Atualmente, no que concerne as problemáticas pertinentes à engenharia estrutural, o Método dos Elementos Finitos (MEF) é a principal ferramenta utilizada para obter soluções aproximadas de Problemas de Valor de Contorno (PVC). No entanto, tal metodologia exige um elevado custo computacional ao demandar malhas muito refinadas para solucionar problemas que apresentam singularidades, ou seja, que apresentam regiões onde ocorrem gradientes de deformação fortemente localizados. Para superar esse inconveniente, o Método dos Elementos Finitos Generalizados (MEFG) propõe a expansão do espaço de aproximação do MEF mediante a inserção de funções (conhecidas como funções de enriquecimento) que melhor representem localmente o comportamento da solução procurada. Tais funções podem apresentar características específicas ou mesmo serem geradas numericamente. Neste caso, dispensam-se malhas muito refinadas. Entretanto, o aumento do espaço de aproximação de modo irrestrito pode introduzir dependências lineares no sistema de equações do MEFG, tornando a solução obtida imprecisa ou mesmo impedindo a solução do sistema por métodos diretos. A chamada versão estável do MEFG explora uma modificação imposta às funções de enriquecimento a fim de melhorar o condicionamento da matriz de rigidez. Contudo, tal modificação não se configura como condição suficiente para garantir uma redução efetiva do número de condição. Neste trabalho, considera-se uma proposição recente para a modificação do espaço das funções de forma do MEFG associadas ao enriquecimento: trata-se do emprego de funções do tipo flat-top e trigonométricas como Partição da Unidade (PU), as quais são empregadas exclusivamente na construção das funções de forma enriquecidas (essas partições são definidas para elementos finitos quadrilaterais e triangulares). Exemplos numéricos são selecionados para evidenciar as vantagens dessas novas versões do MEFG em relação às anteriores e ao MEF convencional. Demonstra-se que tanto a PU flat-top quanto a PU trigonométrica, preservam as excelentes propriedades de convergência do MEFG. Além disso, mostra-se que o condicionamento da matriz de rigidez associada é próximo ao apresentado pelo MEF (uma vez que o enriquecimento, mesmo polinomial, não gera dependências) e que a formulação apresenta-se robusta na consideração de descontinuidades fortes. / Currently, regarding structural engineering issues, the Finite Element Method (FEM) is the main tool used to obtain approximate solutions of Boundary Value Problems (BVP). However, such methodology requires very refined meshes to solve problems that have singularities, i.e., that have regions where strongly localized deformation gradients occur, which leads to a high computational cost. To overcome this drawback, the Generalized Finite Element Method (GFEM) proposes the expansion of the FEM approach space by inserting functions (known as enrichment functions) that best represent locally the behavior of the searched solution. Such functions may have specific characteristics or even be generated numerically. In this case, very refined meshes are dispensed. However, the increase of the unrestricted approach space can introduce linear dependencies in the system of equations of the GFEM, making the solution imprecise or even preventing the solution of the system by direct methods. The so-called stable version of the GFEM exploits a modification imposed on the enrichment functions in order to improve the conditioning of the stiffness matrix. However, such a modification is not a sufficient condition to ensure an effective reduction in the condition number. In this work, it is considered a recent proposition to modify the space of the shape functions of GFEM associated with enrichment: the use of flat-top and trigonometric functions such as Partition of Unity (PU), which are used exclusively in the construction of the enriched shape functions (these partitions are defined for finite elements quadrilateral and triangular). Numerical examples are selected to highlight the advantages of these new versions of the GFEM over the previous ones and the conventional FEM. It is demonstrated that both flat-top PU and trigonometric PU preserve the excellent convergence properties of GFEM. In addition, it is shown that the conditioning of the associated stiffness matrix is close to that presented by FEM (since enrichment, even polynomial, does not generate dependencies) and that the formulation is robust in the consideration of strong discontinuities.

Multiquadric collocation method is highly efficient for solving partial differential equations due to its exponential error convergence rate. More amazingly, there are two ways to reduce the error: the traditional way of refining the grid, and the unexpected way of simply increasing the value of shape constant $c$ contained in the multiquadric basis function, $sqrt{r^2 + c^2}$. The latter is accomplished without increasing computational cost. It has been speculated that in a numerical solution without roundoff error, infinite accuracy can be achieved by letting $c ightarrow infty$. The ability to obtain infinitely accurate solution is limited only by the roundoff error induced instability of matrix solution with large condition number. Using the arbitrary precision computation capability of {it Mathematica}, this paper tests the above conjecture. A sharper error estimate than previously obtained is presented in this paper. A formula for a finite, optimal $c$ value that minimizes the solution error for a given grid size is obtained. Using residual errors, constants in error estimate and optimal $c$ formula can be obtained. These results are supported by numerical examples.

Unbounded domains often appear in engineering applications, such as acoustic or elastic wave radiation from a body immersed in an infinite medium. To simulate the unboundedness of the domain special boundary conditions have to be imposed: the Sommerfeld radiation condition. In the present work we focused on steady-state wave propagation. The objective of this research is to obtain accurate prediction of phenomena occurring in exterior acoustics and elastodynamics and ensure the quality of the solutions even for high wavenumbers. To achieve this aim, we develop higher-order domain-based schemes: Spectral Element Method (SEM) coupled to Dirichlet-to-Neumann (DtN ), Perfectly Matched Layer (PML) and Infinite Element (IEM) methods. Spectral elements combine the rapid convergence rates of spectral methods with the geometric flexibility of the classical finite element methods. The interpolation is based on Chebyshev and Legendre polynomials. This work presents an implementation of these techniques and their validation exploiting some benchmark problems. A detailed comparison between the DtN, PML and IEM is made in terms of accuracy and convergence, conditioning and computational cost.

The Singular Value Decomposition (SVD) has many applications in image processing. The SVD can be used to restore a corrupted image by separating significant information from the noise in the image data set. This thesis outlines broad applications that address current problems in digital image processing. In conjunction with SVD filtering, image compression using the SVD is discussed, including the process of reconstructing or estimating a rank reduced matrix representing the compressed image. Numerical plots and error measurement calculations are used to compare results of the two SVD image restoration techniques, as well as SVD image compression. The filtering methods assume that the images have been degraded by the application of a blurring function and the addition of noise. Finally, we present numerical experiments for the SVD restoration and compression to evaluate our computation.

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The unimodality of a function seems a simple concept. But in the Euclidean space R^m, m=3,4,..., it is not easy to define. We have an easy tool to find the minimum point of a unimodal function. The goal of this project is to formalize and support distinctive strategies that typically guarantee convergence. Support is given both by analytic arguments and simulation study. Application is envisioned in low-dimensional but non-trivial problems. The convergence of the proposed iterated grid search algorithm is presented along with the results of particular application studies. It has been recognized that the derivative methods, such as the Newton-type method, are not entirely satisfactory, so a variety of other tools are being considered as alternatives. Many other tools have been rejected because of apparent manipulative difficulties. But in our current research, we focus on the simple algorithm and the guaranteed convergence for unimodal function to avoid the possible chaotic behavior of the function. Furthermore, in case the loss function to be optimized is not unimodal, we suggest a weaker condition: almost (noisy) unimodality, under which the iterated grid search finds an estimated optimum point. / Ph. D.

L'algorithme de Levenberg-Marquardt (LM) est parmi les algorithmes les plus populaires pour la résolution des problèmes des moindres carrés non linéaire. Motivés par la structure des problèmes de l'assimilation de données, nous considérons dans cette thèse l'extension de l'algorithme LM aux situations dans lesquelles le sous problème linéarisé, qui a la forme min||Ax - b ||^2, est résolu de façon approximative, et/ou les données sont bruitées et ne sont précises qu'avec une certaine probabilité. Sous des hypothèses appropriées, on montre que le nouvel algorithme converge presque sûrement vers un point stationnaire du premier ordre. Notre approche est appliquée à une instance dans l'assimilation de données variationnelles où les modèles stochastiques du gradient sont calculés par le lisseur de Kalman d'ensemble (EnKS). On montre la convergence dans L^p de l'EnKS vers le lisseur de Kalman, quand la taille de l'ensemble tend vers l'infini. On montre aussi la convergence de l'approche LM-EnKS, qui est une variante de l'algorithme de LM avec l'EnKS utilisé comme solveur linéaire, vers l'algorithme classique de LM ou le sous problème est résolu de façon exacte. La sensibilité de la méthode de décomposition en valeurs singulières tronquée est étudiée. Nous formulons une expression explicite pour le conditionnement de la solution des moindres carrés tronqués. Cette expression est donnée en termes de valeurs singulières de A et les coefficients de Fourier de b. / The Levenberg-Marquardt algorithm (LM) is one of the most popular algorithms for the solution of nonlinear least squares problems. Motivated by the problem structure in data assimilation, we consider in this thesis the extension of the LM algorithm to the scenarios where the linearized least squares subproblems, of the form min||Ax - b ||^2, are solved inexactly and/or the gradient model is noisy and accurate only within a certain probability. Under appropriate assumptions, we show that the modified algorithm converges globally and almost surely to a first order stationary point. Our approach is applied to an instance in variational data assimilation where stochastic models of the gradient are computed by the so-called ensemble Kalman smoother (EnKS). A convergence proof in L^p of EnKS in the limit for large ensembles to the Kalman smoother is given. We also show the convergence of LM-EnKS approach, which is a variant of the LM algorithm with EnKS as a linear solver, to the classical LM algorithm where the linearized subproblem is solved exactly. The sensitivity of the trucated sigular value decomposition method to solve the linearized subprobems is studied. We formulate an explicit expression for the condition number of the truncated least squares solution. This expression is given in terms of the singular values of A and the Fourier coefficients of b.

Cette thèse porte sur certains aspects symboliques-numériques du problème des paires invariantes pour les polynômes de matrices. Les paires invariantes généralisent la définition de valeur propre / vecteur propre et correspondent à la notion de sous-espaces invariants pour le cas nonlinéaire. Elles trouvent leurs applications dans le calcul numérique de plusieurs valeurs propres d’un polynôme de matrices; elles présentent aussi un intérêt dans le contexte des systèmes différentiels. En utilisant une approche basée sur les intégrales de contour, nous déterminons des expressions du nombre de conditionnement et de l’erreur rétrograde pour le problème du calcul des paires invariantes. Ensuite, nous adaptons la méthode des moments de Sakurai-Sugiura au calcul des paires invariantes et nous étudions le comportement de la version scalaire et par blocs de la méthode en présence de valeurs propres multiples. Le résultats obtenus à l’aide des approches directes peuvent éventuellement être améliorés numériquement grâce à une méthode itérative: nous proposons ici une comparaison de deux variantes de la méthode de Newton appliquée aux paires invariantes. Le problème des solvants de matrices est très proche de celui des paires invariants. Le résultats présentés ci-dessus sont donc appliqués au cas des solvants pour obtenir des expressions du nombre de conditionnement et de l’erreur, et un algorithme de calcul basé sur la méthode des moments. De plus, nous étudions le lien entre le problème des solvants et la transformation des polynômes de matrices en forme triangulaire. / In this thesis, we study some symbolic-numeric aspects of the invariant pair problem for matrix polynomials. Invariant pairs extend the notion of eigenvalue-eigenvector pairs, providing a counterpart of invariant subspaces for the nonlinear case. They have applications in the numeric computation of several eigenvalues of a matrix polynomial; they also present an interest in the context of differential systems. Here, a contour integral formulation is applied to compute condition numbers and backward errors for invariant pairs. We then adapt the Sakurai-Sugiura moment method to the computation of invariant pairs, including some classes of problems that have multiple eigenvalues, and we analyze the behavior of the scalar and block versions of the method in presence of different multiplicity patterns. Results obtained via direct approaches may need to be refined numerically using an iterative method: here we study and compare two variants of Newton’s method applied to the invariant pair problem. The matrix solvent problem is closely related to invariant pairs. Therefore, we specialize our results on invariant pairs to the case of matrix solvents, thus obtaining formulations for the condition number and backward errors, and a moment-based computational approach. Furthermore, we investigate the relation between the matrix solvent problem and the triangularization of matrix polynomials.

The Generalized Finite Element Method (GFEM) is essentially a partition of unity based method (PUM) that explores the Partition of Unity (PoU) concept to match a set of functions chosen to efficiently approximate the solution locally. Despite its well-known advantages, the method may present some drawbacks. For instance, increasing the approximation space through enrichment functions may introduce linear dependences in the solving system of equations, as well as the appearance of blending elements. To address the drawbacks pointed out above, some improved versions of the GFEM were developed. The Stable GFEM (SGFEM) is a first version hereby considered in which the GFEM enrichment functions are modified. The Higher Order SGFEM proposes an additional modification for generating the shape functions attached to the enriched patch. This research aims to present and numerically test these new versions recently proposed for the GFEM. In addition to highlighting its main features, some aspects about the numerical integration when using the higher order SGFEM, in particular are also addressed. Hence, a splitting rule of the quadrilateral element area, guided by the PoU definition itself is described in detail. The examples chosen for the numerical experiments consist of 2-D panels that present favorable geometries to explore the advantages of each method. Essentially, singular functions with good properties to approximate the solution near corner points and polynomial functions for approximating smooth solutions are examined. Moreover, a comparison among the conventional FEM and the methods herein described is made taking into consideration the scaled condition number and rates of convergence of the relative errors on displacements. Finally, the numerical experiments show that the Higher Order SGFEM is the more robust and reliable among the versions of the GFEM tested. / O Método dos Elementos Finitos Generalizados (MEFG) é essencialmente baseado no método da partição da unidade, que explora o conceito de partição da unidade para compatibilizar um conjunto de funções escolhidas para localmente aproximar de forma eficiente a solução. Apesar de suas vantagens bem conhecidas, o método pode apresentar algumas desvantagens. Por exemplo, o aumento do espaço de aproximação por meio das funções de enriquecimento pode introduzir dependências lineares no sistema de equações resolvente, assim como o aparecimento de elementos de mistura. Para contornar as desvantagens apontadas acima, algumas versões aprimoradas do MEFG foram desenvolvidas. O MEFG Estável é uma primeira versão aqui considerada na qual as funções de enriquecimento do MEFG são modificadas. O MEFG Estável de ordem superior propõe uma modificação adicional para a geração das funções de forma atreladas ao espaço enriquecido. Esta pesquisa visa apresentar e testar numericamente essas novas versões do MEFG recentemente propostas. Além de destacar suas principais características, alguns aspectos sobre a integração numérica quando usado o MEFG Estável de ordem superior, em particular, são também abordados. Por exemplo, detalha-se uma regra de divisão da área do elemento quadrilateral, guiada pela própria definição de sua partição da unidade. Os exemplos escolhidos para os experimentos numéricos consistem em chapas com geometrias favoráveis para explorar as vantagens de cada método. Essencialmente, examinam-se funções singulares com boas propriedades de aproximar a solução nas vizinhanças de vértices de cantos, bem como funções polinomiais para aproximar soluções suaves. Ademais, uma comparação entre o MEF convencional e os métodos aqui descritos é feita levando-se em consideração o número de condição do sistema escalonado e as razões de convergência do erro relativo em deslocamento. Finalmente, os experimentos numéricos mostram que o MEFG Estável de ordem superior é a mais robusta e confiável entre as versões do MEFG testadas.