Abstract High computational complexity constrains truly exhaustive computer searches for good space-time (ST) coded modulations mostly to low constraint length space-time trellis codes (STTrCs). Such codes are primarily devised to achieve maximum transmit diversity gain. Due to their low memory order, optimization based on the design criterion of secondary importance typically results in rather modest coding gains. As another disadvantage of limited freedom, the different low memory order STTrCs are almost exclusively constructed for either slow or fast fading channels. Therefore in practical applications characterized by extremely variable Doppler frequencies, the codes typically fail to demonstrate desired robustness. On the other hand, the main drawback of eventually increased constraint lengths is the prohibitively large decoding complexity, which may increase exponentially if optimal maximum-likelihood decoding (MLD) is applied at the receiver. Therefore, robust ST coded modulation schemes with large equivalent memory orders structured as to allow sub-optimal, low complexity, iterative decoding are needed. To address the aforementioned issues, this thesis proposes parallel concatenated space-time turbo coded modulation (STTuCM). It is among the earliest multiple-input multiple-output (MIMO) coded modulation designs built on the intersection of ST coding and turbo coding. The systematic procedure for building an equivalent recursive STTrC (Rec-STTrC) based on the trellis diagram of an arbitrary non-recursive STTrC is first introduced. The parallel concatenation of punctured constituent Rec-STTrCs designed upon the non-recursive Tarokh et al. STTrCs (Tarokh-STTrCs) is evaluated under different narrow-band frequency flat block fading channels. Combined with novel transceiver designs, the applications for future wide-band code division multiple access (WCDMA) and orthogonal frequency division multiplexing (OFDM) based broadband radio communication systems are considered. The distance spectrum (DS) interpretation of the STTuCM and union bound (UB) performance analysis over slow and fast fading channels reveal the importance of multiplicities in the ST coding design. The modified design criteria for space-time codes (STCs) are introduced that capture the joint effects of error coefficients and multiplicities in the two dimensional DS of a code. Applied to STTuCM, such DS optimization resulted in a new set of constituent codes (CCs) for improved and robust performance over both slow and fast fading channels. A recursive systematic form with a primitive equivalent feedback polynomial is assumed for CCs to assure good convergence in iterative decoding. To justify such assumptions, the iterative decoding convergence analysis based on the Gaussian approximation of the extrinsic information is performed. The DS interpretation, introduced with respect to an arbitrary defined effective Hamming distance (EHD) and effective product distance (EPD), is applicable to the general class of geometrically non-uniform (GNU) CCs. With no constrains on the implemented information interleaving, the STTuCM constructed from newly designed CCs achieves full spatial diversity over quasi-static fading channels, the condition commonly identified as the most restrictive for robust performance over a variety of Doppler spreads. Finally, the impact of bit-wise and symbol-wise information interleaving on the performance of STTuCM is studied.

>Magister Scientiae - MSc / We present the Black-Scholes Merton partial differential equation (BSMPDE) and its analytical solution. We present the Black-Scholes option pricing model and list some limitations of this model. We also present a nonlinear model (the Frey-Patie model) that may improve on one of these limitations. We apply various numerical methods on the BSMPDE and run simulations to compare which method performs best in approximating the value of a European put option based on the maximum errors each method produces when we vary some parameters like the interest rate and the volatility. We re-apply the same finite difference methods on the nonlinear model. / 2025

In this dissertation we discuss bilinear immersed finite elements (IFE) for solving interface problems. The related research works can be categorized into three aspects: (1) the construction of the bilinear immersed finite element spaces; (2) numerical methods based on these IFE spaces for solving interface problems; and (3) the corresponding error analysis. All of these together form a solid foundation for the bilinear IFEs. The research on immersed finite elements is motivated by many real world applications, in which a simulation domain is often formed by several materials separated from each other by curves or surfaces while a mesh independent of interface instead of a body-fitting mesh is preferred. The bilinear IFE spaces are nonconforming finite element spaces and the mesh can be independent of interface. The error estimates for the interpolation of a Sobolev function in a bilinear IFE space indicate that this space has the usual approximation capability expected from bilinear polynomials, which is <i>O</i>(<i>h</i>²) in <i>L</i>² norm and <i>O</i>(<i>h</i>) in <i>H</i>¹ norm. Then the immersed spaces are applied in Galerkin, finite volume element (FVE) and discontinuous Galerkin (DG) methods for solving interface problems. Numerical examples show that these methods based on the bilinear IFE spaces have the same optimal convergence rates as those based on the standard bilinear finite element for solutions with certain smoothness. For the symmetric selective immersed discontinuous Galerkin method based on bilinear IFE, we have established its optimal convergence rate. For the Galerkin method based on bilinear IFE, we have also established its convergence. One of the important advantages of the discontinuous Galerkin method is its flexibility for both <i>p</i> and <i>h</i> mesh refinement. Because IFEs can use a mesh independent of interface, such as a structured mesh, the combination of a DG method and IFEs allows a flexible adaptive mesh independent of interface to be used for solving interface problems. That is, a mesh independent of interface can be refined wherever needed, such as around the interface and the singular source. We also develop an efficient selective immersed discontinuous Galerkin method. It uses the sophisticated discontinuous Galerkin formulation only around the locations needed, but uses the simpler Galerkin formulation everywhere else. This selective formulation leads to an algebraic system with far less unknowns than the immersed DG method without scarifying the accuracy; hence it is far more efficient than the conventional discontinuous Galerkin formulations. / Ph. D.

Submitted by Cláudia Bueno (claudiamoura18@gmail.com) on 2015-10-29T19:04:41Z No. of bitstreams: 2 Tese - Tiberio Bittencourt de Oliveira Martins.pdf: 1155588 bytes, checksum: add1eac74c4397efc29678341b834448 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-11-03T14:25:04Z (GMT) No. of bitstreams: 2 Tese - Tiberio Bittencourt de Oliveira Martins.pdf: 1155588 bytes, checksum: add1eac74c4397efc29678341b834448 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-11-03T14:25:04Z (GMT). No. of bitstreams: 2 Tese - Tiberio Bittencourt de Oliveira Martins.pdf: 1155588 bytes, checksum: add1eac74c4397efc29678341b834448 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-06-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Apresentamos, nesta tese, uma an álise da convergência do m étodo de Newton inexato com tolerância de erro residual relativa e uma an alise semi-local de m etodos de Newton robustos exato e inexato, objetivando encontrar uma singularidade de um campo de vetores diferenci avel de nido em uma variedade Riemanniana completa, baseados no princ pio majorante a m invariante. Sob hip oteses locais e considerando uma fun ção majorante geral, a Q-convergância linear do m etodo de Newton inexato com uma tolerância de erro residual relativa xa e provada. Na ausência dos erros, a an alise apresentada reobtem o teorema local cl assico sobre o m etodo de Newton no contexto Riemanniano. Na an alise semi-local dos m etodos exato e inexato de Newton apresentada, a cl assica condi ção de Lipschitz tamb em e relaxada usando uma fun ção majorante geral, permitindo estabelecer existência e unicidade local da solu ção, uni cando previamente resultados pertencentes ao m etodo de Newton. A an alise enfatiza a robustez, a saber, e dada uma bola prescrita em torno do ponto inicial que satifaz as hip oteses de Kantorovich, garantindo a convergência do m etodo para qualquer ponto inicial nesta bola. Al em disso, limitantes que dependem da função majorante para a taxa de convergência Q-quadr atica do m étodo exato e para a taxa de convergência Q-linear para o m etodo inexato são obtidos. / A local convergence analysis with relative residual error tolerance of inexact Newton method and a semi-local analysis of a robust exact and inexact Newton methods are presented in this thesis, objecting to nd a singularity of a di erentiable vector eld de ned on a complete Riemannian manifold, based on a ne invariant majorant principle. Considering local assumptions and a general majorant function, the Q-linear convergence of inexact Newton method with a xed relative residual error tolerance is proved. In the absence of errors, the analysis presented retrieves the classical local theorem on Newton's method in Riemannian context. In the semi-local analysis of exact and inexact Newton methods presented, the classical Lipschitz condition is also relaxed by using a general majorant function, allowing to establish the existence and also local uniqueness of the solution, unifying previous results pertaining Newton's method. The analysis emphasizes robustness, being more speci c, is given a prescribed ball around the point satisfying Kantorovich's assumptions, ensuring convergence of the method for any starting point in this ball. Furthermore, the bounds depending on the majorant function for Q-quadratic convergence rate of the exact method and Q-linear convergence rate of the inexact method are obtained.

In this thesis, we analyze an adaptive discontinuous finite element method for symmetric second order linear elliptic operators. Moreover, we obtain a fully computable convergence analysis on the broken energy seminorm in first order symmetric interior penalty discontin- uous Galerkin finite element approximations of this problem. The method is formulated on nonconforming meshes made of triangular elements with first order polynomial in two di- mension. We use an estimator which is completely free of unknown constants and provide a guaranteed numerical bound on the broken energy norm of the error. This estimator is also shown to provide a lower bound for the broken energy seminorm of the error up to a constant and higher order data oscillation terms. Consequently, the estimator yields fully reliable, quantitative error control along with efficiency. As a second problem, explicit expression for constants of the inverse inequality are given in 1D, 2D and 3D. Increasing mathematical analysis of finite element methods is motivating the inclusion of mesh dependent terms in new classes of methods for a variety of applications. Several inequalities of functional analysis are often employed in convergence proofs. Inverse estimates have been used extensively in the analysis of finite element methods. It is char- acterized as tools for the error analysis and practical design of finite element methods with terms that depend on the mesh parameter. Sharp estimates of the constants of this inequality is provided in this thesis.

In this thesis we focus on multiscale and multiphysics problems. We derive a posteriori error estimates for a one way coupled multiphysics problem, using the dual weighted residual method. Such estimates can be used to drive local mesh refinement in adaptive algorithms, in order to efficiently obtain good accuracy in a desired goal quantity, which we demonstrate numerically. Furthermore we prove existence and uniqueness of finite element solutions for a two way coupled multiphysics problem. The possibility of deriving dual weighted a posteriori error estimates for two way coupled problems is also addressed. For a two way coupled linear problem, we show numerically that unless the coupling of the equations is to strong the propagation of errors between the solvers goes to zero. We also apply a variational multiscale method to both an elliptic and a hyperbolic problem that exhibits multiscale features. The method is based on numerical solutions of decoupled local fine scale problems on patches. For the elliptic problem we derive an a posteriori error estimate and use an adaptive algorithm to automatically tune the resolution and patch size of the local problems. For the hyperbolic problem we demonstrate the importance of how to construct the patches of the local problems, by numerically comparing the results obtained for symmetric and directed patches.

Philosophiae Doctor - PhD / The global impact of the converging dual epidemics of tuberculosis (TB) and human immunodeficiency virus (HIV) is one of the major public health challenges of our time, because in many countries, human immunodeficiency virus (HIV) and mycobacterium tuberculosis (TB) are among the leading causes of morbidity and mortality. It is found that infection with HIV increases the risk of reactivating latent TB infection, and HIV-infected individuals who acquire new TB infections have high rates of disease progression. Research has shown that these two diseases are enormous public health burden, and unfortunately, not much has been done in terms of modeling the dynamics of HIV-TB co-infection at a population level. In this thesis, we study these models and design and analyze robust numerical methods to solve them. To proceed in this direction, first we study the sub-models and then the full model. The first sub-model describes the transmission dynamics of HIV that accounts for behavior change. The impact of HIV educational campaigns is also studied. Further, we explore the effects of behavior change and different responses of individuals to educational campaigns in a situation where individuals may not react immediately to these campaigns. This is done by considering a distributed time delay in the HIV sub-model. This leads to Hopf bifurcations around the endemic equilibria of the model. These bifurcations correspond to the existence of periodic solutions that oscillate around the equilibria at given thresholds. Further, we show how the delay can result in more HIV infections causing more increase in the HIV prevalence. Part of this study is then extended to study a co-infection model of HIV-TB. A thorough bifurcation analysis is carried out for this model. Robust numerical methods are then designed and analyzed for these models. Comparative numerical results are also provided for each model. / South Africa

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Neste trabalho, uma abordagem micromecânica com aproximação da geometria dada por funções de Bézier triangulares com continuidade geométrica G1 é inserida ao Método dos Elementos de Contorno, o qual é aplicado em problemas da elastostática tridimensional. Para consideração do efeito microestrutural, foi utilizado a teoria gradiente elástica simplificada de Aifantis, a qual é uma particularização da teoria geral de Mindlin. Nesta teoria, um argumento variacional é estabelecido para determinar todas as possíveis condições de contorno, clássica e não-clássica, para o problema de valor de contorno geral. A partir deste argumento, a solução fundamental da elasticidade gradiente é explicitada e com o auxílio da identidade integral recíproca é construído a representação integral de contorno. Para tornar o problema de valor de contorno bem-posto, em adição à representação integral de contorno para deslocamento, uma segunda representação integral para derivada normal do deslocamento foi utilizada. Expressões integrais para deslocamento e tensão em pontos internos são apresentadas. Todos os núcleos das equações integrais são explicitamente desenvolvidos. Para a discretização do MEC foram utilizados elementos triangulares curvos, aproximados tanto para a geometria quanto para os parâmetros físicos por funções de Proriol (com características espectrais) e por funções aqui chamadas de Polinomiais, onde esta última é construída a partir de uma base nodal equidistante e pela imposição da partição da unidade. Entretanto estas funções aproximadoras garantem apenas continuidade C0 entre os elementos triangulares, ou seja, a garantia da continuidade do plano tangente não necessariamente é satisfeita. Com o objetivo de anular o termo de integral de linha presente na formulação microestrutural, a hipótese de superfície suave se faz necessária e assim funções de Bézier com continuidade geométrica G1, a qual depende apenas da posição e das normais dos nós nos vértices da malha triangular é utilizada. Para auxiliar na obtenção das coordenadas e das normais nodais para geometrias complexas foi utilizado o software de computação gráfica BlenderTM 2.7, o qual foi acoplado ao programa do MEC elastostático gradiente. Na sequência foi verificada, por meio de exemplos, a suavidade na intersecção entre os elementos triangulares G1 e estes foram comparados com as aproximações de Proriol e Polinomial. Em seguida, as singularidades presentes nas soluções fundamentais foram tratadas através da expansão em série de Laurent aplicada à técnica de subtração de singularidade. Condições necessárias e suficientes para a convergência das expansões em série das soluções fundamentais, estimador do erro para estas expansões, assim como, a correlação matemática entre o tamanho da malha e o parâmetro micromecânico g foram estabelecidos. Expressões explicitas da série de Laurent dos núcleos das integrais singulares e hipersingulares do MEC clássico e não clássico foram apresentadas. A verificação do tratamento da singularidade aplicado a elementos triangulares curvos foi realizada, tanto na direção radial quanto na direção angular. E pôde ser observado que ocorre uma perda de eficiência no tratamento da singularidade na direção angular, devida a presença do efeito de camada limite para elementos curvos distorcidos. Entretanto, este efeito de quase singularidade pode ser amenizado por meio da abordagem micromecânica, uma vez que foi observado menor presença do efeito da camada limite à medida que o parâmetro g é diminuído. Por último, foi desenvolvido um programa na linguagem FORTRAN 11.0, o qual contempla as abordagens clássica e micromecânica com continuidade geométrica G1. Sua validação foi feita por meio de exemplos considerados Benchmarks. / In this work, a micromechanical approach with approximation of geometry solved by Bézier triangular functions that guaranty continuity G1 is inserted to the Boundary element Method (BEM). This formulation is applied in three-dimensional elastostatic problems. The simplified elastic gradient theory proposed by Aifantis, which is a particularization of the general theory of Mindlin is used to consider the microstructural effect. In this theory a variational argument is established to determine all possible boundary conditions, classical and non-classical, for the general boundary value problem. From this argument, the fundamental solution of the gradient elasticity is explicited and by the reciprocal integral identity the boundary integral representation is achieved. In addition to the boundary integral representation for dispacement, a second integral representation regarding its normal derivative is used to make the well-posed boundary value problem. Integral expressions for displacement and stress on internal points are also presented. All kernels in the integral equations are explicitly developed. Curved triangular elements are used for the discretization of the BEM. The approximation of both the geometry and physical parameters is performed by Proriol functions (with spectral characteristics) and by Polynomial functions. The last is built from an equidistant nodal basis enforcing the partition of unity. However these approximating functions ensure only C0 continuity between the triangular elements, that is, the tangent plane continuity assurance is not necessarily satisfied. In order to cancel line integral terms in the microstructural approach, the hypothesis of smooth surface is required and thus Bézier function with geometric continuity G1, which depends only on the position and the normal of the nodes at the vertices of the triangular mesh is used. In this study the computer graphics software called BlenderTM 2.7 is used to assist in obtaining coordinates and normal vectors at nodes when complex geometries are analyzed. BlenderTM 2.7 is coupled to the gradient elastic BEM program. The smoothness of the resulting mesh using G1 elements is compared to Proriol and Polynomial approximations by means of simple examples. The singularities present in the fundamental solutions are treated by employing the expansion in Laurent series and the singularity subtraction technique. Necessary and sufficient conditions for the convergence of expansions in series of fundamental solutions, error estimator for these expansions, as well as the mathematical correlation between the size of the mesh and the micromechanical parameter, g, are established. Explicit expressions of Laurent series of the classical and micromechanical kernels forthe singular and hipersingular BEM integrals are presented. Treatment of singularity, both in the radial direction and in the angular direction, applied to curved triangular elements is verified. It can be observed that there is a loss of efficiency in the treatment of singularity in the angular direction, due to the presence of the boundary layer effect for distorted curved boundary elements. However, this nearly singularity effect could be alleviated by micromechanics approach, since minor boundary layer effect was observed as the parameter g is decreased. Finally, using FORTRAN 11.0 language, a computational code is developed, which includes the classic and micromechanics approach with geometric continuity G1, and its results are validated by means of Benchmark examples.

Submitted by Maria Aparecida (cidazen@gmail.com) on 2017-08-22T13:19:28Z No. of bitstreams: 1 Patricia Moraes Rêgo.pdf: 1511056 bytes, checksum: 21108136b08107eeb212f5d74ed79ef7 (MD5) / Made available in DSpace on 2017-08-22T13:19:28Z (GMT). No. of bitstreams: 1 Patricia Moraes Rêgo.pdf: 1511056 bytes, checksum: 21108136b08107eeb212f5d74ed79ef7 (MD5) Previous issue date: 2007-08-03 / FAPEMA / In this work are proposed models and a convergence analysis of a hierarchical genetic algorithm for the linear quadratic regulator design loop recovery through LQG/LTR controllers. Models are oriented to the weighting and covariance matrices searching of the performance indices of the LQR and LQG design, respectively, and to the selection of the matrices for the LQR design loop recovery gain. The convergence analysis aims at promoting the enhancement of the algorithm performance, as well as to generate satisfactory solutions and speed up the convergence time. The algorithm performance is evaluated with respect to the e ects of an elitist strategy embodied into the algorithm and to variations in the values of some given parameters of the algorithm. The proposed methodology is evaluated in a multi-variable dynamical system representing an aircraft. / Propõe-se neste trabalho os modelos e a análise de convergência de um algoritmo genético hierárquico para recuperação da malha de projeto do regulador linear quadrático por controladores LQG/LTR (Linear Quadratic Gaussian/Loop Transfer Recovery). Os modelos dedicam-se à busca das matrizes de ponderações e covariâncias dos índices de desempenho dos projetos de controladores LQR (Linear Quadratic Regulator) e LQG (Linear Quadratic Gaussian), respectivamente, e à seleção de matrizes de ajuste para o ganho de recuperação da malha do projeto LQR. O objetivo da análise de convergência é promover melhorias no desempenho do algoritmo no sentido de gerar soluções satisfatórias e acelerar o tempo de convergência. O desempenho do algoritmo é avaliado em relação aos efeitos de uma estratégia elitista incorporada ao algoritmo e à variações nos valores de determinados parâmetros do algoritmo. A metodologia proposta é avaliada em um sistema dinâmico multivariável que representa uma aeronave.