Apresentamos as características de algoritmos para obter a solução de sistemas de equações lineares, utilizando computadores com arquitetura paralela e explorando os recursos de paralelismo neles embutidos. São descritos os modelos de computadores paralelos e os procedimentos para a criação de algoritmos paralelos. / The characteristics of algorithms to solve systems of linear equations, using parallel computers, are presented. The abstract models of these computers are described, as well as the methods to develop parallel algori thms.

Apresentamos as características de algoritmos para obter a solução de sistemas de equações lineares, utilizando computadores com arquitetura paralela e explorando os recursos de paralelismo neles embutidos. São descritos os modelos de computadores paralelos e os procedimentos para a criação de algoritmos paralelos. / The characteristics of algorithms to solve systems of linear equations, using parallel computers, are presented. The abstract models of these computers are described, as well as the methods to develop parallel algori thms.

Os problemas de engenharia atualmente têm aumentado a sua ordem de grandeza, por conta de diversos fatores. A modelagem em ambiente computacional dos mesmos esbarra em limitações, como grandes quantidades de tempo de processamento gastos com diversas simulações da modelagem e a pouca quantidade de memória disponível para alocar propriamente os problemas. A resolução de grandes sistemas de equações lineares, comumente abordado nos problemas atuais de engenharia, necessita da exploração das duas situações mencionadas anteriormente. A subárea computacional que permite explorar a redução do tempo e a possibilidade de alocação na memória de tais problemas é chamada de computação de alto desempenho. O objetivo deste trabalho é ilustrar o uso de softwares de resolução de sistemas de equações lineares, chamados de solvers, projetados para os ambientes computacionais de alto desempenho, testando-os e avaliando-os em um conjunto de matrizes conhecido, bem como abordar os detalhes computacionais envolvidos em tais procedimentos. / Engineering problems today have increased their order of magnitude, due to several factors. Modeling these problems with computers brings up certain limitations, as the amount of processing time needed for several simulations and the lack of available memory to properly allocate them. The resolution of large systems of linear equations, commonly discussed in current engineering problems, needs the exploration of the two situations mentioned above. The subarea that allows exploring the computational time reduction and the possibility of allocating memory in such problems is called high performance computing. The aim of this paper is to illustrate the use of software to solve systems of linear equations, called solvers, designed for high performance computing environments, to test and evaluate them for a set of matrices as well as to address the computational details involved in such procedures.

Apresentamos as características de algoritmos para obter a solução de sistemas de equações lineares, utilizando computadores com arquitetura paralela e explorando os recursos de paralelismo neles embutidos. São descritos os modelos de computadores paralelos e os procedimentos para a criação de algoritmos paralelos. / The characteristics of algorithms to solve systems of linear equations, using parallel computers, are presented. The abstract models of these computers are described, as well as the methods to develop parallel algori thms.

Submitted by isabela.moljf@hotmail.com (isabela.moljf@hotmail.com) on 2016-08-18T15:10:00Z No. of bitstreams: 1 brunosampaiodeoliveira.pdf: 1292975 bytes, checksum: ab81c43620a92db7b1c0c5cf34a64222 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-08-19T11:45:01Z (GMT) No. of bitstreams: 1 brunosampaiodeoliveira.pdf: 1292975 bytes, checksum: ab81c43620a92db7b1c0c5cf34a64222 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-08-19T11:45:31Z (GMT) No. of bitstreams: 1 brunosampaiodeoliveira.pdf: 1292975 bytes, checksum: ab81c43620a92db7b1c0c5cf34a64222 (MD5) / Made available in DSpace on 2016-08-19T11:45:31Z (GMT). No. of bitstreams: 1 brunosampaiodeoliveira.pdf: 1292975 bytes, checksum: ab81c43620a92db7b1c0c5cf34a64222 (MD5) Previous issue date: 2013-03-25 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O objetivo deste trabalho é propor a implementação do software MAXIMA no ensino de sistemas de equações lineares 2 x 2 em turma do 8º ano do Ensino Fundamental e de sistemas de equações lineares 3 x 3 em turmas do 2º ano do Ensino Médio. Além de destacar a importância da utilização de recursos tecnológicos nos dias atuais, o autor sugere algumas atividades e formas de abordar o tema com a utilização do programa. / The objective of this work is to propose the implementation of software MAXIMA in teaching systems of linear equations 2 x 2 in the 8th grade class of elementary school and systems of linear equations in 3 x 3 classes of 2nd year of high school. In addition to highlighting the importance of using technological resources nowadays, the author suggests some activities and ways of approaching the topic using the program.

The Multiplicative Schwarz Preconditioned Inexact Newton (MSPIN) algorithm is presented as a complement to Additive Schwarz Preconditioned Inexact Newton (ASPIN). At an algebraic level, ASPIN and MSPIN are variants of the same strategy to improve the convergence of systems with unbalanced nonlinearities; however, they have natural complementarity in practice. MSPIN is naturally based on partitioning of degrees of freedom in a nonlinear PDE system by field type rather than by subdomain, where a modest factor of concurrency can be sacrificed for physically motivated convergence robustness. ASPIN, originally introduced for decompositions into subdomains, is natural for high concurrency and reduction of global synchronization. The ASPIN framework, as an option for the outermost solver, successfully handles strong nonlinearities in computational fluid dynamics, but is barely explored for the highly nonlinear models of complex multiphase flow with capillarity, heterogeneity, and complex geometry. In this dissertation, the fully implicit ASPIN method is demonstrated for a finite volume discretization based on incompressible two-phase reservoir simulators in the presence of capillary forces and gravity. Numerical experiments show that the number of global nonlinear iterations is not only scalable with respect to the number of processors, but also significantly reduced compared with the standard inexact Newton method with a backtracking technique. Moreover, the ASPIN method, in contrast with the IMPES method, saves overall execution time because of the savings in timestep size. We consider the additive and multiplicative types of inexact Newton algorithms in the field-split context, and we augment the classical convergence theory of ASPIN for the multiplicative case. Moreover, we provide the convergence analysis of the MSPIN algorithm. Under suitable assumptions, it is shown that MSPIN is locally convergent, and desired superlinear or even quadratic convergence can be obtained when the forcing terms are picked suitably. Numerical experiments show that MSPIN can be significantly more robust than Newton methods based on global linearizations, and that MSPIN can be more robust than ASPIN, and maintain fast convergence even for challenging problems, such as high-Reynolds number Navier-Stokes equations.

Magister Educationis - MEd / Mathematical modelling is gaining extensive interest across the schooling sector worldwide, as it is deemed to develop learners with competencies set to deal with the demands of the fourth industrial revolution and being creative problem solvers. As mathematical modelling has only recently gained momentum across the mathematics curricula for schools in South Africa, many teachers may not be aware of the competencies that are needed to be developed in their learners through solving word problems, and even learners may not be aware of these essential modelling competencies. Hence, this mixed-methods approach study adopted a case-study design located within an interpretative paradigm to explore the levels of mathematical modelling competencies a sample of Grade 10 learners attending a Western Cape School demonstrated as they solved a set of word problems associated with the use of simultaneous equations. Additionally, data collected through observations and limited sets of semi-structured interviews were considered in the data analysis processes, which were largely driven by qualitative content analysis methods and supplemented with elementary descriptive statistical methods. The findings of this study showed that most of the learners demonstrated non-competency in modelling mainly because of their inability to understand the problem as evident in their failure to comprehend the context of a problem, inability to recognise important quantities associated with a problem, and muddled relationships if any. The study conjecture that the use of the English language could have been a barrier to the sample of English second language speakers understanding the problem. However, a very limited number of students showed partial modelling competency, as they were only able to understand the problem and build a correct model to solve the problem. Regrettably, these students lacked the knowledge of the heuristics for solving a system of linear equations correctly and completely and did not check or verify their answers. The extremely small number of learners, who demonstrated sufficient modelling competency, demonstrated sufficient understanding of the problem, built and solved the system of simultaneous linear equations successfully without necessarily checking or testing whether their answers satisfied the conditions of the problem. Hence, this study recommends that adequate focus be given to the role of language in understanding a problem, heuristic competencies to solve a system of linear equations should be strengthened, and the habit of checking the reasonableness of the solution should be encouraged and developed continuously across problem-solving tasks. Studying learners’ modelling competencies requires further work to add to the repertoire of this knowledge domain.

Under många år har formativ bedömning blivit ett begrepp som används flitigt i undervisningsdiskussioner. Detta för att resultatet utav arbetsmetoden varit så positivt. Man har sett att man genom att uppmärksamma elever på deras kunskapsnivå kan lägga upp en mer individuell strategi för hur de ska nå sitt slutmål. Syftet med denna studie är att undersöka hur gymnasielärare idag använder formativ bedömning som metod i sin undervisning. Mer specifikt rör denna studie de insatser som genomförts i undervisningsområdet linjära ekvationssystem. Studien är en kvantitativ undersökning som genomförts med hjälp av en enkät som låtit lärare återge sin personliga uppfattning om deras insats, upptäckter samt tankar om framtida utveckling. Resultatet visar att lärare i den svenska gymnasieskolan besitter kunskaper för att implementera formativ bedömning som metod i sin undervisning. De använder i stor utsträckning olika IKT-verktyg för att stärka elevernas utveckling, men även formativa metoder som kamratbedömning, tester under lektionstid, samt värderingsmatriser. De svårigheter lärarna upptäckt med dessa metoder är att de elever som har svårt i arbetet med linjära ekvationssystem framförallt har bristfälliga kunskaper inom aritmetik och förståelse för likhetstecknets betydelse. Lärarna visar tecken på en vilja att utveckla sitt arbete med formativ bedömning. De anser att de vill ha mer fortbildning på området och dessutom arbeta närmare sina kollegor. Det är också tydligt att de är medvetna om att de själva har en stor roll i att denna utveckling äger rum. / For many years, formative assessment has become a term used extensively in teaching discussions. Mostly because the achieved results have been so positive while using this method. It has been shown that by determining students’, and drawing attention to students’, level of knowledge, one can lay down a more individual strategy for how the students will reach their final goals. The purpose of this study is to investigate how upper secondary school teachers implement formative assessment as a method in their teaching. More specifically, this study relates to the efforts undertaken in the field of teaching linear equations. This is a quantitative study conducted using a survey that allowed teachers to present their own views of their personal perceptions of their efforts, discoveries, and thoughts about future development in the work with formative assessment. The result show that teachers in Swedish upper secondary schools have the knowledge to implement formative assessment as a method of teaching. They use a wide range of ICT tools to enhance student development, but also formative methods such as peer assessment, classroom testing, and evaluation charts. The difficulties they discovered by using these methods are that the students who struggle in the field of linear equations have inadequate knowledge of the arithmetic rules and an insufficient understanding of the meaning of the equal sign. The teachers show signs of a willingness to further develop their work in formative assessment. They believe that they in the future will need more education in the field and, moreover, more cooperation with their colleagues. It is also clear that they are aware of the fact that they themselves play a major role in the progress of this development.

Grid environments are inherently heterogeneous. If the computational power provided by collaborations on the Grid is to be harnessed in the true sense, there is a need for applications that can automatically adapt to changes in the execution environment. The application writer should not be burdened with the job of choosing the right algorithm and implementation every time the resources on which the application runs are changed. A lot of research has been done in adapting applications to changing conditions. The existing systems do not address the issue of providing a unified interface to permit algorithm selection at runtime. The goal of this research is to design and develop a unified interface to applications in order to permit seamless access to different algorithms providing similar functionalities. Long running, computationally intensive scientific applications can produce huge amounts of performance data. Often, this data is discarded once the application's execution is complete. This data can be utilized in extracting information about algorithms and their performance. This information can be used to choose algorithms intelligently. The research described in this thesis aims at designing and developing a component based unified interface for runtime algorithm selection in grid environments. This unified interface is necessary so that the application code does not change if a new algorithm is used to solve the problem. The overhead associated with making the algorithm choice transparent to the application is evaluated. We use a data mining approach to algorithm selection and evaluate its potential effectiveness for scientific applications. / Master of Science

No / This study has investigated an artificial intelligence technology ¿ model trees ¿ as a modelling tool applied to an immediate release tablet formulation database. The modelling performance was compared with artificial neural networks that have been well established and widely applied in the pharmaceutical product formulation fields. The predictability of generated models was validated on unseen data and judged by correlation coefficient R2. Output from the model tree analyses produced multivariate linear equations which predicted tablet tensile strength, disintegration time, and drug dissolution profiles of similar quality to neural network models. However, additional and valuable knowledge hidden in the formulation database was extracted from these equations. It is concluded that, as a transparent technology, model trees are useful tools to formulators