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Déformation des feuilletages par variétés complexes / Deformations of foliations by complex manifoldsBurel, Thomas 10 December 2010 (has links)
L'objet de ce travail est de généraliser au cas des variétés feuilletées par variétés complexes la théorie des déformations de variétés complexes compactes développée notamment par les travaux de Kodaira et Spencer vers la fi n des années cinquante. Après avoir défni la notion de famille de déformations de variétés feuilletées par variétés complexes compactes, nous avons pu obtenir un analogue des théorèmes de rigidité, de complétude et d'existence dans notre cadre. Les méthodes de démonstration usant de la théorie du potentiel ne sont pas généralisables car les opérateurs différentiels considérés ici ne sont plus elliptiques. On se tourne alors vers des techniques de séries majorantes pour obtenir ces résultats, en particulier pour le théorème d'existence qui généralise la démonstration faite par Forster et Knorr en 1974. / The aim of this work is to generalise the study of deformations of complex manifolds by kodaira and Spencer to the case of manifolds foliated by complex manifolds. After defning the notion of family of deformations of compact manifold foliated by complex manifolds, we prove a theorem of rigidity, one of completeness and one of existence in our framework. We can not apply one potential theory here, so we have to use power series technics.
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Sur l'estimation adaptative d'une densité multivariée sous l'hypothèse de la structure d'indépendance / On adaptive estimation of a multivariate density under independence hypothesis.Rebelles, Gilles 10 December 2015 (has links)
Les résultats obtenus dans cette thèse concernent l'estimation non paramétrique de densités de probabilité. Principalement, nous nous intéressons à estimer une densité de probabilité multidimensionnelle de régularité anisotrope et inhomogène. Nous proposons des procédures d'estimation qui sont adaptatives, non seulement par rapport aux paramètres de régularité, mais aussi par rapport à la structure d'indépendance de la densité de probabilité estimée. Cela nous permet de réduire l'influence de la dimension du domaine d'observation sur la qualité d'estimation et de faire en sorte que cette dernière soit la meilleure possible. Pour analyser la performance de nos méthodes nous adoptons le point de vue minimax et nous généralisons un critère d'optimalité pour l'estimation adaptative. L'utilisation du critère que nous proposons s'impose lorsque le paramètre d'intérêt est estimé en un point fixé car, dans ce cas, il y a un "prix à payer" pour l'adaptation par rapport à la régularité et à la structure d'indépendance. Cela n'est plus vrai lorsque l'estimation est globale. Dans le modèle de densité (avec des observations directes) nous considérons le problème de l'estimation ponctuelle et celui de l'estimation en norme $bL_p$, $pin[1,infty)$. Dans le modèle de déconvolution (avec des observations bruitées) nous étudions le problème de l'estimation en norme $bL_p$, $pin[1,infty]$, dans le cas où la fonction caractéristique du bruit décroît polynomialement à l'infini. Chaque estimateur que nous proposons est obtenu par une procédure de sélection aléatoire dans une famille d'estimateurs à noyau. / The results obtained in this thesis concern the non parametric estimation of probability densities. Primarily, we are interested in estimating a multivariate probability density which is anisotropic and inhomogeneous. We propose estimation procedures that enable us to take into account the regularity properties of the underlying probability density and its independence structure simultaneously. This allows us to reduce the influence of the dimension of the observation space on the accuracy of estimation and then to improve it. To analyze the performance of our methods we adopt the minimax point of view and we generalize a criterion of optimality for adaptive estimation. The use of the criterion we propose is necessary for estimation at a fixed point. Indeed, in this setting, there is a "penalty" for adaptation with respect to the regularity and to the independence structure. This is no longer true for global estimation. In the density model (with direct observations) we consider both the problem of pointwise estimation and the problem of estimation under $bL_p$-loss ($pin[1,infty)$). In the deconvolution model (with noisy observations) we study the problem of estimation with an $bL_p$-risk ($pin[1,infty]$) when the characteristic function of the noise decreases polynomially at infinity. Any estimator that we propose is obtained by a random selection procedure in a family of kernel estimators.
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Unificando o análise local do método de Newton em variedades Riemannianas / Unifying local analysis of Newton's method in Riemannian manifoldsGuevara, Stefan Alberto Gómez 08 March 2017 (has links)
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Previous issue date: 2017-03-08 / In this work we consider the problem of finding a singularity of a field of differentiable vectors X on a Riemannian manifold. We present a local analysis of the convergence of Newton's method to
find a singularity of field X on an increasing condition. The analysis shows a relationship between the major function and the vector field X. We also present a semi-local Kantorovich type analysis in the Riemannian context under a major condition. The two results allow to unify some previously unrelated results. / Neste trabalho consideramos o problema de encontrar uma singularidade de um campo de vetores diferenciável X sobre uma variedade Riemanniana. Apresentamos uma análise local da convergência do método de Newton para encontrar uma singularidade do Campo X sobre uma condição majorante. A análise mostra uma relação entre a função majorante e o campo de vetores X. Também apresentamos uma análise semi-local do tipo Kantorovich no contexto Riemanniana sob uma condição majorante. Os dois resultados permitem unificar alguns resultados não previamente.
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Análise semi-local do método de Gauss-Newton sob uma condição majorante / Semi-local analysis of the Gauss-Newton under a majorant conditionAguiar, Ademir Alves 18 December 2014 (has links)
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Previous issue date: 2014-12-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this dissertation we present a semi-local convergence analysis for the Gauss-Newton
method to solve a special class of systems of non-linear equations, under the hypothesis
that the derivative of the non-linear operator satisfies a majorant condition. The proofs
and conditions of convergence presented in this work are simplified by using a simple
majorant condition. Another tool of demonstration that simplifies our study is to identify
regions where the iteration of Gauss-Newton is “well-defined”. Moreover, special cases
of the general theory are presented as applications. / Nesta dissertação apresentamos uma análise de convergência semi-local do método de
Gauss-Newton para resolver uma classe especial de sistemas de equações não-lineares,
sob a hipótese que a derivada do operador não-linear satisfaz uma condição majorante. As
demonstrações e condições de convergência apresentadas neste trabalho são simplificadas
pelo uso de uma simples condição majorante. Outra ferramenta de demonstração que
simplifica o nosso estudo é a identificação de regiões onde a iteração de Gauss-Newton
está “bem-definida”. Além disso, casos especiais da teoria geral são apresentados como
aplicações.
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Newton's methods under the majorant principle on Riemannian manifolds / Métodos de Newton sob o princípio majorante em variedades riemannianasMartins, Tiberio Bittencourt de Oliveira 26 June 2015 (has links)
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Previous issue date: 2015-06-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Apresentamos, nesta tese, uma an álise da convergência do m étodo de Newton inexato
com tolerância de erro residual relativa e uma an alise semi-local de m etodos de Newton
robustos exato e inexato, objetivando encontrar uma singularidade de um campo de vetores diferenci avel de nido em uma variedade Riemanniana completa, baseados no princ pio majorante a m invariante. Sob hip oteses locais e considerando uma fun ção majorante geral, a Q-convergância linear do m etodo de Newton inexato com uma tolerância de erro residual relativa xa e provada. Na ausência dos erros, a an alise apresentada reobtem o teorema
local cl assico sobre o m etodo de Newton no contexto Riemanniano. Na an alise semi-local
dos m etodos exato e inexato de Newton apresentada, a cl assica condi ção de Lipschitz tamb em
e relaxada usando uma fun ção majorante geral, permitindo estabelecer existência e unicidade
local da solu ção, uni cando previamente resultados pertencentes ao m etodo de Newton. A
an alise enfatiza a robustez, a saber, e dada uma bola prescrita em torno do ponto inicial
que satifaz as hip oteses de Kantorovich, garantindo a convergência do m etodo para qualquer
ponto inicial nesta bola. Al em disso, limitantes que dependem da função majorante para a
taxa de convergência Q-quadr atica do m étodo exato e para a taxa de convergência Q-linear
para o m etodo inexato são obtidos. / A local convergence analysis with relative residual error tolerance of inexact Newton
method and a semi-local analysis of a robust exact and inexact Newton methods are presented
in this thesis, objecting to nd a singularity of a di erentiable vector eld de ned on a
complete Riemannian manifold, based on a ne invariant majorant principle. Considering
local assumptions and a general majorant function, the Q-linear convergence of inexact
Newton method with a xed relative residual error tolerance is proved. In the absence
of errors, the analysis presented retrieves the classical local theorem on Newton's method
in Riemannian context. In the semi-local analysis of exact and inexact Newton methods
presented, the classical Lipschitz condition is also relaxed by using a general majorant
function, allowing to establish the existence and also local uniqueness of the solution,
unifying previous results pertaining Newton's method. The analysis emphasizes robustness,
being more speci c, is given a prescribed ball around the point satisfying Kantorovich's
assumptions, ensuring convergence of the method for any starting point in this ball.
Furthermore, the bounds depending on the majorant function for Q-quadratic convergence
rate of the exact method and Q-linear convergence rate of the inexact method are obtained.
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Newton's method for solving strongly regular generalized equation / Método de Newton para resolver equações generalizadas fortemente regularesSilva, Gilson do Nascimento 13 March 2017 (has links)
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Previous issue date: 2017-03-13 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We consider Newton’s method for solving a generalized equation of the form
f(x) + F(x) 3 0,
where f : Ω → Y is continuously differentiable, X and Y are Banach spaces, Ω ⊆ X is open
and F : X ⇒ Y has nonempty closed graph. Assuming strong regularity of the equation
and that the starting point satisfies Kantorovich’s conditions, we show that the method
is quadratically convergent to a solution, which is unique in a suitable neighborhood of
the starting point. In addition, a local convergence analysis of this method is presented.
Moreover, using convex optimization techniques introduced by S. M. Robinson (Numer.
Math., Vol. 19, 1972, pp. 341-347), we prove a robust convergence theorem for inexact
Newton’s method for solving nonlinear inclusion problems in Banach space, i.e., when
F(x) = −C and C is a closed convex set. Our analysis, which is based on Kantorovich’s
majorant technique, enables us to obtain convergence results under Lipschitz, Smale’s and
Nesterov-Nemirovskii’s self-concordant conditions. / N´os consideraremos o m´etodo de Newton para resolver uma equa¸c˜ao generalizada da forma
f(x) + F(x) 3 0,
onde f : Ω → Y ´e continuamente diferenci´avel, X e Y s˜ao espa¸cos de Banach, Ω ⊆ X ´e
aberto e F : X ⇒ Y tem gr´afico fechado n˜ao-vazio. Supondo regularidade forte da equa¸c˜ao
e que o ponto inicial satisfaz as hip´oteses de Kantorovich, mostraremos que o m´etodo ´e
quadraticamente convergente para uma solu¸c˜ao, a qual ´e ´unica em uma vizinhan¸ca do ponto
inicial. Uma an´alise de convergˆencia local deste m´etodo tamb´em ´e apresentada. Al´em disso,
usando t´ecnicas de otimiza¸c˜ao convexa introduzida por S. M. Robinson (Numer. Math., Vol.
19, 1972, pp. 341-347), provaremos um robusto teorema de convergˆencia para o m´etodo de
Newton inexato para resolver problemas de inclus˜ao n˜ao–linear em espa¸cos de Banach, i.e.,
quando F(x) = −C e C ´e um conjunto convexo fechado. Nossa an´alise, a qual ´e baseada
na t´ecnica majorante de Kantorovich, nos permite obter resultados de convergˆencia sob as
condi¸c˜oes Lipschitz, Smale e Nesterov-Nemirovskii auto-concordante.
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