Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-02-18T13:23:35Z No. of bitstreams: 1 martaaparecidaferreiradeoliveirabritto.pdf: 973294 bytes, checksum: 3bb46c887b254be0f099b06ad0b53d7c (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-02-26T13:30:03Z (GMT) No. of bitstreams: 1 martaaparecidaferreiradeoliveirabritto.pdf: 973294 bytes, checksum: 3bb46c887b254be0f099b06ad0b53d7c (MD5) / Made available in DSpace on 2016-02-26T13:30:03Z (GMT). No. of bitstreams: 1 martaaparecidaferreiradeoliveirabritto.pdf: 973294 bytes, checksum: 3bb46c887b254be0f099b06ad0b53d7c (MD5) Previous issue date: 2014-03-18 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho, abordamos algumas aplicações de matrizes que julgamos ser possível inserir na educação básica, com o intuito de fornecer ao aluno uma visão da utilidade da matemática no mundo real, contribuindo para tornar o seu ensino mais dinâmico e atraente. As aplicações que apontamos são criptografia, cadeias de Markov, grafos, transformações no plano e sistemas lineares. Percebemos que o tratamento dado a este tópico aparece, em geral, de maneira muito tímida nos livros didáticos de ensino médio e que raramente aparecem atividades que as envolvam. No entanto, o tema é muito abrangente e rico, podendo ser relacionado a inúmeras áreas do conhecimento humano, como administração, economia, biologia, computação e física, podendo ser uma ferramenta útil para as atividades interdisciplinares. Notamos ser possível explorar o conceito de matriz, sua representação, suas operações, propriedades e definições através de problemas contextualizados. No decorrer deste trabalho, coletamos sugestões de atividades presentes em artigos, dissertações e livros de Álgebra Linear. / In this paper, we intent to make an approach of some matrix application that we judge possible to introduce in basic level education, in order to give students a broader vision of the real world mathematical utility, contributing to make a more dynamic and attractive mathematics teaching. The applications are cryptography, Markov chains, graphs, plane transforms and linear systems. We realized that the treatment given in basic textbooks to this topic is, frequently, sketchy and superficial, scarcely happening to make activities encompassing these topics. However, this is a very rich and broadening topic, that can be related to many areas of human knowledge such as Administration, Economy, Biology, Computer Science and Physics, working as an useful tool to educational interdisciplinarity . Then, it is possible to explore the matrix concept, its representations, its operations, properties and definitions to contextualized problems. Along this paper, we collect several activity suggestions found in articles, essays and Linear Algebra textbooks.

Álgebra Linear e particularmente a teoria das matrizes e dos sistemas lineares são tópicos da Matemática que têm aplicações, não só dentro da própria Matemática, mas também em várias outras áreas do conhecimento humano. Neste trabalho, além de estudar estas teorias, estudamos algumas de suas aplicações na área da Economia, como em modelos lineares de produção, modelos de Markov para emprego e modelos de benefícios obtidos no pagamento de impostos após realizarmos contribuições filantrópicas. / Linear Algebra and particularly matrices and linear systems theory are topics in Mathematics with many applications in several branches of science. In this work we study this theory and some of its applications in Economy as in linear models of production, Markov models of employment and tax benefits of charitable contributions.

Le travail de cette thèse porte sur les raisons structurales derrière la formation de cristaux maclés. Ce travail ouvre une voie pour un futur développement de protocoles de synthèse afin de réduire l'occurrence de macles. La motivation de cette étude est que la présence de macles affecte négativement les propriétés physico-chimiques des matériaux d'intérêts technologiques et réduit aussi la qualité des données expérimentales sur lesquelles se fonde l'analyse structurelle. Ce dernier problème est particulièrement sensible dans le cas de cristaux ayant des paramètres de maille importantes, comme les macromolécules biologiques. Les principes de symétrie responsables du phénomène de maclage dans le cas d’une macle de transformation ou d'origine mécanique sont bien connues. En revanche dans le cas d’une macle de croissance, le maclage est toujours considéré comme un accident lié aux conditions aléatoires de croissance cristalline où à la cinétique, plutôt qu'à la thermodynamique. Une approche générale connue comme la « théorie réticulaire des macles » a été développée depuis le XIXe siècle, fondée sur l'existence d'un sous-réseau commun aux cristaux maclés, qui donne les conditions  nécessaires pour l'apparition d'une macle. Cette approche est cependant insuffisante pour déterminer la différence entre les macles avec le même degré de chevauchement des réseaux mais montrant une fréquence d'occurrence assez différente. Une approche structurale, fondée sur l'analyse de la symétrie propre des orbites cristallographiques a été proposée il y a plus d'un demi-siècle (Donnay et Curien, 1960), mais est restée à l'état embryonnaire, malgré une certaine reprise récente (Nespolo et Ferraris, 2009). En outre, l'idée qu'une interface commune aux cristaux maclés puisse contenir une opération reliant ces individus a été proposée (Holser, 1958) mais n'a jamais été portée à un plein développement. Dans cette thèse, nous présentons un développement algébrique de ces idées. Nous montrons que les conditions structurales nécessaires pour la formation d'une macle de croissance peuvent être formulées en se basant, notamment, sur la symétrie propre des orbites cristallographiques et sur le groupe sous-périodique de la couche transversale donnant la symétrie d'une couche commune. L'analyse détaillée dans cette thèse de trois macles fréquentes démontre une corrélation claire entre le degré de restauration de la structure par l'opération de maclage et la fréquence d'occurrence des macles. Un exemple négatif, à savoir une macle hypothétique dont on pourrait prévoir la formation  sur la base de la théorie réticulaire a aussi été analysé. Le fait que cette macle n'ait jamais été observée, en raison d’une faible restauration de la structure qui serait produite par l'opération de macle, confirme le bien fondé de l'approche. Nous nous attendons à ce que la généralisation de l'approche présentée dans cette thèse fournisse une procédure semi-automatique pour prévoir la probabilité de formation d'une macle. Cela permettrait aux personnes travaillant dans la synthèse cristalline démoduler la fréquence de maclage. Le procédé fait appel à la modification de la morphologie du cristal pour une plus grande exposition et le développement des faces cristallines qui présentent une interface défavorable pour le maclage. / This thesis addresses the structural rationale behind the formation of growth twins, with the purpose of opening a route to the future development of synthesis protocols to reduce the occurrence frequency of twinning. The reason for this effort is that twinning affects negatively the physico-chemical properties of materials and biomaterials of technological interests and reduces the quality of the experimental data on which the structural investigation is based. While on the one hand the reasons for twinning in transformation and mechanical twins are well understood, in the case of growth twins twinning is still seen as an accident linked to aleatory conditions where kinetics, rather than thermodynamics, plays a principal role. A general approach known as the reticular theory of twinning has been developed since the XIX century, based on the existence of a sublattice common to the twinned crystals, which gives the minimal necessary conditions for the occurrence of a twin. This approach is, however, insufficient to discriminate between twins with the same degree of lattice overlap but showing a fairly different occurrence frequency. A structural approach, based on the analysis of the eigensymmetry of the crystallographic orbits building a crystal structure was proposed more than half a century ago (Donnay and Curien, 1960) but remained at an embryonic state, despite some recent revival (Nespolo and Ferraris, 2009). Also, the idea that a slice common to the twinned individuals may contain an operation mapping these individuals was proposed (Holser, 1958) but never brought to a full development. In this thesis, we present a full development of these ideas and show that the structurally necessary conditions for the formation of a growth twin can be described on the basis of the eigensymmetry of the crystallographic orbits and on the sectional layer group giving the symmetry of the common slice. The detailed analysis of three well-know twins demonstrates a clear correlation between the degree of structural restoration by the twin operation and the occurrence frequency of the twins. The analysis of a negative example, i.e. of a hypothetical twin which one would expect on the basis of the reticular theory but has never been observed, strengthens the evidence of this correlation, because of the low structural restoration one would observe in that twin. We expect that the generalisation of the approach presented in this thesis through a semi-automatic procedure will provide crystal growers with a powerful tool to modulate the occurrence frequency of twinning through a modification of the crystal morphologies towards a larger exposure and development of crystal faces which represent an unfavorable interface for twinning.

This thesis aims at building a numerical linear algebra library with appropriate software engineering abstractions. Three areas of knowledge, namely, Numerical Linear Algebra (NLA), Software Engineering and Compiler Optimisation Techniques, are involved. Numerical simulation is widely used in a large number of distinct disciplines to help scientists understand and discover the world. The solutions to frequently occurring numerical problems have been implemented in subroutines, which were then grouped together to form libraries for ease of use. The design, implementation and maintenance of a NLA library require a great deal of work so that the other two topics, namely, software engineering and compiler optimisation techniques have emerged. Generally speaking, these both try to divide the system into smaller and controllable concerns, and allow the programmer to deal with fewer concerns at one time. Band matrix operation, as a new level of abstraction, is proposed for simplifying library implementation and enhancing extensibility for future functionality upgrades. Iteration Space Partitioning (ISP) is applied, in order to make the performance of this generalised implementation for band matrices comparable to that of the specialised implementations for dense and triangular matrices. The optimisation of ISP can be either programmed using the pointcut-advice model of Aspect-Oriented Programming, or integrated as part of a compiler. This naturally leads to a comparison of these two different techniques for resolving one fundamental problem. The thesis shows that software engineering properties of a library, such as modularity and extensibility, can be improved by the use of the appropriate level of abstraction, while performance is either not sacrificed at all, or at least the loss of performance is limited. In other words, the perceived trade-off between the use of high-level abstraction and fast execution is made less significant than previously assumed.

A range of numerical linear algebra problems that arise in finite element-based structural analysis are considered. These problems were encountered when implementing the finite element method in the software package Oasys GSA. We present novel solutions to these problems in the form of a new method for error detection, algorithms with superior numerical effeciency and algorithms with scalable performance on parallel computers. The solutions and their corresponding software implementations have been integrated into GSA's program code and we present results that demonstrate the use of these implementations by engineers to solve real-world structural analysis problems.

Orientador: Samuel Rocha de Oliveira / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Educação / Made available in DSpace on 2018-08-26T16:42:27Z (GMT). No. of bitstreams: 1 Cardoso_ValdineiCezar_D.pdf: 4997650 bytes, checksum: 1a1744fbebd33d7857dac964fbba6f66 (MD5) Previous issue date: 2014 / Resumo: Neste trabalho, buscamos investigar em que medida os vídeos digitais e a metodologia de ensino podem contribuir para a conceitualização em Álgebra Linear. Para isso, ministramos dois cursos, com 68 horas de duração cada um, em dois cenários: o primeiro com uma turma presencial e a gravação de pequenas partes das aulas e o segundo utilizando a metodologia das aulas reversas. Nosso referencial teórico foram as Teorias: dos Campos Conceituais, dos Registros de Representação Semiótica e Cognitiva da Aprendizagem Multimídia. Por meio deste estudo, identificamos e analisamos teoremas em ação que emergem durante a resolução de situações-problemas. A abordagem utilizada na investigação foi a pesquisa qualitativa, seguindo a abordagem de Campbell e Stanley (1979). Entre os resultados encontrados, destacamos que a forma como os estudantes utilizam os vídeos digitais para estudar Álgebra Linear está diretamente relacionada com a metodologia de ensino adotada pelo professor. Em particular, percebemos que o uso de vídeos, associado às aulas reversas, contribui para a aproximação entre estudantes e professor durante as aulas, o que facilita a mediação docente durante o processo de conceitualização nessa disciplina / Abstract: In this work, we sought to investigate to what extent digital videos and teaching methodology can contribute to the conceptualization in Linear Algebra. For this, we ministered two courses, which were 68 (sixty-eight) hours long, in two scenarios. The first class was with attendance and recordings of small parts of the lessons, the second class using methodology of the reverse lessons. Our theoretical framework was the theories of conceptual fields and semiotic representation registers and the cognitive theory of multimedia learning. Through this study, we identified and analyzed theorems in action that emerges during the resolution of problem situations. The approach used in the research was qualitative research, following the approach of Campbell and Stanley (1979). Between the results, we highlight that the way the students use the digital videos to study Linear Algebra is directly related with the methodology of teaching adopted by the teacher, in particular, we realized the use of the videos, associated to the reversed lessons contribute to the approach between students and teacher, during the lessons, which makes the teacher mediation easier during the process of conceptualization in this subject / Doutorado / Ensino de Ciencias e Matematica / Doutor em Multiunidades em Ensino de Ciências e Matemática

[pt] O produto tensorial é o formalismo adequado para desenvolver a técnica de separação de variáveis em sua generalidade. São estudadas representações tensoriais decompostas de transformações lineares e algumas aplicações recentes em análise numérica (o algoritmo de Beylkin). Os exemplos tratam da discretização do laplaciano em malhas retangulares, suas propriedades espectrais e seu cálculo funcional, com ênfase na função sinal. / [en] Separation of variables is adequately understood and extended by making use of tensor products. We consider linear transformations admitting tensor decompositions and some recent applications in numerical analysis (Beylkin s algorithm). The examples concern the discretization of the Laplacian on rectangular meshes, its spectral properties and functional calculus, with emphasis on its sign function.

abstract: Dimensionality reduction methods are examined for large-scale discrete problems, specifically for the solution of three-dimensional geophysics problems: the inversion of gravity and magnetic data. The matrices for the associated forward problems have beneficial structure for each depth layer of the volume domain, under mild assumptions, which facilitates the use of the two dimensional fast Fourier transform for evaluating forward and transpose matrix operations, providing considerable savings in both computational costs and storage requirements. Application of this approach for the magnetic problem is new in the geophysics literature. Further, the approach is extended for padded volume domains. Stabilized inversion is obtained efficiently by applying novel randomization techniques within each update of the iteratively reweighted scheme. For a general rectangular linear system, a randomization technique combined with preconditioning is introduced and investigated. This is shown to provide well-conditioned inversion, stabilized through truncation. Applying this approach, while implementing matrix operations using the two dimensional fast Fourier transform, yields computationally effective inversion, in memory and cost. Validation is provided via synthetic data sets, and the approach is contrasted with the well-known LSRN algorithm when applied to these data sets. The results demonstrate a significant reduction in computational cost with the new algorithm. Further, this new algorithm produces results for inversion of real magnetic data consistent with those provided in literature. Typically, the iteratively reweighted least squares algorithm depends on a standard Tikhonov formulation. Here, this is solved using both a randomized singular value de- composition and the iterative LSQR Krylov algorithm. The results demonstrate that the new algorithm is competitive with these approaches and offers the advantage that no regularization parameter needs to be found at each outer iteration. Given its efficiency, investigating the new algorithm for the joint inversion of these data sets may be fruitful. Initial research on joint inversion using the two dimensional fast Fourier transform has recently been submitted and provides the basis for future work. Several alternative directions for dimensionality reduction are also discussed, including iteratively applying an approximate pseudo-inverse and obtaining an approximate Kronecker product decomposition via randomization for a general matrix. These are also topics for future consideration. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2020

Large dense matrices are ubiquitous in scientific computing, arising from the discretization of integral operators associated with elliptic pdes, Schur complement methods, covariances in spatial statistics, kernel-based machine learning, and numerical optimization problems. Hierarchical matrices are an efficient way for storing the dense matrices of very large dimension that appear in these and related settings. They exploit the fact that the underlying matrices, while formally dense, are data sparse. They have a structure consisting of blocks many of which can be well-approximated by low rank factorizations. A hierarchical organization of the blocks avoids superlinear growth in memory requirements to store n × n dense matrices in a scalable manner, requiring O(n) units of storage with a constant depending on a representative rank k for the low rank blocks. The asymptotically optimal storage requirement of the resulting hierarchical matrices is a critical advantage, particularly in extreme computing environments, characterized by low memory per processing core. The challenge then becomes to develop the parallel linear algebra operations that can be performed directly on this compressed representation. In this dissertation, I implement a set of hierarchical basic linear algebra subroutines (HBLAS) optimized for GPUs, including hierarchical matrix vector multiplication, orthogonalization, compression, low rank updates, and matrix multiplication. I develop a library of open source batched kernel operations previously missing on GPUs for the high performance implementation of the H2 operations, while relying wherever possible on existing open source and vendor kernels to ride future improvements in the technology. Fast marshaling routines extract the batch operation data from an efficient representation of the trees that compose the hierarchical matrices. The methods developed for GPUs extend to CPUs using the same code base with simple abstractions around the batched routine execution. To demonstrate the scalability of the hierarchical operations I implement a distributed memory multi-GPU hierarchical matrix vector product that focuses on reducing communication volume and hiding communication overhead and areas of low GPU utilization using low priority streams. Two demonstrations involving Hessians of inverse problems governed by pdes and space-fractional diffusion equations show the effectiveness of the hierarchical operations in realistic applications.

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