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This paper presents detailed aspects regarding the implementation of the Finite Element Method (FEM) to solve a Poisson’s equation with homogeneous boundary conditions. The aim of this paper is to clarify details of this implementation, such as the construction of algorithms, implementation of numerical experiments, and their results. For such purpose, the continuous problem is described, and a classical FEM approach is used to solve it. In addition, a multilevel technique is implemented for an efficient resolution of the corresponding linear system, describing and including some diagrams to explain the process and presenting the implementation codes in MATLAB®. Finally, codes are validated using several numerical experiments. Results show an adequate behavior of the preconditioner since the number of iterations of the PCG method does not increase, even when the mesh size is reduced. En este artículo se presenta, en forma detallada, aspectos sobre la implementación del Método de Elementos Finitos (FEM, por sus siglas en inglés), para resolver una ecuación de Poisson con condiciones de frontera homogéneas. El objetivo de este trabajo es clarificar los detalles de esta implementación, tales como la construcción de los algoritmos, creación de experimentos numéricos y los resultados acerca de estos. Por ello, se describe el problema continuo y se muestra un enfoque FEM clásico para resolverlo. Después, se establece una técnica multiniveles para la resolución eficiente del sistema lineal correspondiente, que describe e incluye algunos diagramas para explicar el proceso y presenta los códigos de la implementación en MATLAB®. Finalmente, se realiza una validación de los códigos con varios experimentos numéricos. Los resultados muestran un comportamiento adecuado del precondicionador debido a que el número de iteraciones del método PCG no se incrementa, incluso cuando el tamaño de la malla se reduce. Este artigo apresenta, em detalhes, aspectos sobre a implementação do Método dos Elementos Finitos (MEF) para resolver uma equação de Poisson com condições de contorno homogêneas. O objetivo deste trabalho é esclarecer os detalhes dessa implementação, tais como a construção dos algoritmos, a criação de experimentos numéricos e os resultados sobre eles. Descreve-se, portanto, o problema contínuo e mostra-se uma abordagem clássica do MEF para resolvê-lo. Em seguida, estabelece-se uma técnica multinível para a resolução eficiente do sistema linear correspondente, que descreve e inclui alguns diagramas para explicar o processo e apresenta os códigos de implementação no MATLAB®. Finalmente, realiza-se uma validação dos códigos com várias experiências numéricas. Os resultados mostram um comportamento adequado do pré-condicionador devido ao número de iterações do método PCG não aumentar, mesmo quando o tamanho da malha é reduzido. Universidad Nacional, Costa Rica Escuela de Matemática

Solving optimization models (including parameters fitting) for support vector machines on large-scale training data is often an expensive computational task. This paper proposes a multilevel algorithmic framework that scales efficiently to very large data sets. Instead of solving the whole training set in one optimization process, the support vectors are obtained and gradually refined at multiple levels of coarseness of the data. Our multilevel framework substantially improves the computational time without loosing the quality of classifiers. The algorithms are demonstrated for both regular and weighted support vector machines for balanced and imbalanced classification problems. Quality improvement on several imbalanced data sets has been observed.

For industrial aerodynamic applications to compressible flow simulation, finite volume methods with low order of consistency are a mature technology, usually combined with shock capturing techniques. However, high accuracy is needed for those problems which involve a wide range of spatial and temporal scales. High-order methods are potentially able to deliver high accuracy while at the same time avoiding excessive grid resolution. In spite of that, these methods have not made their way into industrial flow solvers for aircraft design yet. One of the reasons for this is the lack of tailor-suited, best-practice solution techniques that compare favorably to highly-tuned loworder methods. Currently, reliable, efficient, high-order numerical solutions can be guaranteed only in a limited number of cases. One issue to address in order to make these methods competitive is efficiency, which involves different aspects of the solution strategy. This thesis contributes to the development of efficient strategies for solving compressible flows via high-order discretizations. Systems of nonlinear hyperbolic conservation laws govern the dynamics of these flows. When discretizing steady problems, one obtains a large system of nonlinear algebraic equations. We focus on convergence accelerators for the solution of steady problems. Nonetheless, since implicit time discretization produces a very similar set of equations, the techniques presented can be applied to time-dependent problems as well. A combination of implicit Newton’s relaxation and explicit multilevel methods is proposed. The resulting technique is reliable and easy to tune, exceptionally simple compared to other ‘expert systems’. In the framework of implicit relaxation, an important issue is the storage of the system matrix. Strategies alternative to explicit storage of the matrix are introduced and analyzed. We also investigate the underlying high-order spatial discretization by considering and directly comparing the performance of three different methods, all employing a local, discontinuous solution space. Finally, a novel method for multiwavelet-based mesh adaptivity is extended to systems of conservation laws so as to adapt the mesh to the flow features. On the one hand, high-order approximations on rather large-size cells are used in regions where the solution is smooth. On the other hand, low-order approximations together with very fine cells are applied in regions where shocks or large gradients appear. Such an adaptive grid can allocate the resources efficiently, in that cells are concentrated in areas where they are needed, as opposed to uniform mesh refinement.

This thesis aims at analysing the attitudes of youngsters in Brussels towards sociological minorities. The term “minorities” is used to refer to the main social groups that suffer from subordination and misrecognition by the wider society according to the philosophical theory of recognition: women, lesbians and gay men, and ethnic minorities. Our dataset is composed of a sample of seventy schools in the Brussels Capital Region. In total, three thousand one hundred and twenty one pupils attending in 2007 the last grade of secondary education participated in the study. About half of the sample consists of pupils with a migrant background originating from about 100 different countries. This cultural diversity, reflecting one of the main characteristics of the population of the Brussels Capital Region, is at the centre of the thesis. Because of the hierarchical structure of the sample (pupils aggregated within schools), the culturally diverse population of our sample and the multidimensionality of prejudice, multilevel multivariate linear responses models were performed. In brief, these models allowed us to interpret items regrouped according to their common variation across social (and ethnic) groups and not according to their a priori content similarities. Furthermore, these models allowed us to integrate three different research traditions on prejudice: social psychology on the dimensionality of prejudice, sociology on the impact of socio demographic characteristics on prejudice and school effectiveness research on the role schools may play in reducing pupils’ prejudice. With these models, we could demonstrate the capacity of multilevel techniques to encompass the complexity of prejudice and norms, and to provide an interdisciplinary approach of social processes. Besides the impact of gender and socio economic differences on prejudice, the association between ethnic origin and prejudice was the focus of the analysis at the individual level. Hence, the empirical literature showed that respondents of foreign descent and respondents from the receiving society do not hold similar attitudes towards minorities. This association was investigated in a twofold strategy: after having assessed ethnic differences on the different kinds of prejudice, the explanatory power of possible mediators -such as the experience of group-level institutional discrimination or the bidimensional identification- on this association was tested. The choice of these mediators was influenced by different disciplines of the social sciences. Hence, besides the empirical literature specific to the topic of prejudice, these mediators are derived from theories of political sciences, of sociology of immigration, of social psychology and of cross-cultural psychology. The results showed that these mediators could indeed explain to a large extent ethnic differences on prejudice towards minorities. On the school level, we have shown that the impact schools may have on pupils’ prejudice is a differentiated one. Hence, this impact varies according to both the targets and the dimensions of prejudice. Moreover, besides school institutional characteristics, several contextual characteristics were investigated such as the cultural and social diversity within a school. Our results showed that the impact on prejudice of social and cultural diversity within schools was non-significant. This is, however, most probably related to a masking effect by the specificities of the education landscape in Brussels: differences between schools are huge and homogeneity within schools is important, given that the educational field is highly segregated both in social and in cultural terms. The implications of these results based on an interdisciplinary approach for future research and for policymakers are discussed., Doctorat en Sciences politiques et sociales, info:eu-repo/semantics/nonPublished

In this paper we use the multigrid algorithm - commonly used to improve the efficiency of the flow solver - to improve the efficiency of partitioned fluid-structure interaction iterations. Coupling not only the structure with the fine flow mesh, but also with the coarse flow mesh (often present due to the multigrid scheme) leads to a significant efficiency improvement. As solution of the flow equations typically takes much longer than the structure solve, and as multigrid is not standard in structure solvers, we do not coarsen the structure or the interface. As a result, the two level method can be easily implemented into existing solvers. Two types of two level algorithms were implemented: 1) Coarse grid correction of the partitioning error and 2) Coarse grid prediction or full multigrid to generate a better initial guess. The resulting schemes are combined with a fourth-order Runge-Kutta implicit time integration scheme. For the linear, one-dimensional piston problem with compressible flow the superior stability, accuracy and efficiency of the two level algorithms is shown. The parameters of the piston problem were chosen such that both a weak and a strong interaction case were obtained. Even the strong interaction case, with a flexible structure, could be solved with our new two level partitioned scheme with just one iteration on the fine grid. This is a major accomplishment as most weakly coupled methods fail in this case. Of the two algorithms the coarse grid prediction or full multigrid method was found to perform best. The resulting efficiency gain for our one-dimensional problem is around a factor of ten for the coarse to intermediate time steps at which the high order time integration methods should be run. For two- and three-dimensional problems the efficiency gain is expected to be even larger.

In this paper we use the multigrid algorithm - commonly used to improve the efficiency of the flow solver - to improve the efficiency of partitioned fluid-structure interaction iterations. Coupling not only the structure with the fine flow mesh, but also with the coarse flow mesh (often present due to the multigrid scheme) leads to a significant efficiency improvement. As solution of the flow equations typically takes much longer than the structure solve, and as multigrid is not standard in structure solvers, we do not coarsen the structure or the interface. As a result, the two level method can be easily implemented into existing solvers. Two types of two level algorithms were implemented: 1) Coarse grid correction of the partitioning error and 2) Coarse grid prediction or full multigrid to generate a better initial guess. The resulting schemes are combined with a fourth-order Runge-Kutta implicit time integration scheme. For the linear, one-dimensional piston problem with compressible flow the superior stability, accuracy and efficiency of the two level algorithms is shown. The parameters of the piston problem were chosen such that both a weak and a strong interaction case were obtained. Even the strong interaction case, with a flexible structure, could be solved with our new two level partitioned scheme with just one iteration on the fine grid. This is a major accomplishment as most weakly coupled methods fail in this case. Of the two algorithms the coarse grid prediction or full multigrid method was found to perform best. The resulting efficiency gain for our one-dimensional problem is around a factor of ten for the coarse to intermediate time steps at which the high order time integration methods should be run. For two- and three-dimensional problems the efficiency gain is expected to be even larger.

In this paper, we review several recent developments centering upon the application of multiscale basis methods for the numerical solution of operator equations with special emphasis on complexity questions. In particular, issues like preconditioning, matrix compression, construction of special wavelet bases and adapted error estimators are addressed.