A Variational Transport Theory Method for Two-Dimensional Reactor Core Calculations Scott W. Mosher 110 Pages Directed by Dr. Farzad Rahnema It seems very likely that the next generation of reactor analysis methods will be based largely on neutron transport theory, at both the assembly and core levels. Signifi-cant progress has been made in recent years toward the goal of developing a transport method that is applicable to large, heterogeneous coarse-meshes. Unfortunately, the ma-jor obstacle hindering a more widespread application of transport theory to large-scale calculations is still the computational cost. In this dissertation, a variational heterogeneous coarse-mesh transport method has been extended from one to two-dimensional Cartesian geometry in a practical fashion. A generalization of the angular flux expansion within a coarse-mesh was developed. This allows a far more efficient class of response functions (or basis functions) to be employed within the framework of the original variational principle. New finite element equations were derived that can be used to compute the expansion coefficients for an individual coarse-mesh given the incident fluxes on the boundary. In addition, the non-variational method previously used to converge the expansion coefficients was developed in a new and more thorough manner by considering the implications of the fission source treat-ment imposed by the response expansion. The new coarse-mesh method was implemented for both one and two-dimensional (2-D) problems in the finite-difference, multigroup, discrete ordinates approximation. An efficient set of response functions was generated using orthogonal boundary conditions constructed from the discrete Legendre polynomials. Several one and two-dimensional heterogeneous light water reactor benchmark problems were studied. Relatively low-order response expansions were used to generate highly accurate results using both the variational and non-variational methods. The expansion order was found to have a far more significant impact on the accuracy of the results than the type of method. The varia-tional techniques provide better accuracy, but at substantially higher computational costs. The non-variational method is extremely robust and was shown to achieve accurate re-sults in the 2-D problems, as long as the expansion order was not very low.

A novel video observational method for the space-time stereoscopic reconstruction of dynamic surfaces representable as graphs, such as ocean waves, is developed. Variational optimization algorithms combining image processing, computer vision and partial differential equations are designed to address the problem of the recovery of the shape of an object's surface from sequences of synchronized multi-view images. Several theoretical and numerical paths are discussed to solve the problem. The variational stereo method developed in this thesis has several advantages over existing 3-D reconstruction algorithms. Our method follows a top-down approach or object-centered philosophy in which an explicit model of the target object in the scene is devised and then related to image measurements. The key advantages of our method are the coherence (smoothness) of the reconstructed surface caused by a coherent object-centered design, the robustness to noise due to a generative model of the observed images, the ability to handle surfaces with smooth textures where other methods typically fail to provide a solution, and the higher resolution achieved due to a suitable graph representation of the object's surface. The method provides competitive results with respect to existing variational reconstruction algorithms. However, our method is based upon a simplified but complete physical model of the scene that allows the reconstruction process to include physical properties of the object's surface that are otherwise difficult to take into account with existing reconstruction algorithms. Some initial steps are taken toward incorporating the physics of ocean waves in the stereo reconstruction process. The developed method is applied to empirical data of ocean waves collected at an off-shore oceanographic platform located off the coast of Crimea, Ukraine. An empirically-based physical model founded upon current ocean engineering standards is used to validate the results. Our findings suggest that this remote sensing observational method has a broad impact on off-shore engineering to enrich the understanding of sea states, enabling improved design of off-shore structures. The exploration of ways to incorporate dynamical properties, such as the wave equation, in the reconstruction process is discussed for future research.

Computer Vision aims at developing techniques to extract and exploit information from images. The successful applications of computer vision approaches are multiple and have benefited diverse fields such as manufacturing, medicine or defense. Some of the most challenging tasks performed by computer vision systems are arguably segmentation and tracking. Segmentation can be defined as the partitioning of an image into homogeneous or meaningful regions. Tracking also aims at extracting meaning or information from images, however, it is a dynamic task that operates on temporal (video) sequences. Active contours have been proven to be quite valuable at performing the two aforementioned tasks. The active contours framework is an example of variational approaches, in which a problem is compactly (and elegantly) described and solved in terms of energy functionals. The objective of the proposed research is to develop statistical and shape-based tools inspired from or completing the geometric active contours methodology. These tools are designed to perform segmentation and tracking. The approaches developed in the thesis make an extensive use of partial differential equations and differential geometry to address the problems at hand. Most of the proposed approaches are cast into a variational framework. The contributions of the thesis can be summarized as follows: 1. An algorithm is presented that allows one to robustly track the position and the shape of a deformable object. 2. A variational segmentation algorithm is proposed that adopts a shape-driven point of view. 3. Diverse frameworks are introduced for including prior knowledge on shapes in the geometric active contour framework. 4. A framework is proposed that combines statistical information extracted from images with shape information learned a priori from examples 5. A technique is developed to jointly segment a 3D object of arbitrary shape in a 2D image and estimate its 3D pose with respect to a referential attached to a unique calibrated camera. 6. A methodology for the non-deterministic evolution of curves is presented, based on the theory of interacting particles systems.

[pt] Desde a década de 1990, o Método Fast Multipole (FMM) tem sido usado em conjunto com o Métodos dos Elementos de Contorno (BEM) para a simulação de problemas de grande escala. Este método utiliza expansões em série de Taylor para aglomerar pontos da discretização do contorno, de forma a reduzir o tempo computacional necessário para completar a simulação. Ele se tornou uma ferramenta bastante importante para os BEMs, pois eles apresentam matrizes cheias e assimétricas, o que impossibilita a utilização de técnicas de otimização de solução de sistemas de equação. A aplicação do FMM ao BEM é bastante complexa e requer muita manipulação matemática. Este trabalho apresenta uma formulação do FMM que é independente da solução fundamental utilizada pelo BEM, o Método Fast Multipole Generalizado (GFMM), que se aplica a elementos de contorno curvos e de qualquer ordem. Esta característica é importante, já que os desenvolvimentos de fast multipole encontrados na literatura se restringem apenas a elementos constantes. Todos os aspectos são abordados neste trabalho, partindo da sua base matemática, passando por validação numérica, até a solução de problemas de potencial com muitos milhões de graus de liberdade. A aplicação do GFMM a problemas de potencial e elasticidade é discutida e validada, assim como os desenvolvimentos necessários para a utilização do GFMM com o Método Híbrido Simplificado de Elementos de Contorno (SHBEM). Vários resultados numéricos comprovam a eficiência e precisão do método apresentado. A literatura propõe que o FMM pode reduzir o tempo de execução do algoritmo do BEM de O(N2) para O(N), em que N é o número de graus de liberdade do problema. É demonstrado que esta redução é de fato possível no contexto do GFMM, sem a necessidade da utilização de qualquer técnica de otimização computacional. / [en] The Fast Multipole Method (FMM) has been used since the 1990s with the Boundary Elements Method (BEM) for the simulation of large-scale problems. This method relies on Taylor series expansions of the underlying fundamental solutions to cluster the nodes on the discretised boundary of a domain, aiming to reduce the computational time required to carry out the simulation. It has become an important tool for the BEMs, as they present matrices that are full and nonsymmetric, so that the improvement of storage allocation and execution time is not a simple task. The application of the FMM to the BEM ends up with a very intricate code, and usually changing from one problem s fundamental solution to another is not a simple matter. This work presents a kernel-independent formulation of the FMM, here called the General Fast Multipole Method (GFMM), which is also able to deal with high order, curved boundary elements in a straightforward manner. This is an important feature, as the fast multipole implementations reported in the literature only apply to constant elements. All necessary aspects of this method are presented, starting with the mathematical basics of both FMM and BEM, carrying out some numerical assessments, and ending up with the solution of large potential problems. The application of the GFMM to both potential and elasticity problems is discussed and validated in the context of BEM. Furthermore, the formulation of the GFMM with the Simplified Hybrid Boundary Elements Method (SHBEM) is presented. Several numerical assessments show that the GFMM is highly efficient and may be as accurate as arbitrarily required, for problems with up to many millions of degrees of freedom. The literature proposes that the FMM is capable of reducing the time complexity of the BEM algorithms from O(N2) to O(N), where N is the number of degrees of freedom. In fact, it is shown that the GFMM is able to arrive at such time reduction without resorting to techniques of computational optimisation.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-13T18:38:15Z No. of bitstreams: 1 JOÃO PAULO FORMIGA DE MENESES - DISSERTAÇÃO PPGMAT 2016..pdf: 1613708 bytes, checksum: 5f49f16ec6b9bdf21a073af08bdf1006 (MD5) / Made available in DSpace on 2018-08-13T18:38:15Z (GMT). No. of bitstreams: 1 JOÃO PAULO FORMIGA DE MENESES - DISSERTAÇÃO PPGMAT 2016..pdf: 1613708 bytes, checksum: 5f49f16ec6b9bdf21a073af08bdf1006 (MD5) Previous issue date: 2016-11-25 / Neste trabalho, utilizando sub e supersoluções e métodos variacionais sobre espaços de Orlicz-Sobolev, estudamos a existência de múltiplas soluções positivas para uma classe de problemas elípticos quaselineares. / In this work, using sub and supersolutions and variational methods on Orlicz-Sobolev spaces, we study the existence of multiple positive solutions for a class of quasilinear elliptic problems.

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Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-25T17:05:28Z No. of bitstreams: 1 DENILSON DA SILVA PEREIRA - DISSERTAÇÃO PPGMAT 2010..pdf: 630527 bytes, checksum: 8a6ec5b5fb5e2a462945183d2180a573 (MD5) / Made available in DSpace on 2018-07-25T17:05:28Z (GMT). No. of bitstreams: 1 DENILSON DA SILVA PEREIRA - DISSERTAÇÃO PPGMAT 2010..pdf: 630527 bytes, checksum: 8a6ec5b5fb5e2a462945183d2180a573 (MD5) Previous issue date: 2010-12 / Neste trabalho estudamos a existência e multiplicidade de soluções para uma certa classe de problemas elípticos. Utilizaremos métodos variacionais juntamente com a teoria de Morse em dimensão infinita. / In this work, we study the existence and multiplicity of solution for a large class of Elliptic problems. The main tools used are variational methods together with the infinite dimensional Morse Theory.

Made available in DSpace on 2015-05-15T11:46:10Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 705090 bytes, checksum: 6259c1312a92c4f8f051446d8ad30afc (MD5) Previous issue date: 2012-04-13 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we obtain existence, nonexistence and multiplicity of solutions for the elliptic partial differential equation u 1 2 (x:ru) + "jujp1u = u; x 2 RN; where N 3, " = 1, > 0 and 1 < p (N + 2)=(N 2). Such equation is obtained when we look for self-similar solutions for certain nonlinear heat equations. To obtain the main results, we use variational methods, more precisely, minimization arguments, Lagrange multipliers theorem and elliptic regularity results. / Neste trabalho, obtemos resultados de existência, não existência e multiplicidade de soluções para a equação diferencial parcial elíptica u 1/2(x:ru) + "jujp1u = u; x 2 RN; em que N 3, " =1, > 0 e 1 < p (N + 2)=(N 2). Tal equação é obtida quando procuramos soluções autossimilares para certas equações do calor não-lineares. Para a obtenção dos resultados principais, usamos métodos variacionais, mais precisamente, argumentos de minimização, Teorema dos Multiplicadores de Lagrange e resultados de regularidade elíptica.

Neste trabalho estudamos existência, multiplicidade e não existência de soluções não triviais para o seguinte problema elíptico: { - \'DELTA\' = f(x, u), em \'OMEGA\' u = 0, sobre \'\\PARTIAL\' \'OMEGA\', onde \'OMEGA\' é um conjunto limitado de \'R POT. 2\' com fronteira suave e a função f possui crescimento exponencial. Para a existência de soluções são aplicados métodos variacionais combinados com as desigualdades de Trudinger-Moser. O resultado de não-existência é restrito ao caso de soluções radiais positivas e \'OMEGA\' = \'B IND.1\'(0). A prova usa técnicas de equações diferenciais ordinárias / In this work we study the existence, multiplicity and non-existence of non-trivial solutions to the following elliptic problem: { - \'DELTA\' u = f(x; u); in \'OMEGA\', ; u = 0; on \'\\PARTIAL\' \'OMEGA\' where \"OMEGA\' is a bounded and smooth domain in \'R POT. 2\' and f possesses exponential growth. The existence results are proved by using variational methods and the Trudinger- Moser inequalities. The non-existence result is restricted to the case of positive radial solutions and \'OMEGA\' = \'B IND. 1\'(0). The proof uses techniques of the theory of ordinary differential equations.

Les méthodes d'assimilation de données sont en constante évolution pour s'adapter aux problèmes à résoudre dans les multiples domaines d’application. En sciences de l'atmosphère, chaque nouvel algorithme a d'abord été implémenté sur des modèles de prévision numérique du temps avant d'être porté sur des modèles de chimie atmosphérique. Ce fut le cas des méthodes variationnelles 4D et des filtres de Kalman d'ensemble par exemple. La nouvelle génération d'algorithmes variationnels d'ensemble quadridimensionnels (EnVar 4D) ne fait pas exception. Elle a été développée pour tirer partie des deux approches variationnelle et ensembliste et commence à être appliquée au sein des centres opérationnels de prévision numérique du temps, mais n'a à ce jour pas été testée sur des modèles opérationnels de chimie atmosphérique.En effet, la complexité de ces modèles rend difficile la validation de nouvelles méthodes d’assimilation. Il est ainsi nécessaire d'avoir à disposition des modèles d’ordre réduit, qui doivent être en mesure de synthétiser les phénomènes physiques à l'{oe}uvre dans les modèles opérationnels tout en limitant certaines des difficultés liées à ces derniers. Un tel modèle, nommé L95-GRS, a donc été développé. Il associe la météorologie simpliste du modèle de Lorenz-95 à un module de chimie de l'ozone troposphérique avec 7 espèces chimiques. Bien que de faible dimension, il reproduit des phénomènes physiques et chimiques observables en situation réelle. Une méthode d'assimilation de donnée, le lisseur de Kalman d'ensemble itératif (IEnKS), a été appliquée sur ce modèle. Il s'agit d'une méthode EnVar 4D itérative qui résout le problème non-linéaire variationnel complet. Cette application a permis de valider les méthodes EnVar 4D dans un contexte de chimie atmosphérique non-linéaire, mais aussi de soulever les premières limites de telles méthodes.Fort de cette expérience, les résultats ont été étendus au cas d’un modèle réaliste de prévision de pollution atmosphérique. Les méthodes EnVar 4D, via l'IEnKS, ont montré leur potentiel pour tenir compte de la non-linéarité du modèle de chimie dans un contexte maîtrisé, avec des observations synthétiques. Cependant, le passage à des observations réelles d'ozone troposphérique mitige ces résultats et montre la difficulté que représente l'assimilation de données en chimie atmosphérique. En effet, une très forte erreur est associée à ces modèles, provenant de sources d'incertitudes variées. Deux démarches doivent alors être entreprises pour pallier ce problème.Tout d’abord, la méthode d’assimilation doit être en mesure de tenir compte efficacement de l’erreur modèle. Cependant, la majorité des méthodes sont développées en supposant au contraire un modèle parfait. Pour se passer de cette hypothèse, une nouvelle méthode a donc été développée. Nommée IEnKF-Q, elle étend l'IEnKS au cas avec erreur modèle. Elle a été validée sur un modèle jouet, démontrant sa supériorité par rapport à des méthodes d'assimilation adaptées naïvement pour tenir compte de l’erreur modèle.Toutefois, une telle méthode nécessite de connaître la nature et l'amplitude exacte de l'erreur modèle qu'elle doit prendre en compte. Aussi, la deuxième démarche consiste à recourir à des outils statistiques pour quantifier cette erreur modèle. Les algorithmes d'espérance-maximisation, de emph{randomize-then-optimize} naïf et sans biais, un échantillonnage préférentiel fondé sur l'approximation de Laplace, ainsi qu'un échantillonnage avec une méthode de Monte-Carlo par chaînes de Markov, y compris transdimensionnelle, ont ainsi été évalués, étendus et comparés pour estimer l'incertitude liée à la reconstruction du terme source des accidents des centrales nucléaires de Tchernobyl et Fukushima-Daiichi.Cette thèse a donc enrichi le domaine de l'assimilation de données EnVar 4D par ses apports méthodologiques et en ouvrant la voie à l’application de ces méthodes sur les modèles de chimie atmosphérique / Data assimilation methods are constantly evolving to adapt to the various application domains. In atmospheric sciences, each new algorithm has first been implemented on numerical weather prediction models before being ported to atmospheric chemistry models. It has been the case for 4D variational methods and ensemble Kalman filters for instance. The new 4D ensemble variational methods (4D EnVar) are no exception. They were developed to take advantage of both variational and ensemble approaches and they are starting to be used in operational weather prediction centers, but have yet to be tested on operational atmospheric chemistry models.The validation of new data assimilation methods on these models is indeed difficult because of the complexity of such models. It is hence necessary to have at our disposal low-order models capable of synthetically reproducing key physical phenomenons from operational models while limiting some of their hardships. Such a model, called L95-GRS, has therefore been developed. It combines the simple meteorology from the Lorenz-95 model to a tropospheric ozone chemistry module with 7 chemical species. Even though it is of low dimension, it reproduces some of the physical and chemical phenomenons observable in real situations. A data assimilation method, the iterative ensemble Kalman smoother (IEnKS), has been applied to this model. It is an iterative 4D EnVar method which solves the full non-linear variational problem. This application validates 4D EnVar methods in the context of non-linear atmospheric chemistry, but also raises the first limits of such methods.After this experiment, results have been extended to a realistic atmospheric pollution prediction model. 4D EnVar methods, via the IEnKS, have once again shown their potential to take into account the non-linearity of the chemistry model in a controlled environment, with synthetic observations. However, the assimilation of real tropospheric ozone concentrations mitigates these results and shows how hard atmospheric chemistry data assimilation is. A strong model error is indeed attached to these models, stemming from multiple uncertainty sources. Two steps must be taken to tackle this issue.First of all, the data assimilation method used must be able to efficiently take into account the model error. However, most methods are developed under the assumption of a perfect model. To avoid this hypothesis, a new method has then been developed. Called IEnKF-Q, it expands the IEnKS to the model error framework. It has been validated on a low-order model, proving its superiority over data assimilation methods naively adapted to take into account model error.Nevertheless, such methods need to know the exact nature and amplitude of the model error which needs to be accounted for. Therefore, the second step is to use statistical tools to quantify this model error. The expectation-maximization algorithm, the naive and unbiased randomize-then-optimize algorithms, an importance sampling based on a Laplace proposal, and a Markov chain Monte Carlo simulation, potentially transdimensional, have been assessed, expanded, and compared to estimate the uncertainty on the retrieval of the source term of the Chernobyl and Fukushima-Daiichi nuclear power plant accidents.This thesis therefore improves the domain of 4D EnVar data assimilation by its methodological input and by paving the way to applying these methods on atmospheric chemistry models