O objetivo deste trabalho é construir rigorosamente variedades de soluções definidas implicitamente por equações não-lineares em dimensão infinita. Usando um método de continuação a múltiplos parâmetros aplicado a uma projeção em dimensão finita, uma triangulação da variedade é construída e usada para construir localmente a variedade no espaço de dimensão infinita. Aplicamos este método para encontrar equilíbrio da equação de Cahn-Hilliard. Estudamos também bifurcações cúspides, com o objetivo de encontrar as condições necessárias para a existência das mesmas em qualquer dimensão finita. / The goal of this research is to rigorously compute implicitly defined manifolds of solutions of infinite dimensional nonlinear equations. Using a multi-parameter continuation method on a finite dimensional projection, a triangulation of the manifold is computed and is then used to construct local charts of the global manifold in the infinite dimensional domain of the operator. We apply this method to find the equilibria of the Cahn-Hilliard equation. We also studied cusp bifurcations, in order to find the necessary conditions for the existence of the same in any finite dimension.

O objetivo deste trabalho é construir rigorosamente variedades de soluções definidas implicitamente por equações não-lineares em dimensão infinita. Usando um método de continuação a múltiplos parâmetros aplicado a uma projeção em dimensão finita, uma triangulação da variedade é construída e usada para construir localmente a variedade no espaço de dimensão infinita. Aplicamos este método para encontrar equilíbrio da equação de Cahn-Hilliard. Estudamos também bifurcações cúspides, com o objetivo de encontrar as condições necessárias para a existência das mesmas em qualquer dimensão finita. / The goal of this research is to rigorously compute implicitly defined manifolds of solutions of infinite dimensional nonlinear equations. Using a multi-parameter continuation method on a finite dimensional projection, a triangulation of the manifold is computed and is then used to construct local charts of the global manifold in the infinite dimensional domain of the operator. We apply this method to find the equilibria of the Cahn-Hilliard equation. We also studied cusp bifurcations, in order to find the necessary conditions for the existence of the same in any finite dimension.

Hydraulic fracturing plays an important role in increasing production rate in tight reservoirs. The performance of the reservoir after fracturing can be observed from the productivity index. This parameter is dependent on the fracture geometry; height, length and width. Unified fracture design (UFD) offers a method to determine the fracture dimensions providing the maximum productivity index for a specific proppant amount. Then, in order to achieve the maximum productivity index, the treatment schedules including the amount of liquid and proppant used for each stage must be determined according to the fracture dimensions obtained from the UFD. The proppant number is necessary for determining the fracture geometry using the UFD. This number is used to find the maximum productivity index for a given proppant amount. Then, the dimensionless fracture conductivity index corresponding to the maximum productivity index can be computed. The penetration ration, the fracture length, and the propped fracture width can be computed from the dimensionless fracture conductivity. However, calculating the proppant number used in UFD requires the fracture height as an input. The most convenient way to estimate fracture height to input to the UFD is to assume that the fracture height is restricted by stress contrast between the pay zone and over and under-lying layers. In other words, the fracture height is assumed to be constant, independent of net pressure and equal to the thickness of the layer which has the least minimum principal stress. However, in reality, the fracture may grow out from the target formation and the height of fracture is dependent on the net pressure during the treatment. Therefore, it is necessary to couple determination of the fracture height with determination of the other fracture parameters. In this research, equilibrium height theory is applied to rigorously determine the height of fracture. Solving the problem iteratively, it is possible to incorporate the rigorous fracture height determination into the unified fracture design.

Tailoring the radiative properties of periodic micro/nanostructures can be used as an efficient way to create devices which have applications in energy harvesting, bioengineering and optical sensing. These structures are analyzed by a rigorous solution of the electromagnetic wave phenomena at the interfaces. The thesis explores the application of rigorous coupled-wave analysis (RCWA) method to study the optical responses of microstructure arrays. First section of the thesis elucidates the various mechanisms which are responsible for causing enhanced light absorption in inclined parallel plate grating arrays. Illustrative evidences of surface plasmon and magnetic resonances are provided by one and two-dimensional plots prepared by RCWA. Analytical agreement with visual data is obtained through use of LC circuit models. Finally, the effects of different geometric parameters on the resonance conditions are investigated. The second part of the thesis deals with application of RCWA to study the effect of light scattering on inclined silver nanorod (AgNR) arrays grown on compact disc (CD) gratings. Depending on the manner in which AgNRs are oriented with respect to CD gratings, they exhibit different optical behavior to incoming light. Effects of both incident light polarization and AgNR orientation with respect to the grating have been studied through the use of RCWA and effective medium theory. Calculated results are compared with experimental values and good agreements are observed for total reflection as well as trends of individual diffraction orders.

The PhD thesis considers four topics in dynamical systems and is based on one paper and three manuscripts. In Paper I we apply methods of interval analysis in order to compute the rigorous enclosure of rotation number. The described algorithm is supplemented with a method of proving the existence of periodic points which is used to check rationality of the rotation number. In Manuscript II we provide a numerical algorithm for computing critical points of the multiplier map for the quadratic family (i.e., points where the derivative of the multiplier with respect to the complex parameter vanishes). Manuscript III concerns continued fractions of quadratic irrationals. We show that the generating function corresponding to the sequence of denominators of the best rational approximants of a quadratic irrational is a rational function with integer coefficients. As a corollary we can compute the Lévy constant of any quadratic irrational explicitly in terms of its partial quotients. Finally, in Manuscript IV we develop a method for finding rigorous enclosures of all odd periodic solutions of the stationary Kuramoto-Sivashinsky equation. The problem is reduced to a bounded, finite-dimensional constraint satisfaction problem whose solution gives the desired information about the original problem. Developed approach allows us to exclude the regions in L2, where no solution can exist.

Impact evaluations in development interventions has been growing in recent years. The increasing demand for evidence-based outcomes has led to a debate of what methodology is best to evaluate the impact of development interventions. Accordingly, Randomised Controlled Trial (RCT) has been labeled as a gold standard for impact evaluations. The RCT method functions in a unique way, as it removes the selection bias and ensure high validity of a study. The aim of this research study is to critically assess the RCT as an alternative approach for impact assessment in relation to post-conflict countries; whether this claim holds in a conflict-affected environment or that the context-specific factors of post-conflict countries challenge the implementation of an RCT. This study implements mixed method approach by using simple descriptive statistics and semi-structured interview to answer the research questions. The findings of this study indicate that context-specific factors of post-conflict environments pose challenges on the implementation of an RCT.  As a result, these challenges threaten the quality of the RCT method which lies in reliability, internal validity and external validity. The findings also indicate that feasibility of RCT which lies in ethics, logistics and security, cannot be addressed individually, as the feasibility has a direct impact on the quality of the RCT method.

Cette thèse démontre que l'exercice de la justice criminelle des juges de Maupeou résulte de différentes tensions qui influent sur les jugements des nouveaux parlementaires parisiens. L'analyse des décisions des juges en matières criminelles reflète les multiples défis qui se posent à la nouvelle magistrature et la volonté de contenter différents publics. Les nouveaux juges doivent assurer leur légitimité judiciaire et asseoir l'autorité de la nouvelle Cour. Ils exercent une justice criminelle différente de celle des anciens parlementaires parisiens. Devant les différentes pressions politiques et judiciaires qui se dressent devant les magistrats, une forme de réformisme conservateur se dégage des pratiques des hommes de Maupeou. La thèse démontre que les pratiques judiciaires des parlementaires s'inscrivent au sein d'une tension importante : la nécessité de préserver la sévérité des châtiments de façon à rendre la justice plus terrible et la volonté de réduire la douleur sur le corps des coupables. Ainsi, les juges tentent de substituer des peines corporelles plus douloureuses pour des châtiments publics plus spectaculaires qui permettront au Parlement de publiciser sa nouvelle autorité judiciaire. / This thesis argues that Maupeou's Parisian parlementaires exercised criminal justice in the context of «judicial tensions». An analysis of the« Parlement Maupeou »'s criminal judgements reflects the challenges regarding a need to please different publics. Maupeou's magistrates wanted to impose their legitimacy and authority over the French population. They challenged political pressure that influenced their administration of criminal justice and contributed to forge a judiciary « conservative reformism ». On one hand, Maupeou's magistrates wanted to exercise a terrifying justice over the French population with rigorous judgements and, on the other band, they wanted to reduce pain on criminal's bodies. These parlementaires pronounced spectacular public executions that could publicized and reinforced their authority over the public, but they also reduced dolorous corporal punishments.

An enhanced thermal imaging concept made possible through the development of a gradient-indexed metamaterial infrared detector that offers broadband transmission and reflection in THz waves. This thesis proposes a proof of feasibility for a metamaterial infrared detector containing an anti-reflective coating with various geometrically varying periodic metasurfaces, a gradient-indexed dielectric multilayer for near-perfect longpass filtering, and a gradient index of refraction (GRIN) metalens for enhanced focal plane thermal imaging. 2D Rigorous Coupled-Wave Analysis (RCWA) is used for understanding the photonic gratings performance based on material selection and varying geometric structure. Finite Difference Time Domain (FDTD) is used to characterize performance for a diffractive metalens by optimizing the radius and arrangement of cylindrical nanorods to create a desired phase profile that can achieve a desired focal distance for projections on a detector for near- to far-infrared thermal imaging. Through combining a micromachined anti-reflective coating, a near-perfect longpass filter, and metamaterial GRIN metalens, infrared/THz focal plane thermal imaging can obtain faster photo-response and image quality at targeted wavelengths, which allows for scientific advancements in electro-optical devices for the Department of Defense, aerospace, and biochemical detection applications.

Modern numerical ordinary differential equation initial-value problem (ODE-IVP) solvers compute a piecewise polynomial approximate solution to the mathematical problem. Evaluating the mathematical problem at this approximate solution defines the defect. Corless and Corliss proposed rigorous defect control of numerical ODE-IVP. This thesis automates rigorous defect control for explicit, first-order, nonlinear ODE-IVP. Defect control is residual-based backward error analysis for ODE, a special case of Wilkinson's backward error analysis. This thesis describes a complete software implementation of the Corless and Corliss algorithm and extensive numerical studies. Basic time-stepping software is adapted to defect control and implemented. Advances in software developed for validated computing applications and advances in programming languages supporting operator overloading enable the computation of a tight rigorous enclosure of the defect evaluated at the approximate solution with Taylor models. Rigorously bounding a norm of the defect, the Corless and Corliss algorithm controls to mathematical certainty the norm of the defect to be less than a user specified tolerance over the integration interval. The validated computing software used in this thesis happens to compute a rigorous supremum norm. The defect of an approximate solution to the mathematical problem is associated with a new problem, the perturbed reference problem. This approximate solution is often the product of a numerical procedure. Nonetheless, it solves exactly the new problem including all errors. Defect control accepts the approximate solution whenever the sup-norm of the defect is less than a user specified tolerance. A user must be satisfied that the new problem is an acceptable model. / Thesis / Master of Science (MSc) / Many processes in our daily lives evolve in time, even the weather. Scientists want to predict the future makeup of the process. To do so they build models to model physical reality. Scientists design algorithms to solve these models, and the algorithm implemented in this project was designed over 25 years ago. Recent advances in mathematics and software enabled this algorithm to be implemented. Scientific software implements mathematical algorithms, and sometimes there is more than one software solution to apply to the model. The software tools developed in this project enable scientists to objectively compare solution techniques. There are two forces at play; models and software solutions. This project build software to automate the construction of the exact solution of a nearby model. That's cool.

Le calcul rigoureux vise à produire des représentations certifiées pour les solutions de nombreux problèmes, notamment en analyse fonctionnelle, comme des équations différentielles ou des problèmes de contrôle optimal. En effet, certains domaines particuliers comme l’ingénierie des systèmes critiques ou les preuves mathématiques assistées par ordinateur ont des exigences de fiabilité supérieures à ce qui peut résulter de l’utilisation d’algorithmes relevant de l’analyse numérique classique.Notre objectif consiste à développer des algorithmes à la fois efficaces et validés / certifiés, dans le sens où toutes les erreurs numériques (d’arrondi ou de méthode) sont prises en compte. En particulier, nous recourons aux approximations polynomiales rigoureuses combinées avec des méthodes de validation a posteriori à base de points fixes. Ces techniques sont implémentées au sein d’une bibliothèque écrite en C, ainsi que dans un développement de preuve formelle en Coq, offrant ainsi le plus haut niveau de confiance, c’est-à-dire une implémentation certifiée.Après avoir présenté les opérations élémentaires sur les approximations polynomiales rigoureuses, nous détaillons un nouvel algorithme de validation pour des approximations sous forme de séries de Tchebychev tronquées de fonctions D-finies, qui sont les solutions d’équations différentielles ordinaires linéaires à coefficients polynomiaux. Nous fournissons une analyse fine de sa complexité, ainsi qu’une extension aux équations différentielles ordinaires linéaires générales et aux systèmes couplés de telles équations. Ces méthodes dites symboliques-numériques sont ensuite utilisées dans plusieurs problèmes reliés : une nouvelle borne sur le nombre de Hilbert pour les systèmes quartiques, la validation de trajectoires de satellites lors du problème du rendez-vous linéarisé, le calcul de polynômes d’approximation optimisés pour l’erreur d’évaluation, et enfin la reconstruction du support et de la densité pour certaines mesures, grâce à des techniques algébriques. / Rigorous numerics aims at providing certified representations for solutions of various problems, notably in functional analysis, e.g., differential equations or optimal control. Indeed, specific domains like safety-critical engineering or computer-assisted proofs in mathematics have stronger reliability requirements than what can be achieved by resorting to standard numerical analysis algorithms. Our goal consists in developing efficient algorithms, which are also validated / certified in the sense that all numerical errors (method or rounding) are taken into account. Specifically, a central contribution is to combine polynomial approximations with a posteriori fixed-point validation techniques. A C code library for rigorous polynomial approximations (RPAs) is provided, together with a Coq formal proof development, offering the highest confidence at the implementation level.After providing basic operations on RPAs, we focus on a new validation algorithm for Chebyshev basis solutions of D-finite functions, i.e., solutions of linear ordinary differential equations (LODEs) with polynomial coefficients. We give an in-depth complexity analysis, as well as an extension to general LODEs, and even coupled systems of them. These symbolic-numeric methods are finally used in several related problems: a new lower bound on the Hilbert number for quartic systems; a validation of trajectories arising in the linearized spacecraft rendezvous problem; the design of evaluation error efficient polynomial approximations; and the support and density reconstruction of particular measures using algebraic techniques.