with labelled partitions. Then, we focus on natural structures of labelled partitions for actions of some group constructions : direct sum ; semi-direct product ; wreath product and free product. We establish stability results for property PLp by these constructions. Especially, we generalize a result of Cornulier, Stalder and Valette in the following way : the wreath product of a group having property PLp by a Haagerup group has property PLp. In the second chapter, we focus on the notion of quasi-median metric spaces - a generalization of both Gromov hyperbolic spaces and median spaces - and its properties. After the study of some examples, we show that a δ-median space is δ′-median for all δ′ ≥ δ. This result gives us a way to establish the stability of the quasi-median property by direct product and by free product of metric spaces - notion that we develop at the same time. The third chapter is devoted to the definition and the study of an explicit proper, left-invariant metric which generates the topology on locally compact, compactly generated groups. Having showed these properties, we prove that this metric is quasi-isometric to the word metric and that the volume growth of the balls is exponentially controlled.

Cette thèse concerne le développement, l'extension et l'application d'une formulation stochastique des équations de la mécanique des fluides introduite par Mémin (2014). La vitesse petite échelle, non-résolue, est modélisée au moyen d'un champ aléatoire décorrélé en temps. Cela modifie l'expression de la dérivée particulaire et donc les équations de la mécanique des fluides. Les modèles qui en découlent sont dénommés modèles sous incertitude de position. La thèse s'articulent autour de l'étude successive de modèles réduits, de versions stochastiques du transport et de l'advection à temps long d'un champ de traceur par une vitesse mal résolue. La POD est une méthode de réduction de dimension, pour EDP, rendue possible par l'utilisation d'observations. L'EDP régissant l'évolution de la vitesse du fluide est remplacée par un nombre fini d'EDOs couplées. Grâce à la modélisation sous incertitude de position et à de nouveaux estimateurs statistiques, nous avons dérivé et simulé des versions réduites, déterministe et aléatoire, de l'équation de Navier-Stokes. Après avoir obtenu des versions aléatoires de plusieurs modèles océaniques, nous avons montré numériquement que ces modèles permettaient de mieux prendre en compte les petites échelles des écoulements, tout en donnant accès à des estimés de bonne qualité des erreurs du modèle. Ils permettent par ailleurs de mieux rendre compte des évènements extrêmes, des bifurcations ainsi que des phénomènes physiques réalistes absents de certains modèles déterministes équivalents. Nous avons expliqué, démontré et quantifié mathématiquement l'apparition de petites échelles de traceur, lors de l'advection par une vitesse mal résolu. Cette quantification permet de fixer proprement des paramètres de la méthode d'advection Lagrangienne, de mieux le comprendre le phénomène de mélange et d'aider au paramétrage des simulations grande échelle en mécanique des fluides. / This thesis develops, analyzes and demonstrates several valuable applications of randomized fluid dynamics models referred to as under location uncertainty. The velocity is decomposed between large-scale components and random time-uncorrelated small-scale components. This assumption leads to a modification of the material derivative and hence of every fluid dynamics models. Through the thesis, the mixing induced by deterministic low-resolution flows is also investigated. We first applied that decomposition to reduced order models (ROM). The fluid velocity is expressed on a finite-dimensional basis and its evolution law is projected onto each of these modes. We derive two types of ROMs of Navier-Stokes equations. A deterministic LES-like model is able to stabilize ROMs and to better analyze the influence of the residual velocity on the resolved component. The random one additionally maintains the variability of stable modes and quantifies the model errors. We derive random versions of several geophysical models. We numerically study the transport under location uncertainty through a simplified one. A single realization of our model better retrieves the small-scale tracer structures than a deterministic simulation. Furthermore, a small ensemble of simulations accurately predicts and describes the extreme events, the bifurcations as well as the amplitude and the position of the ensemble errors. Another of our derived simplified model quantifies the frontolysis and the frontogenesis in the upper ocean. This thesis also studied the mixing of tracers generated by smooth fluid flows, after a finite time. We propose a simple model to describe the stretching as well as the spatial and spectral structures of advected tracers. With a toy flow but also with satellite images, we apply our model to locally and globally describe the mixing, specify the advection time and the filter width of the Lagrangian advection method, as well as the turbulent diffusivity in numerical simulations.

Cette thèse présente l'étude de la réduction de modèles pour les réseaux électriques et les réseaux de transmission. Un point de vue mathématique a été adopté pour la réduction de modèles. Les réseaux électriques sont des réseaux immenses et complexes, dont l'analyse et la conception nécessite la simulation et la résolution de grands modèles non-linéaires. Dans le cadre du développement de réseaux électriques intelligents (smart grids) avec une génération distribuée de puissance, l'analyse en temps réel de systèmes complexes tels que ceux-ci nécessite des modèles rapides,fiables et précis. Dans la présente étude, nous proposons des méthodes de réduction de de modèles à la fois a priori et a posteriori, adaptées aux modèles dynamiques des réseaux électriques.Un accent particulier a été mis sur la dynamique transitoire des réseaux électriques, décrite par un modèle oscillant non linéaire et complexe. La non-linéarité de ce modèle nécessite une attention particulière pour bénéficier du maximum d'avantages des techniques de réduction de modèles.Initialement, des méthodes comme POD et LATIN ont été adoptées avec des degrés de succès divers. La méthode de TPWL, qui combine la POD avec des approximations linéaires multiples, a été prouvée comme étant la méthode de réduction de modèles la mieux adaptée pour le modèle dynamique oscillant.Pour les lignes de transmission, un modèle de paramètres distribués en domaine fréquentiel est utilisé. Des modèles réduits de type PGD sont proposés pour le modèle DP des lignes de transmission. Un problème multidimensionnel entièrement paramétrique a été formulé, avec les paramètres électriques des lignes de transmission inclus comme coordonnées additionnelles de la représentation séparée. La méthode a été étendue pour étudier la solution du modèle des lignes de transmission pour laquelle les paramètres dépendent de la fréquence. / This thesis presents the study of the model order reduction for power grids and transmission networks. The specific focus has been the transient dynamics. A mathematical viewpoint has been adopted for model reduction. Power networks are huge and complex network, simulation for power grid analysis and design require large non-linearmodels to be solved. In the context of developing “SmartGrids” with the distributed generation of power, real time analysis of complex systems such as these needs fast,reliable and accurate models. In the current study we propose model order reduction methods both a-priori and aposteriori suitable for dynamic models of power grids.The model that describes the transient dynamics of the power grids is complex non-linear swing dynamics model. The non-linearity of the swing dynamics model necessitates special attention to achieve maximum benefit from the model order reduction techniques. In the current research, POD and LATIN methods were applied initially with varying degrees of success. The method of TPWL has been proved as the best-suited model reduction method for swing dynamics model ; this method combines POD with multiple linear approximations.For the transmission lines, a distributed parameters model infrequency-domain is used. PGD based reduced-order models are proposed for the DP model of transmission lines. A fully parametric problem with electrical parameters of transmission lines included as coordinates of the separated representation. The method was extended to present the solution of frequency-dependent parameters model for transmission lines.

Les simulations numériques représentent un outil fondamental pour la conception et l'optimisation de procédés industriels de fabrication tels que le soudage. Malgré le développement impressionnant des méthodes numériques et des moyens de calcul utilisables, la complexité des procédés de fabrication et les nouvelles exigences des industries les plus avancées obligent à repenser les méthodes, les stratégies et les algorithmes de simulation disponibles. Dans cette thèse, de nouvelles méthodes numériques avec une approche de Réduction des Modèles sont proposées, une discipline consolidée qui a fourni des solutions étonnantes dans différentes applications, comme les procédés de fabrication avancés. Tout d'abord, différentes stratégies sont proposées pour la simulation efficace des procédés de soudage conventionnel, à cet effet, l'utilisation de Computational Vademecums est introduite. L’introduction de ces abaques numériques améliorent des méthodes telles que : les Éléments Finis Généralisés pour le calcul thermique, l'approche local-global pour le calcul mécanique et enfin, la construction directe des abaques numériques utiles pour la phase de pré-design. En second lieu, un solveur PGD efficace est présenté pour les simulations thermo-mécaniques de soudage par friction-malaxage. Cette thèse montre comment la réduction des modèles,en plus d'être une fin en soi, peut être un excellent ingrédient pour améliorer l'efficacité des méthodes numériques traditionnelles. Cela représente un grand intérêt pour l'industrie. / Numerical simulations represent a fundamental tool for the design and optimization of industrial manufacturing processes such as welding. Despite the impressive development of the numerical methods and the means of calculation, the complexity of these processes and the new demands of the more advanced industries make it necessary to rethink the available methods, strategies and simulation algorithms. In this thesis, we propose new numerical methods with a Model Order Reduction approach, a consolidated discipline that has provided surprising solutions indifferent applications, such as advanced manufacturing processes. First, different strategies for the efficient simulation of conventional welding processes are proposed. To this end, the use of Computational Vademecums is introduced for the improvement of methods such as the Generalized Finite Element for thermal calculation, the local-global approach for the mechanical calculation or the direct construction of vademecums useful for predesign phases. Then, an efficient PGD solver for thermomechanical simulations for friction stir welding is presented. This thesis shows how Model Reduction, besides being an end, it can be an excellent ingredient to improve the efficiency of traditional numerical methods, with great interest for the industry.

Network reconstruction is the process of recovering a unique structured representation of some dynamic system using input-output data and some additional knowledge about the structure of the system. Many network reconstruction algorithms have been proposed in recent years, most dealing with the reconstruction of strictly proper networks (i.e., networks that require delays in all dynamics between measured variables). However, no reconstruction technique presently exists capable of recovering both the structure and dynamics of networks where links are proper (delays in dynamics are not required) and not necessarily strictly proper.The ultimate objective of this dissertation is to develop algorithms capable of reconstructing proper networks, and this objective will be addressed in three parts. The first part lays the foundation for the theory of mathematical representations of proper networks, including an exposition on when such networks are well-posed (i.e., physically realizable). The second part studies the notions of abstractions of a network, which are other networks that preserve certain properties of the original network but contain less structural information. As such, abstractions require less a priori information to reconstruct from data than the original network, which allows previously-unsolvable problems to become solvable. The third part addresses our original objective and presents reconstruction algorithms to recover proper networks in both the time domain and in the frequency domain.

The composite right to meaningful and informed participation in the criminal process comprises the right to information, the right to understand, the right to be prepared, the right to be present, the right to confrontation and the right to present one’s case. The sub-rights are not of an overarching nature such as the right to legal representation and the right of access to the law. The various rights are grouped together because they show some connection with the ability of the suspect or the accused to participate in the criminal proceedings as a legal subject, and not as an object of the proceedings as in primitive times. These rights ensure that the accused will not participate in the criminal process from an unfavourable position. The heading ‟meaningful and informed participation” is therefore a collective term for these rights. These sub-rights form part of the comprehensive right to a fair trial. The thesis examines aspects of the position of the accused in South Africa and in foreign jurisdictions such as the United States of America, Canada, New Zealand, Australia, Germany, the United Kingdom and Islamic systems. International instruments such as the European Convention for the Protection of Human Rights and decisions of the United Nations Human Rights Committee are also considered. The thesis first considers the historical perspective of the accused in primitive times when he was regarded as an object of the criminal proceedings, to the present time when he is regarded as a subject of the proceedings. The study on foreign jurisdictions reveals that for the most part, our law is in line with the law of other countries. The study also demonstrates that the various rights are not absolute. In exceptional circumstances, some diminution of the accused’s rights is necessary to protect the interests of society. Nevertheless, the courts should act cautiously and not allow the exceptions to overtake the rule. The judiciary should strive to find a better balance between the constitutional rights of the accused and the interests of society. To this end, the judicial system must be objective yet vigilant. / Criminal & Procedural Law / LL.D.

金融機構健全經營、信用分配決策之公平性，經營階層能否以誠實、正當的方式，信守並履行對客戶之承諾，對人民財產保障、金融秩序穩定、國家經濟發展之影響甚鉅。職是之故，各國金融監理主管機關乃對金融機構的各項營業活動與以監理並採行必要之行政處分或導正措施，甚且於相關法規中，課予其善良管理人之法定義務，以確保民眾之財產權。然而，一家金融機構能否有效健全經營，主要關鍵仍在於經營者之良窳，即其負責人是否具備充足的專業知識與良好的道德品性，因此主管機關對金融機構負責人所訂的「適格性」條件，必然較一般企業的負責人更為嚴格。 本文從金融監理機關之行政處分與爭訟出發，先了解主管機關的組織架構、運作機制及行政權限，進而探討其行政處分範圍。考量行政機關之行政行為必須符合一般法律原則與行政程序法之規定，因此進一步檢視金融處分應如何適用一般法律原則，及如何確保能踐正當法律程序；本文亦從日、德、英、美等四個金融市場發展歷程較久、且法制較為完備的國家，窺探其相關法令中，對於金融機構負責人資格審查之規範重點與實務操作，並與我國相關法規中之規定與監理實務進行比較，冀從中獲得啟發。 為了解相關行政處分之司法實務觀點，進而將透過臺北高等行政法院與最高行政法院，就金融監督管理委員會對金控公司、銀行、保險公司、證券公司之負責人，所作成不適格的若干行政處分案件之司法判決，加以整理。冀能從司法實務中釐清相關爭點。最後擬從主管機關對金融機構負責人適格性審查之前、中、後三個階段，透過相關制度的設計即審查前能建立相關審查原則，且對於若干不確定法律概念之法規要件，力求客觀與明確；於審查時，能就個案事實與法規要件加以調和，作成妥適的行政決定；而對於通過適格性審查後之負責人，其後如有不符合適格性條件，應如何為適當之處置，冀就此一類型之行政事件，建立最佳行政處理程序。