In a series of work initiated by Nemirovsky and Yudin, and later extended by Nesterov, first-order algorithms for unconstrained minimization with optimal theoretical complexity bound have been proposed. On the other hand, conjugate gradient algorithms as one of the widely used first-order techniques suffer from the lack of a finite complexity bound. In fact their performance can possibly be quite poor. This dissertation is partially on tightening the gap between these two classes of algorithms, namely the traditional conjugate gradient methods and optimal first-order techniques. We derive conditions under which conjugate gradient methods attain the same complexity bound as in Nemirovsky-Yudin's and Nesterov's methods. Moreover, we propose a conjugate gradient-type algorithm named CGSO, for Conjugate Gradient with Subspace Optimization, achieving the optimal complexity bound with the payoff of a little extra computational cost. We extend the theory of CGSO to convex problems with linear constraints. In particular we focus on solving $l_1$-regularized least square problem, often referred to as Basis Pursuit Denoising (BPDN) problem in the optimization community. BPDN arises in many practical fields including sparse signal recovery, machine learning, and statistics. Solving BPDN is fairly challenging because the size of the involved signals can be quite large; therefore first order methods are of particular interest for these problems. We propose a quasi-Newton proximal method for solving BPDN. Our numerical results suggest that our technique is computationally effective, and can compete favourably with the other state-of-the-art solvers.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Fala-se hoje muito sobre as possÃveis mudanÃas climÃticas como resultado das emissÃes de gases na atmosfera. Embora nÃo seja de comum acordo, grande parte da comunidade cientÃfica acredita que acontecerà uma elevaÃÃo na temperatura mÃdia global, o qual pode provocar, por exemplo, um aumento mÃdio nos nÃveis dos oceanos. Associada a isto, hà tambÃm um consenso de que haverà significativas transformaÃÃes no regime pluvial e de evaporaÃÃo em vÃrias partes do planeta. Entretanto, a quantificaÃÃo dessas mudanÃas, em locais especÃficos, ainda necessitam de mais estudos, os quais permitirÃo avaliaÃÃes mais conclusivas a respeito do tema. Sendo assim, o presente trabalho procurou jogar um pouco de luz neste tema e, considerando alguns cenÃrios de alteraÃÃo nos padrÃes de precipitaÃÃo e evaporaÃÃo (considerando a evaporaÃÃo lÃquida = E-P), avaliou os impactos destas mudanÃas climÃticas na vazÃo regularizada de reservatÃrios no Nordeste semi-Ãrido, cujos rios apresentam altos coeficiente de variaÃÃo dos deflÃvios anuais. Observou-se que em rios menos variÃveis â CV iguais a 0,6 e 0,8 â à necessÃrio que a lÃmina lÃquida (EL) aumente 60% para que o reservatÃrio perca cerca de 10% de sua vazÃo regularizada. Jà para rios de maior variabilidade - CV iguais a 1,4 e 1,6 â a lÃmina lÃquida precisaria de um aumento de apenas 30%, para que o reservatÃrio fosse objeto desta mesma reduÃÃo. Concluiu-se, portanto, que como a maioria dos reservatÃrios do Nordeste semi-Ãrido localiza-se em rios que apresentam coeficientes de variaÃÃo dos deflÃvios anuais em torno de 1,2 a 1,6, uma atenÃÃo maior deverà ser dada aos mesmos, uma vez que se mostraram bastante sensÃveis as variaÃÃes climÃticas. / It is much talked nowadays about the possible climatic changes as a result of the gas emission on the atmosphere. Although it is not a common agreement, a great part of the scientific community believe that an elevation of the measured global temperature will occur, which can provoke, for example, a medium increase on the ocean levels. Associated to this, there is also a consensus that significant transformations will happen on the pluvial and evaporation regimen around the world. However, the quantification of these changes, on specific locations, is still needed more studies, which can allow more conclusive evaluations about the theme. With that been said, this study tried to put some light on this theme, and considering some scenarios of alteration on the precipitation patterns and evaporation (considering liquid evaporation = E-P) it was evaluated the impact of these climatic changes in reservoirs yield at Northeast Semi-Arid of Brazil. whose rivers present high coefficient of variation of the annual emanations. It was observed that in rivers less variable â CV equal to 0,6 and 0,8 â it is necessary that the net evaporation EL increase 60% so that the reservoir can lose about 10% of its regularized outflow. But for rivers with high variability â CV equal to 1,4 and 1,6 â the net evaporation â would need a increase of just 30% so that the reservoir could be the object of the same reduction. It was concluded that, as most of the half-barren northeastâs reservoirs are located on rivers that present variation coefficients of the annual emanations in lathe of 1,2 to 1,6 , a greater attention should be given to these, since it showed sensible enough to the climatic variations.

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This thesis describes several approaches for stabilizing POD-ROMs (that is, reduced order models based on basis functions derived from the proper orthogonal decomposition) for both the CDR (convection-diffusion-reaction) equation and the NSEs (Navier-Stokes equations). Stabilization is necessary because standard POD-ROMs of convection-dominated problems usually display numerical instabilities. The first stabilized ROM investigated is a streamline-upwind Petrov-Galerkin ROM (SUPG-ROM). I prove error estimates for the SUPG-ROM and derive optimal scalings for the stabilization parameter. I test the SUPG-ROM with the optimal parameter in the numerical simulation of a convection-dominated CDR problem. The SUPG-ROM yields more accurate results than the standard Galerkin ROM (G-ROM) by eliminating the inherent numerical artifacts (noise) in the data and dampening spurious oscillations. I next propose two regularized ROMs (Reg-ROMs) based on ideas from large eddy simulation and turbulence theory: the Leray ROM (L-ROM) and the evolve-then-filter ROM (EF-ROM). Both Reg-ROMs use explicit POD spatial filtering to regularize (smooth) some of the terms in the standard G-ROM. I propose two different POD spatial filters: one based on the POD projection and a novel POD differential filter. These two new Reg-ROMs and the two spatial filters are investigated in the numerical simulation of the three-dimensional flow past a circular cylinder problem at Re = 100. The numerical results show that EF-ROM-DF is the most accurate Reg-ROM and filter combination and the differential filter generally yields better results than the projection filter. The Reg-ROMs perform significantly better than the standard G-ROM and decrease the CPU time (compared against the direct numerical simulation) by orders of magnitude (from about four days to four minutes). / Ph. D.

Este trabalho foi baseado na realização de simulações computacionais 3D de parâmetros de RQD obtidos de classificação geomecânica de sondagens rotativas realizadas nas fases de investigação geológico-geotécnica e construção da barragem de Itaipu (PR), Brasil. Tais simulações foram desenvolvidas em duas áreas dispostas no entorno da barragem principal da usina, utilizando como ferramenta o programa GRASS, que executa interpolações tridimensionais pelo método numérico Spline Regularizado com Tensão (RST). Os resultados obtidos foram comparados com dados previamente interpretados e obtidos por outros métodos como os geoestatísticos e apresentaram resultados razoáveis devido as dimensões dos voxels adotados, diferenças dos métodos e distribuição espacial dos dados. / This work was based on three-dimensional computer simulations of the parameters RQD of the boreholes on geological and geotechnical research phases and construction of the Itaipu dam (PR), Brazil. These simulations were carried out in two areas arranged around the plant the main dam, using as tool the GRASS program, which performs three-dimensional interpolation by mathematical method Regularized Spline with Tension (RST) . The results were compared with data previously obtained and interpreted by other methods such as geostatistical and had reasonable results because the dimensions of the voxels adopted, differences in methods and spatial distribution of the data.

Apresentamos dois modelos para o elétron na eletrodinâmica clássica que incorporam alguns efeitos da eletrodinâmica quântica. No primeiro modelo, o elétron é tratado como uma partícula extensa como conseqüência das oscilações de alta-freqüência (Zitterbewegung) que sua carga elétrica realiza. Mostramos que este modelo prevê corretamente a magnitude do spin do elétron e lhe atribui o mesmo fator giromagnético previsto pela equação de Dirac sem correções radiativas. Neste modelo, a auto-energia do elétron diverge logaritmicamente como resultado da distribuição extensa de sua carga elétrica. No segundo modelo, a criação de pares virtuais em torno do elétron é levada em conta por uma generalização da lagrangeana do campo eletromagnético que respeita as simetrias da eletrodinâmica clássica. Esta generalização altera a interação entre o elétron e o campo eletromagnético em pequenas distâncias e permite que a auto-força de uma partícula puntiforme seja determinada de modo consistente. Mostramos que as soluções da equação de movimento resultante não apresentam auto-aceleração nem pré-aceleração, sendo consistentes com a causalidade. / We present two models for the electron in classical electrodynamics, which include some effects from quantum electrodynamics. In the first model, the electron is treated as an extended particle owing to the high-frequency oscillations (Zitterbewegung) of its electrical charge. We show that this model predicts correctly the magnitude of the electron spin and it gives the electron the same gyromagnetic factor as predicted by Dirac equation without radiative corrections. In this model, the electron self-energy has a logarithmic divergence due to the extended distribution of its electric charge. In the second model, virtual pair creation around the electron is taken into account by a generalization of the lagrangian for the electromagnetic field that preserves the symmetries of classical electrodynamics. This generalization changes the interaction of the electron with the electromagnetic field at small distances and allows us to evaluate the self-force of a point particle in a consistent way. We show that the solutions of the derived equation of motion do not exhibit self-acceleration nor pre-acceleration, being consistent with causality.

Imageamento acústico é um problema computacionalmente caro e mal-condicionado, que envolve estimar distribuições de fontes com grandes arranjos de microfones. O método clássico para imageamento acústico utiliza beamforming, e produz a distribuição de fontes de interesse convoluída com a função de espalhamento do arranjo. Esta convolução borra a imagem ideal, significativamente diminuindo sua resolução. Convoluções podem ser evitadas com técnicas de ajuste de covariância, que produzem estimativas de alta resolução. Porém, estas têm sido evitadas devido ao seu alto custo computacional. Nesta tese, admitimos um arranjo bidimensional com geometria separável, e desenvolvemos transformadas rápidas para acelerar imagens acústicas em várias ordens de grandeza. Estas transformadas são genéricas, e podem ser aplicadas para acelerar beamforming, algoritmos de deconvolução e métodos de mínimos quadrados regularizados. Assim, obtemos imagens de alta resolução com algoritmos estado-da-arte, mantendo baixo custo computacional. Mostramos que arranjos separáveis produzem estimativas competitivas com as de geometrias espirais logaritmicas, mas com enormes vantagens computacionais. Finalmente, mostramos como estender este método para incorporar calibração, um modelo para propagação em campo próximo e superfícies focais arbitrárias, abrindo novas possibilidades para imagens acústicas. / Acoustic imaging is a computationally intensive and ill-conditioned inverse problem, which involves estimating high resolution source distributions with large microphone arrays. The classical method for acoustic imaging consists of beamforming, and produces the source distribution of interest convolved with the array point spread function. This convolution smears the image of interest, significantly reducing its effective resolution. Convolutions can be avoided with covariance fitting methods, which have been known to produce robust high-resolution estimates. However, these have been avoided due to prohibitive computational costs. In this thesis, we assume a 2D separable array geometry, and develop fast transforms to accelerate acoustic imaging by several orders of magnitude with respect to previous methods. These transforms are very generic, and can be applied to accelerate beamforming, deconvolution algorithms and regularized least-squares solvers. Thus, one can obtain high-resolution images with state-of-the-art algorithms, while maintaining low computational cost. We show that separable arrays deliver accuracy competitive with multi-arm spiral geometries, while producing huge computational benefits. Finally, we show how to extend this approach with array calibration, a near-field propagation model and arbitrary focal surfaces, opening new and exciting possibilities for acoustic imaging.

Imageamento acústico é um problema computacionalmente caro e mal-condicionado, que envolve estimar distribuições de fontes com grandes arranjos de microfones. O método clássico para imageamento acústico utiliza beamforming, e produz a distribuição de fontes de interesse convoluída com a função de espalhamento do arranjo. Esta convolução borra a imagem ideal, significativamente diminuindo sua resolução. Convoluções podem ser evitadas com técnicas de ajuste de covariância, que produzem estimativas de alta resolução. Porém, estas têm sido evitadas devido ao seu alto custo computacional. Nesta tese, admitimos um arranjo bidimensional com geometria separável, e desenvolvemos transformadas rápidas para acelerar imagens acústicas em várias ordens de grandeza. Estas transformadas são genéricas, e podem ser aplicadas para acelerar beamforming, algoritmos de deconvolução e métodos de mínimos quadrados regularizados. Assim, obtemos imagens de alta resolução com algoritmos estado-da-arte, mantendo baixo custo computacional. Mostramos que arranjos separáveis produzem estimativas competitivas com as de geometrias espirais logaritmicas, mas com enormes vantagens computacionais. Finalmente, mostramos como estender este método para incorporar calibração, um modelo para propagação em campo próximo e superfícies focais arbitrárias, abrindo novas possibilidades para imagens acústicas. / Acoustic imaging is a computationally intensive and ill-conditioned inverse problem, which involves estimating high resolution source distributions with large microphone arrays. The classical method for acoustic imaging consists of beamforming, and produces the source distribution of interest convolved with the array point spread function. This convolution smears the image of interest, significantly reducing its effective resolution. Convolutions can be avoided with covariance fitting methods, which have been known to produce robust high-resolution estimates. However, these have been avoided due to prohibitive computational costs. In this thesis, we assume a 2D separable array geometry, and develop fast transforms to accelerate acoustic imaging by several orders of magnitude with respect to previous methods. These transforms are very generic, and can be applied to accelerate beamforming, deconvolution algorithms and regularized least-squares solvers. Thus, one can obtain high-resolution images with state-of-the-art algorithms, while maintaining low computational cost. We show that separable arrays deliver accuracy competitive with multi-arm spiral geometries, while producing huge computational benefits. Finally, we show how to extend this approach with array calibration, a near-field propagation model and arbitrary focal surfaces, opening new and exciting possibilities for acoustic imaging.

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No description available.