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A finalidade deste trabalho e calcular a composição de modelos no FBST (the Full Bayesian Signicance Test) descrito por Borges e Stern [6]. Nosso objetivo foi encontrar um método de aproximação numérica mais eficiente que consiga substituir o método de condensação descrita por Kaplan. Três técnicas foram comparadas: a primeira é a aproximação da convolução de Mellin usando discretização e condensação descrita por Kaplan [11], a segunda é a aproximação da convolução de Mellin usando mistura de exponenciais, descrita por Dufresne [8], para calcular a convolução de Fourier mediante a aproximação de mistura de convoluções exponenciais, usando a estrutura algébrica descrita por Hogg [10], mais a aplicação do operador descrito por Collins [7], para transformar a convolução de Fourier para a convolução de Mellin, a terceira é a aproximação da convolução de Mellin usando mistura de exponenciais, descrita por Dufresne [8], para aproximar diretamente via mistura de exponenciais a convolução de Fourier, mais a aplicação do operador descrito por Collins [7], para transformar a convolução de Fourier para a convolução de Mellin. / The purpose of this work is to calculate the compositional models of FBST (the Full Bayesian Signicance Test) studied by Borges and Stern [6]. The objective of this work was to find an approximation method numerically eficient that can replace the condensation methods described by Kaplan. Three techniques were compared: First, the approximation of Mellin convolution using discretization and condensation described by Kaplan [11], second, the approximation of Mellin convolution using mixtures of exponentials, described by Dufresne [8], to calculate the Fourier convolution by approximation of mixtures of exponential convolutions, using the algebraic structure described by Hogg [10], and then to apply the operator described by Collins [7], to transform the usual convolution to Mellin convolution, third, the approximation of Mellin convolution using mixtures of exponentials, described by Dufresne [8], to calculate the Fourier convolution by direct approximation of mixtures of exponentials, and then to apply the operator described by Collins [7], to transform the usual convolution to Mellin convolution.

In this dissertation we describe a coordinate scaling technique for the numerical approximation of solutions to certain problems posed on unbounded domains in two and three dimensions. This technique amounts to introducing variable coefficients into the problem, which results in defining a solution coinciding with the solution to the original problem inside a bounded domain of interest and rapidly decaying outside of it. The decay of the solution to the modified problem allows us to truncate the problem to a bounded domain and subsequently solve the finite element approximation problem on a finite domain. The particular problems that we consider are exterior problems for the Laplace equation and the time-harmonic acoustic and elastic wave scattering problems. We introduce a real scaling change of variables for the Laplace equation and experimentally compare its performance to the performance of the existing alternative approaches for the numerical approximation of this problem. Proceeding from the real scaling transformation, we introduce a version of the perfectly matched layer (PML) absorbing boundary as a complex coordinate shift and apply it to the exterior Helmholtz (acoustic scattering) equation. We outline the analysis of the continuous PML problem, discuss the implementation of a numerical method for its approximation and present computational results illustrating its efficiency. We then discuss in detail the analysis of the elastic wave PML problem and its numerical discretiazation. We show that the continuous problem is well-posed for sufficiently large truncation domain, and the discrete problem is well-posed on the truncated domain for a sufficiently small PML damping parameter. We discuss ways of avoiding the latter restriction. Finally, we consider a new non-spherical scaling for the Laplace and Helmholtz equation. We present computational results with such scalings and conduct numerical experiments coupling real scaling with PML as means to increase the efficiency of the PML techniques, even if the damping parameters are small.

Photonic cavities are able to confine light to a volume of the order of wavelength of light and this ability can be described in terms of the cavity’s quality factor, which in turn, is proportional to the confinement time in units of optical period. This property of the photonic cavities have been found to be very useful in cavity quantum electrodynamics, for e.g., controlling emission from strongly coupled single photon sources like quantum dots. The smallest possible mode volume attainable by a dielectric cavity, however, poses a limit to the degree of coupling and therefore to the Purcell effect. As metal nanoparticles with plasmonic properties can have mode volumes far below the diffraction limit of light, these can be used to achieve stronger coupling, but the lossy nature of the metals can result in extremely poor quality factors. Hence a hybrid approach, where a high-quality dielectric cavity is combined with a low-quality metal nanoparticle, is being actively pursued. Such structures have been shown to have the potential to preserve the best of both worlds. This thesis describes the design, fabrication and characterization of hybrid plasmonic – photonic nanobeam cavities. Experimentally, we were able to achieve a quality factor of 1200 with the hybrid approach, which suggests that the results are promising for future single photon emission studies. It was found that modeling the behaviour (resonant frequencies, quality factors) of these hybrid cavities with conventional computation methods like FDTD can be tedious, for e.g., a comprehensive study of the electromagnetic fields inside a hybrid photonic nanobeam cavity has been found to take up to 48 hours with FDTD. Hence, we also present an alternate method of analysis using perturbation theory, showing good agreement with FDTD. / Graduate

A finalidade deste trabalho e calcular a composição de modelos no FBST (the Full Bayesian Signicance Test) descrito por Borges e Stern [6]. Nosso objetivo foi encontrar um método de aproximação numérica mais eficiente que consiga substituir o método de condensação descrita por Kaplan. Três técnicas foram comparadas: a primeira é a aproximação da convolução de Mellin usando discretização e condensação descrita por Kaplan [11], a segunda é a aproximação da convolução de Mellin usando mistura de exponenciais, descrita por Dufresne [8], para calcular a convolução de Fourier mediante a aproximação de mistura de convoluções exponenciais, usando a estrutura algébrica descrita por Hogg [10], mais a aplicação do operador descrito por Collins [7], para transformar a convolução de Fourier para a convolução de Mellin, a terceira é a aproximação da convolução de Mellin usando mistura de exponenciais, descrita por Dufresne [8], para aproximar diretamente via mistura de exponenciais a convolução de Fourier, mais a aplicação do operador descrito por Collins [7], para transformar a convolução de Fourier para a convolução de Mellin. / The purpose of this work is to calculate the compositional models of FBST (the Full Bayesian Signicance Test) studied by Borges and Stern [6]. The objective of this work was to find an approximation method numerically eficient that can replace the condensation methods described by Kaplan. Three techniques were compared: First, the approximation of Mellin convolution using discretization and condensation described by Kaplan [11], second, the approximation of Mellin convolution using mixtures of exponentials, described by Dufresne [8], to calculate the Fourier convolution by approximation of mixtures of exponential convolutions, using the algebraic structure described by Hogg [10], and then to apply the operator described by Collins [7], to transform the usual convolution to Mellin convolution, third, the approximation of Mellin convolution using mixtures of exponentials, described by Dufresne [8], to calculate the Fourier convolution by direct approximation of mixtures of exponentials, and then to apply the operator described by Collins [7], to transform the usual convolution to Mellin convolution.

This thesis investigates the possibility of approximating stationary solutions of stochastic differential equations using numerical methods. We consider a particular class of stochastic differential equations, which are known to generate random dynamical systems. The existence of stochastic stationary solution is proved using global attractor approach. Euler's numerical method, applied to the stochastic differential equation, is proved to generate a discrete random dynamical system. The existence of stationary solution is proved again using global attractor approach. At last we prove that the approximate stationary point converges in mean-square sense to the exact one as the time step of the numerical scheme diminishes.

In this thesis, we explore two important aspects of study of differential equations: analytical and computational aspects. We first consider a partial differential equation model for a static liquid surface (capillary surface). We prove through mathematical analyses that the solution of this mathematical model (the Laplace-Young equation) in a cusp domain can be bounded or unbounded depending on the boundary conditions. By utilizing the knowledge we have obtained about the singular behaviour of the solution through mathematical analysis, we then construct a numerical methodology to accurately approximate unbounded solutions of the Laplace-Young equation. Using this accurate numerical methodology, we explore some remaining open problems on singular solutions of the Laplace-Young equation. Lastly, we consider ordinary differential equation models used in the pharmaceutical industry and develop a numerical method for estimating model parameters from incomplete experimental data. With our numerical method, the parameter estimation can be done significantly faster and more robustly than with conventional methods.

In this thesis, we explore two important aspects of study of differential equations: analytical and computational aspects. We first consider a partial differential equation model for a static liquid surface (capillary surface). We prove through mathematical analyses that the solution of this mathematical model (the Laplace-Young equation) in a cusp domain can be bounded or unbounded depending on the boundary conditions. By utilizing the knowledge we have obtained about the singular behaviour of the solution through mathematical analysis, we then construct a numerical methodology to accurately approximate unbounded solutions of the Laplace-Young equation. Using this accurate numerical methodology, we explore some remaining open problems on singular solutions of the Laplace-Young equation. Lastly, we consider ordinary differential equation models used in the pharmaceutical industry and develop a numerical method for estimating model parameters from incomplete experimental data. With our numerical method, the parameter estimation can be done significantly faster and more robustly than with conventional methods.

Maximum likelihood estimation of the concentration parameter of von Mises-Fisher distributions involves inverting the ratio R_nu = I_{nu+1} / I_nu of modified Bessel functions. Computational issues when using approximative or iterative methods were discussed in Tanabe et al. (Comput Stat 22(1):145-157, 2007) and Sra (Comput Stat 27(1):177-190, 2012). In this paper we use Amos-type bounds for R_nu to deduce sharper bounds for the inverse function, determine the approximation error of these bounds, and use these to propose a new approximation for which the error tends to zero when the inverse of R is evaluated at values tending to 1 (from the left). We show that previously introduced rational bounds for R_nu which are invertible using quadratic equations cannot be used to improve these bounds. / Series: Research Report Series / Department of Statistics and Mathematics

L’opérateur du p-Laplacien non local, l’équation d’évolution et la régularisation variationnelle associées régies par un noyau donné ont des applications dans divers domaines de la science et de l’ingénierie. En particulier, ils sont devenus des outils modernes pour le traitement massif des données (y compris les signaux, les images, la géométrie) et dans les tâches d’apprentissage automatique telles que la classification. En pratique, cependant, ces modèles sont implémentés sous forme discrète (en espace et en temps, ou en espace pour la régularisation variationnelle) comme approximation numérique d’un problème continu, où le noyau est remplacé par la matrice d’adjacence d’un graphe. Pourtant, peu de résultats sur la consistence de ces discrétisations sont disponibles. En particulier, il est largement ouvert de déterminer quand les solutions de l’équation d’évolution ou du problème variationnel des tâches basées sur des graphes convergent (dans un sens approprié) à mesure que le nombre de sommets augmente, vers un objet bien défini dans le domaine continu, et si oui, à quelle vitesse. Dans ce manuscrit, nous posons les bases pour aborder ces questions.En combinant des outils de la théorie des graphes, de l’analyse convexe, de la théorie des semi- groupes non linéaires et des équations d’évolution, nous interprétons rigoureusement la limite continue du problème d’évolution et du problème variationnel du p-Laplacien discrets sur graphes. Plus précisé- ment, nous considérons une suite de graphes (déterministes) convergeant vers un objet connu sous le nom de graphon. Si les problèmes d’évolution et variationnel associés au p-Laplacien continu non local sont discrétisés de manière appropriée sur cette suite de graphes, nous montrons que la suite des solutions des problèmes discrets converge vers la solution du problème continu régi par le graphon, lorsque le nombre de sommets tend vers l’infini. Ce faisant, nous fournissons des bornes d’erreur/consistance.Cela permet à son tour d’établir les taux de convergence pour différents modèles de graphes. En parti- culier, nous mettons en exergue le rôle de la géométrie/régularité des graphons. Pour les séquences de graphes aléatoires, en utilisant des inégalités de déviation (concentration), nous fournissons des taux de convergence nonasymptotiques en probabilité et présentons les différents régimes en fonction de p, de la régularité du graphon et des données initiales. / The non-local p-Laplacian operator, the associated evolution equation and variational regularization, governed by a given kernel, have applications in various areas of science and engineering. In particular, they are modern tools for massive data processing (including signals, images, geometry), and machine learning tasks such as classification. In practice, however, these models are implemented in discrete form (in space and time, or in space for variational regularization) as a numerical approximation to a continuous problem, where the kernel is replaced by an adjacency matrix of a graph. Yet, few results on the consistency of these discretization are available. In particular it is largely open to determine when do the solutions of either the evolution equation or the variational problem of graph-based tasks converge (in an appropriate sense), as the number of vertices increases, to a well-defined object in the continuum setting, and if yes, at which rate. In this manuscript, we lay the foundations to address these questions.Combining tools from graph theory, convex analysis, nonlinear semigroup theory and evolution equa- tions, we give a rigorous interpretation to the continuous limit of the discrete nonlocal p-Laplacian evolution and variational problems on graphs. More specifically, we consider a sequence of (determin- istic) graphs converging to a so-called limit object known as the graphon. If the continuous p-Laplacian evolution and variational problems are properly discretized on this graph sequence, we prove that the solutions of the sequence of discrete problems converge to the solution of the continuous problem governed by the graphon, as the number of graph vertices grows to infinity. Along the way, we provide a consistency/error bounds. In turn, this allows to establish the convergence rates for different graph models. In particular, we highlight the role of the graphon geometry/regularity. For random graph se- quences, using sharp deviation inequalities, we deliver nonasymptotic convergence rates in probability and exhibit the different regimes depending on p, the regularity of the graphon and the initial data.