This paper concerns the solution of plate bending problems in domains composed of rectangles. Domain decomposition (DD) is the basic tool used for both the parallelization of the conjugate gradient method and the construction of efficient parallel preconditioners. A so-called Dirich- let DD preconditioner for systems of linear equations arising from the fi- nite element approximation by non-conforming Adini elements is derived. It is based on the non-overlapping DD, a multilevel preconditioner for the Schur-complement and a fast, almost direct solution method for the Dirichlet problem in rectangular domains based on fast Fourier transform. Making use of Xu's theory of the auxiliary space method we construct an optimal preconditioner for plate problems discretized by conforming Bogner-Fox-Schmidt rectangles. Results of numerical experiments carried out on a multiprocessor sys- tem are given. For the test problems considered the number of iterations is bounded independent of the mesh sizes and independent of the number of subdomains. The resulting parallel preconditioned conjugate gradient method requiresO(h^-2 ln h^-1 ln epsilon^-11) arithmetical operations per processor in order to solve the finite element equations with the relative accuracy epsilon.

Orientadores: Aurélio Ribeiro Leite de Oliveira, Marta Ines Velazco Fontova / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T12:49:39Z (GMT). No. of bitstreams: 1 Coelho_AlessandroFonsecaEsteves_M.pdf: 2659631 bytes, checksum: fc1bec925179612ee07a4aaef7092d8a (MD5) Previous issue date: 2011 / Resumo: Um método frequentemente utilizado para a solução de problemas de programação linear é o método de pontos interiores. Nestes métodos precisamos resolver sistemas lineares para calcular a direção de Newton a cada iteração. A solução desses sistemas consiste no passo de maior esforço computacional nos métodos de pontos interiores. A fatoração de Cholesky é a opção mais utilizada para resolver estes sistemas. Contudo, quando trabalhamos com problemas de grande porte, esta fatoração pode ser densa e torna-se inviável trabalhar com esses métodos. Nestes casos, uma boa opção consiste no uso de métodos iterativos precondicionados. Estudos anteriores utilizam o método dos gradientes conjugados precondicionado para obter uma solução destes sistemas. Particularmente, os sistemas originados dos métodos de pontos interiores, são, naturalmente, sistemas de equações normais. Porém, a versão padrão do método dos gradientes conjugados, não considera a estrutura de equações normais do sistema. Neste trabalho propomos a utilização de duas versões do método de gradientes conjugados precondicionado que consideram a estrutura de equações normais destes sistemas. Estas versões serão comparadas com a versão de gradientes conjugados precondicionada que não considera a estrutura de equações normais do sistema. Resultados numéricos com problemas de grande porte mostram que uma dessas versões é competitiva em relação à versão padrão / Abstract: An often used method for solving linear programming problems is the interior point method. In these methods we need to solve linear systems to compute the Newton search direction at each iteration. The solution of these systems is the procedure of most computational effort in interior point methods. The Cholesky factorization is the most often used method to solve these systems. However, when dealing with large scale problems, this factorization can be dense and it become impossible to apply such methods. In such cases, a good option is the use of preconditioned iterative methods. Previous studies have used the preconditioned conjugate gradient method to find the solution of these systems. Particularly, the systems arising from interior point methods are, naturally, systems of normal equations type. Nevertheless, the standard version of the conjugate gradient method, does not take into account the normal equations system structure. This study proposes the use of two versions of preconditioned conjugate gradient method considering the normal equations structure of these systems. These versions are compared with the preconditioned conjugate gradient version that does not consider that structure. Numerical results with large scale problems show that one of these versions is competitive with the standard one / Mestrado / Matematica Aplicada / Mestre em Matemática Aplicada

Two alternative technologies of communication inside a parallelized Conjugate-Gradient algorithm are presented and compared to the well known hypercubecommunication. The amount of communication is diskussed in detail. A large range of numerical results corroborate the theoretical investigations.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第17968号 / 情博第512号 / 新制||情||91(附属図書館) / 30798 / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 中村 佳正, 教授 西村 直志, 准教授 山下 信雄 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

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Jump-diffusion models can under certain assumptions be expressed as partial integro-differential equations (PIDE). Such a PIDE typically involves a convection term and a nonlocal integral like for the here considered models of Merton and Kou. We transform the PIDE to eliminate the convection term, discretize it implicitly using finite differences and the second order backward difference formula (BDF2) on a uniform grid. The arising dense linear system is solved by an iterative method, either a splitting technique or a circulant preconditioned conjugate gradient method. Exploiting the Fast Fourier Transform (FFT) yields the solution in only $O(n\log n)$ operations and just some vectors need to be stored. Second order accuracy is obtained on the whole computational domain for Merton's model whereas for Kou's model first order is obtained on the whole computational domain and second order locally around the strike price. The solution for the PIDE with convection term can oscillate in a neighborhood of the strike price depending on the choice of parameters, whereas the solution obtained from the transformed problem is stabilized. / Master of Science

In this thesis, we develop block-decomposition (BD) methods and variants of accelerated *9gradient methods for large-scale conic programming and convex optimization, respectively. The BD methods, discussed in the first two parts of this thesis, are inexact versions of proximal-point methods applied to two-block-structured inclusion problems. The adaptive accelerated methods, presented in the last part of this thesis, can be viewed as new variants of Nesterov's optimal method. In an effort to improve their practical performance, these methods incorporate important speed-up refinements motivated by theoretical iteration-complexity bounds and our observations from extensive numerical experiments. We provide several benchmarks on various important problem classes to demonstrate the efficiency of the proposed methods compared to the most competitive ones proposed earlier in the literature. In the first part of this thesis, we consider exact BD first-order methods for solving conic semidefinite programming (SDP) problems and the more general problem that minimizes the sum of a convex differentiable function with Lipschitz continuous gradient, and two other proper closed convex (possibly, nonsmooth) functions. More specifically, these problems are reformulated as two-block monotone inclusion problems and exact BD methods, namely the ones that solve both proximal subproblems exactly, are used to solve them. In addition to being able to solve standard form conic SDP problems, the latter approach is also able to directly solve specially structured non-standard form conic programming problems without the need to add additional variables and/or constraints to bring them into standard form. Several ingredients are introduced to speed-up the BD methods in their pure form such as: adaptive (aggressive) choices of stepsizes for performing the extragradient step; and dynamic updates of scaled inner products to balance the blocks. Finally, computational results on several classes of SDPs are presented showing that the exact BD methods outperform the three most competitive codes for solving large-scale conic semidefinite programming. In the second part of this thesis, we present an inexact BD first-order method for solving standard form conic SDP problems which avoids computations of exact projections onto the manifold defined by the affine constraints and, as a result, is able to handle extra large-scale SDP instances. In this BD method, while the proximal subproblem corresponding to the first block is solved exactly, the one corresponding to the second block is solved inexactly in order to avoid finding the exact solution of a linear system corresponding to the manifolds consisting of both the primal and dual affine feasibility constraints. Our implementation uses the conjugate gradient method applied to a reduced positive definite dual linear system to obtain inexact solutions of the latter augmented primal-dual linear system. In addition, the inexact BD method incorporates a new dynamic scaling scheme that uses two scaling factors to balance three inclusions comprising the optimality conditions of the conic SDP. Finally, we present computational results showing the efficiency of our method for solving various extra large SDP instances, several of which cannot be solved by other existing methods, including some with at least two million constraints and/or fifty million non-zero coefficients in the affine constraints. In the last part of this thesis, we consider an adaptive accelerated gradient method for a general class of convex optimization problems. More specifically, we present a new accelerated variant of Nesterov's optimal method in which certain acceleration parameters are adaptively (and aggressively) chosen so as to: preserve the theoretical iteration-complexity of the original method; and substantially improve its practical performance in comparison to the other existing variants. Computational results are presented to demonstrate that the proposed adaptive accelerated method performs quite well compared to other variants proposed earlier in the literature.

<div>The objective of this doctoral research was to develop a versatile inverse heat transfer approach, that would enable the solution of small scale problems present in microelectronics, as well as the analysis of the complex heat flux in turbines. An inverse method is a mathematical approach which allows the resolution of problems starting from the solution. In a direct problem, the boundary conditions are given, and using the governing physics principles and equations you can calculate the solution or physical effect. In an inverse method, the solution is provided and through the physical equations, the boundary conditions can be determined. Therefore, the inverse method applied to heat transfer means that we know the variation of temperature (effect) over time and space. With the temperature input, the geometry, thermal properties of the test article and the heat diffusion equation, we can compute the spatially- and temporally-varying heat flux that generated the temperature map.</div><div><br></div><div>This doctoral dissertation develops two inverse methodologies: (1) an optimization methodology based on the conjugate gradient method and (2) a function specification method combined with a regularization technique, which is less robust but much faster. We implement these methodologies with commercial codes for solving conductive heat transfer with COMSOL and for conjugate heat transfer with ANSYS Fluent.</div><div><br></div><div>The goal is not only the development of the methods but also the validation of the techniques in two different fields with a common purpose: quantifying heat dissipation. The inverse methods were applied in the micro-scale to the dissipation of heat in microelectronics and in the macro-scale to the gas turbine engines.<br></div><div><br></div><div>In microelectronics, we performed numerical and experimental studies of the two developed inverse methodologies. The intent was to predict where heat is being dissipated and localized hot spots inside of the chip from limited measurements of the temperature outside of the chip. Here, infrared thermography of the chip surface is the input to the inverse methods leveraging thermal model of the chip. Furthermore, we combined the inverse methodology with a Kriging interpolation technique with genetic algorithm optimization to optimize the location and number of the temperature sensors inside of the chip required to accurately predict the thermal behavior of the microchip at each moment of time and everywhere.<br></div><div><br></div><div>In the application for gas turbine engines, the inverse method can be useful to detect or predict the conditions inside of the turbine by taking measurements in the outer casing. Therefore, the objective is the experimental validation of the technique in a wind tunnel especially designed with optical access for non-contact measurement techniques. We measured the temperature of the outer casing of the turbine rotor with an infrared camera and surface temperature sensors and this information is the input of the two methodologies developed in order to predict which the heat flux through the turbine casing. A new facility, specifically, an annular turbine cascade, was designed to be able to measure the relative frame of the rotor from the absolute frame. In order to get valuable data of the heat flux in a real engine, we need to replicate the Mach, Reynolds, and temperature ratios between fluid and solid. Therefore, the facility can reproduce a large range of pressures and flow temperatures. Because some regions of interest are not accessible, this researchprovides a significant benefit for understanding the system performance from limited data. With inverse methods, we can measure the outside of objects and provide an accurate prediction of the behavior of the complete system. This information is relevant not only for new designs of gas turbines or microchips, but also for old designs where due to lack of prevision there are not enough sensors to monitor the thermal behavior of the studied system.<br></div><div><br></div>