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Maroofi, Hamed. "Applications of the Monge - Kantorovich theory." Diss., Georgia Institute of Technology, 2002. http://hdl.handle.net/1853/29197.
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Östman, Martin. "Video Coding Based on the Kantorovich Distance." Thesis, Linköping University, Department of Electrical Engineering, 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-2330.
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In this Master Thesis, a model of a video coding system that uses the transportation plan taken from the calculation of the Kantorovich distance is developed. The coder uses the transportation plan instead of the differential image and sends it through blocks of transformation, quantization and coding.
The Kantorovich distance is a rather unknown distance metric that is used in optimization theory but is also applicable on images. It can be defined as the cheapest way to transport the mass of one image into another and the cost is determined by the distance function chosen to measure distance between pixels. The transportation plan is a set of finitely many five-dimensional vectors that show exactly how the mass should be moved from the transmitting pixel to the receiving pixel in order to achieve the Kantorovich distance between the images. A vector in the transportation plan is called an arc.
The original transportation plan was transformed into a new set of four-dimensional vectors called the modified difference plan. This set replaces the transmitting pixel and the receiving pixel with the distance from the transmitting pixel of the last arc and the relative distance between the receiving pixel and the transmitting pixel. The arcs where the receiving pixels are the same as the transmitting pixels are redundant and were removed. The coder completed an eleven frame sequence of size 128x128 pixels in eight to ten hours.
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Aguiar, Guilherme Ost de. "O Problema de Monge-Kantorovich para o custo quadrático." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2011. http://hdl.handle.net/10183/32384.
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Abordamos o problema do transporte otimo de Monge-Kantorovich no caso em que o custo e dado pelo quadrado da distância. Tal custo tem uma estrutura que permite a obtenção de resultados mais ricos do que o caso geral. Nosso objetivo e determinar se h a soluções para tal problema e caracteriza-las. Al em disso, tratamos informalmente do problema de transporte otimo para um custo geral.We analyze the Monge-Kantorovich optimal transportation problem in the case where the cost function is given by the square of the Euclidean norm. Such cost has a structure which allow us to get more interesting results than the general case. Our main purpose is to determine if there are solutions to such problem and characterize them. We also give an informal treatment to the optimal transportation problem in the general case.
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Russo, Daniele. "Introduzione alla Teoria del Trasporto Ottimale e Dualità di Kantorovich." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amslaurea.unibo.it/21788/.
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In questa tesi viene introdotta la teoria del trasporto ottimale a partire dal problema originale di Monge, che consiste nel trovare la strategia ottimale per trasportare una certa quantità di massa da uno scavo a una fortificazione. Si cerca cioè una mappa tra due spazi di probabilità che “trasporta” una misura nell’altra (quest’ultima viene detta quindi misura “push-forward”) e minimizza un funzionale detto “costo”. Dopo alcuni esempi, si introduce il caso discreto, coincidente con un problema di programmazione lineare; successivamente vengono dati alcuni risultati su funzioni semicontinue e spazi polacchi; vengono inoltre introdotte le nozioni di c-convessità e c-ciclica monotonia che permettono di enunciare e dimostrare il risultato principale della tesi: il teorema di Kantorovich, grazie al quale è possibile cercare il minimo del funzionale risolvendo un problema duale. Si danno quindi alcuni cenni di analisi convessa per poi applicare il teorema e costruire una mappa ottimale per una funzione costo quadratica e, in generale, strettamente convessa. Infine, si nota che dalla costruzione della mappa ottimale si può dedurre la cosiddetta decomposizione polare di un campo vettoriale, da cui si ricava una versione non lineare della decomposizione di Helmholtz; come ultima applicazione si risolve un problema di minimo riguardo un modello che descrive la configurazione di equilibrio di un gas utilizzando una misura “push-forward”.
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Zimmer, Raphael [Verfasser]. "Couplings and Kantorovich contractions with explicit rates for diffusions / Raphael Zimmer." Bonn : Universitäts- und Landesbibliothek Bonn, 2017. http://d-nb.info/1140525913/34.
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Alpargu, Gülhan. "The Kantorovich inequality, with some extensions and with some statistical applications." Thesis, McGill University, 1996. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=23985.
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In this thesis we focus on the "Kantorovich Inequality": {t prime At cdot t prime A sp{-1}t} over({t prime t) sp2}} le {{( lambda sb1+ lambda sb{n}) sp2} over {4 lambda sb1 lambda sb{n}}}, here t is a real $n times 1$ vector and A is a real $n times n$ symmetric positive definite matrix, with $ lambda sb1$ and $ lambda sb{n},$ respectively, its (fixed) largest and smallest, necessarily positive, eigenvalues. We begin the thesis with five different proofs of the Kantorovich Inequality and continue by showing that it is equivalent to five closely related inequalities due, respectively, to Schweitzer (1914), Polya-Szego (1925), Krasnosel'skii-Krei n (1952), Cassels (1955) and Greub-Rheinboldt (1959). We also examine several related inequalities which admit the Kantorovich Inequality as a special case, including the Bloomfield-Watson-Knott Inequality, for which we give a proof based on that presented by Bloomfield and Watson (1975). We also show that there appears to be a lacuna in the "brief proof" given by Yang (1990). Some statistical applications conclude the thesis with special emphasis on the efficiency of the Ordinary Least Squares Estimator in the Gauss-Markov linear statistical model.
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Oliveira, Aline Duarte de. "O teorema da dualidade de Kantorovich para o transporte de ótimo." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2011. http://hdl.handle.net/10183/32470.
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Abordaremos a teoria do transporte otimo demonstrando o teorema da dualidade de Kantorovich para uma classe ampla de funções custo. Tal resultado desempenha um papel de suma importância na teoria do transporte otimo. Uma ferramenta importante utilizada e o teorema da dualidade de Fenchel-Rockafellar, aqui enunciado e demonstrado em bastante generalidade. Demonstramos tamb em o teorema da dualidade de Kantorovich-Rubinstein, que trata do caso particular da função custo distância.We analyze the optimal transport theory proving the Kantorovich duality theorem for a wide class of cost functions. Such result plays an extremely important role in the optimal transport theory. An important tool used here is the Fenchel-Rockafellar duality theorem, which we state and prove in a general case. We also prove the Kantorovich-Rubinstein duality theorem, which deals with the particular case of cost function given by the distance.
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Engvall, Sebastian. "Kaijsers algoritm för beräkning av Kantorovichavstånd parallelliserad i CUDA." Thesis, Linköpings universitet, Informationskodning, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-102867.
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This thesis processes the work of developing CPU code and GPU code for Thomas Kaijsers algorithm for calculating the kantorovich distance and the performance between the two is compared. Initially there is a rundown of the algorithm which calculates the kantorovich distance between two images. Thereafter we go through the CPU implementation followed by GPGPU written in CUDA. Then the results are presented. Lastly, an analysis about the results and a discussion with possible improvements is presented for possible future applications.
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Agueh, Martial Marie-Paul. "Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory." Diss., Georgia Institute of Technology, 2002. http://hdl.handle.net/1853/29180.
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Zhao, Junqing. "The computer oriented Kantorovich-finite difference method and its application on bridge engineering." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/nq21980.pdf.
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Souza, Estefano Alves de. "O problema de Monge-Kantorovich para duas medidas de probabilidade sobre um conjunto finito." Universidade de São Paulo, 2009. http://www.teses.usp.br/teses/disponiveis/45/45133/tde-04052009-162654/.
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Apresentamos o problema do transporte ótimo de Monge-Kantorovich com duas medidas de probabilidade conhecidas e que possuem suporte em um conjunto de cardinalidade finita. O objetivo é determinar condições que permitam construir um acoplamento destas medidas que minimiza o valor esperado de uma função de custo conhecida e que assume valor nulo apenas nos elementos da diagonal. Apresentamos também um resultado relacionado com a solução do problema de Monge-Kantorovich em espaços produto finitos quando conhecemos soluções para o problema nos espaços marginais.We present the Monge-Kantorovich optimal problem with two known probability measures on a finite set. The objective is to obtain conditions that allow us to build a coupling of these measures that minimizes the expected value of a cost function that is known and is zero only on the diagonal elements. We also present a result that is related with the solution of the Monge-Kantorovich problem in finite product spaces in the case that solutions to the problem in the marginal spaces are known.
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Lissing, Johan. "Video coding using compressed transportation plans." Thesis, Linköping University, Department of Electrical Engineering, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-8335.
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A transportation plan is a byproduct from the calculation of the Kantorovich distance between two images. It describes a transformation from one of the images to the other. This master thesis shows how transportation plans can be used for video coding and how to process the transportation plans to achieve a good bitrate/quality ratio. Various parameters are evaluated using an implemented transportation plan video coder.
The introduction of transform coding with DCT proves to be very useful, as it reduces the size of the resulting transportation plans. DCT coding roughly gives a 10-fold decrease in bitrate with maintained quality compared to the nontransformed transportation plan coding.
With the best settings for transportation plan coding, I was able to code a test sequence at about 5 times the bitrate for MPEG coding of the same sequence with similar quality.
As video coding using transportation plans is a very new concept, the thesis is ended with conclusions on the test results and suggestions for future research in this area.
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Fioriti, Emanuele. "La metrica di Wasserstein." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/23477/.
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L'elaborato tratta alcune proprietà topologiche della metrica di Wasserstein, in particolare la convergenza di misure di probabilità in relazione alla convergenza debole, la struttura dello spazio metrico di Wasserstein e la formula duale di Kantorovich.
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GONAÇALVES, Max Leandro Nobre. "Convergência local do método de Newton inexato e suas variações do ponto de vista do princípio majorante de kantorovich." Universidade Federal de Goiás, 2007. http://repositorio.bc.ufg.br/tede/handle/tde/1961.
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Previous issue date: 2007-12-14The search for solutions of nonlinear equations in the Euclidean spaces is object of interest in some areas of science and engineerings. Due the speed of convergence and computational efficiency, the inexact Newton method and its variations have been suficiently used to obtain solutions of these equations. In this dissertation we present a local analysis of convergence of the inexact Newton method and some of its variations, more specifically the inexact Newton-like method and the inexact modified Newton method. This analysis has the disadvantage to demand the previous knowledge of a zero of the operator in consideration and the hypotheses on the behavior of the operator at this zero, but on the other hand it supplies to information on the convergence rate and convergence radius.
A busca por soluçõeses de equaçõess não-lineares nos espaços Euclidianos é objeto de interesse em várias áreas da ciência e das engenharias. Devido a sua velocidade de convergência e e¯eficiência computacional, o método de Newton inexato e suas variações têm sido bastante utilizados para o propósito de obter solu»c~oes dessas equações. Nesta dissertação apresentamos uma anáalise de convergência local do método de Newton inexato e algumas de suas variações, mais especificamente, o método quase-Newton inexato e o método de Newton modificado inexato. Esta análise tem a desvantagem de exigir o conhecimento prévio de um zero do operador em consideração e hipóteses sobre o comportamento do operador nesse zero, mas por outro lado ela fornece informações sobre a taxa e o raio de convergência.
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Shao, Jinghai. "Inégalité du coût de transport et le problème de Monge-Kantorovich sur le groupe des lacets et l'espace des chemins." Dijon, 2006. http://www.theses.fr/2006DIJOS020.
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Nous nous intéressons principalement à l'inégalité du coût de transport (ICT) et au problème de Monge-Kantorovich sur PeG l'espace des chemins et le groupe des lacets LeG. Nous établissions ICT pour la H-distance sur PeG et pour la distance uniforme sur LeG. Dans un espace polonais, on definit un semi-groupe et montre qu'il satisfait à l'inégalité de Hamilton-Jacobi sous quelques conditions convenables. Ce résultat nous permet d'établir ICT sur LeG pour la “distance riemannienne”. Enfin, nous montrons qu'il existe uniquement une application optimale T de LeG dans LeG qui pousse la mesure de la chaleur ν en avant vers F ν, où F est une densité telle que l'entropie Ent ν (F) est finieWe mainly consider to establish the transportation cost inequality (TCI ) and to solve the Monge-Kantorovich problem on path space PeG and loop group LeG. An H-distance is defined on PeG and enables us to establish TCI on PeG and generalize this method to the path space over LeG to obtain TCI w. R. T. Uniform distance on LeG. In a Polish space, we define a semigroup and prove it satisfies Hamilton-Jacobi inequality under suitable conditions, then we apply it to establish TCI w. R. T. “Riemannian distance” on LeG. At last, we prove there exists a unique optimal map T from LeG to LeG which pushes heat kernel measure ν forward to F ν , where F is a density such that the entropy Ent ν (F) is finite
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Guittet, Kévin. "Contributions à la résolution numérique de problèmes de transport optimal de masse." Paris 6, 2003. http://www.theses.fr/2003PA066380.
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Buckley, Donovan O. "Solution of Nonlinear Transient Heat Transfer Problems." FIU Digital Commons, 2010. http://digitalcommons.fiu.edu/etd/302.
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In the presented thesis work, meshfree method with distance fields was extended to obtain solution of nonlinear transient heat transfer problems. The thesis work involved development and implementation of numerical algorithms, data structure, and software. Numerical and computational properties of the meshfree method with distance fields were investigated. Convergence and accuracy of the methodology was validated by analytical solutions, and solutions produced by commercial FEM software (ANSYS 12.1). The research was focused on nonlinearities caused by temperature-dependent thermal conductivity. The behavior of the developed numerical algorithms was observed for both weak and strong temperature-dependency of thermal conductivity. Oseen and Newton-Kantorovich linearization techniques were applied to linearized the governing equation and boundary conditions. Results of the numerical experiments showed that the meshfree method with distance fields has the potential to produced fast accurate solutions. The method enables all prescribed boundary conditions to be satisfied exactly.
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Hurtado, Elard Juárez. "Semigrupos gerados pelo p-Laplaciano e um estudo do limite p→∞." Universidade Federal de São Carlos, 2012. https://repositorio.ufscar.br/handle/ufscar/5884.
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Previous issue date: 2012-05-02Financiadora de Estudos e Projetos
We study the problem: (2) _∂tup − Δpup = 0, (0,∞) × Rd up(0, x) = g(x), {t = 0} × Rd ∞ > p ≥ d+1, where the initial data up(0, x) = g(x) are Lipschitz continuous, non-negative and it have compact support. Solutions of this problem provide a simplistic model for collapse of an initially unstable sandpile . We regard the limit up when p→∞as a solution for instantaneous mass transfer problem governed by Monge-Kantorovich theory. We study the case d = 1 for which we obtain explicit solutions. Keywords: p-laplacian, Monge-Kantorovich Theory, Monotone operator theory.
Neste trabalho, nós estudamos o problema: (1) _∂tup − Δpup = 0, (0,∞) × Rd up(0, x) = g(x), {t = 0} × Rd ∞ > p ≥ d+1, no caso em que o dado inicial up(0, x) = g(x) é Lipschitz contínuo, não negativo e com suporte compacto. As soluções deste problema fornecem um modelo rudimentar para o colapso de pilhas de areia com uma configuração inicialmente instável . Tomando o limite de up quando p→∞ obtemos uma solução para o problema de transferência de massa instantânea governado pela Teoria de Monge-Kantorovich. Como exemplo de aplicação estudamos o caso d = 1, para o qual obtemos soluções explícitas. Palavras-chave: p-laplaciano, Teoria de Monge-Kantorovich, Operadores Monótonos.
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Kaijser, Thomas. "Convergence in distribution for filtering processes associated to Hidden Markov Models with densities." Linköpings universitet, Matematik och tillämpad matematik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-92590.
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A Hidden Markov Model generates two basic stochastic processes, a Markov chain, which is hidden, and an observation sequence. The filtering process of a Hidden Markov Model is, roughly speaking, the sequence of conditional distributions of the hidden Markov chain that is obtained as new observations are received. It is well-known, that the filtering process itself, is also a Markov chain. A classical, theoretical problem is to find conditions which implies that the distributions of the filtering process converge towards a unique limit measure. This problem goes back to a paper of D Blackwell for the case when the Markov chain takes its values in a finite set and it goes back to a paper of H Kunita for the case when the state space of the Markov chain is a compact Hausdor space. Recently, due to work by F Kochmann, J Reeds, P Chigansky and R van Handel, a necessary and sucient condition for the convergence of the distributions of the filtering process has been found for the case when the state space is finite. This condition has since been generalised to the case when the state space is denumerable. In this paper we generalise some of the previous results on convergence in distribution to the case when the Markov chain and the observation sequence of a Hidden Markov Model take their values in complete, separable, metric spaces; it has though been necessary to assume that both the transition probability function of the Markov chain and the transition probability function that generates the observation sequence have densities.
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Achille, Matteo d'. "Statistical properties of the Euclidean random assignment problem." Thesis, université Paris-Saclay, 2020. http://www.theses.fr/2020UPASQ003.
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Étant donné 2n points, n ``rouge'' et n ``bleu'', dans un espace euclidien,la résolution du problème d'assignation euclidienne associé consiste à trouver la bijection entre les points rouges et bleus qui minimise une fonctionnelle des positions de points. Dans la version stochastique de ce problème, les points sont un processus de point de Poisson, et un certain intérêt a développé au fil des ans sur les propriétés typiques et moyennes de la solution dans la limite n to ∞. Cette thèse de doctorat porte sur ce problème dans un certain nombre de cas (;plusieurs résultats exacts en d=1, la dérivation de certaines propriétés fines en d=2, en partie encore conjecturales, un étude des fractales auto-similaires avec 1Given 2n points, n ``red'' and n ``blue'', in a Euclidean space,solving the associated Euclidean Assignment Problem consists infinding the bijection between red and blue points that minimizes afunctional of the point positions. In the stochastic version of this problem, the points are a Poisson Point Process, and some interest has developed over the years on the typical and average properties of thesolution in the limit n to ∞. This PhD thesis investigates this problem in a number of cases (many exact results in d=1, the derivation of some fine properties in d=2, in part still conjectural, an investigation on self-similar fractals with 1
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Nguyen, Van Thanh. "Problèmes de transport partiel optimal et d'appariement avec contrainte." Thesis, Limoges, 2017. http://www.theses.fr/2017LIMO0052.
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Cette thèse est consacrée à l'analyse mathématique et numérique pour les problèmes de transport partiel optimal et d'appariement avec contrainte (constrained matching problem). Ces deux problèmes présentent de nouvelles quantités inconnues, appelées parties actives. Pour le transport partiel optimal avec des coûts qui sont donnés par la distance finslerienne, nous présentons des formulations équivalentes caractérisant les parties actives, le potentiel de Kantorovich et le flot optimal. En particulier, l'EDP de condition d'optimalité permet de montrer l'unicité des parties actives. Ensuite, nous étudions en détail des approximations numériques pour lesquelles la convergence de la discrétisation et des simulations numériques sont fournies. Pour les coûts lagrangiens, nous justifions rigoureusement des caractérisations de solution ainsi que des formulations équivalentes. Des exemples numériques sont également donnés. Le reste de la thèse est consacré à l'étude du problème d'appariement optimal avec des contraintes pour le coût de la distance euclidienne. Ce problème a un comportement différent du transport partiel optimal. L'unicité de solution et des formulations équivalentes sont étudiées sous une condition géométrique. La convergence de la discrétisation et des exemples numériques sont aussi établis. Les principaux outils que nous utilisons dans la thèse sont des combinaisons des techniques d'EDP, de la théorie du transport optimal et de la théorie de dualité de Fenchel--Rockafellar. Pour le calcul numérique, nous utilisons des méthodes du lagrangien augmentéThe manuscript deals with the mathematical and numerical analysis of the optimal partial transport and optimal constrained matching problems. These two problems bring out new unknown quantities, called active submeasures. For the optimal partial transport with Finsler distance costs, we introduce equivalent formulations characterizing active submeasures, Kantorovich potential and optimal flow. In particular, the PDE of optimality condition allows to show the uniqueness of active submeasures. We then study in detail numerical approximations for which the convergence of discretization and numerical simulations are provided. For Lagrangian costs, we derive and justify rigorously characterizations of solution as well as equivalent formulations. Numerical examples are also given. The rest of the thesis presents the study of the optimal constrained matching with the Euclidean distance cost. This problem has a different behaviour compared to the partial transport. The uniqueness of solution and equivalent formulations are studied under geometric condition. The convergence of discretization and numerical examples are also indicated. The main tools which we use in the thesis are some combinations of PDE techniques, optimal transport theory and Fenchel--Rockafellar dual theory. For numerical computation, we make use of augmented Lagrangian methods
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Martins, Tiberio Bittencourt de Oliveira. "Newton's methods under the majorant principle on Riemannian manifolds." Universidade Federal de Goiás, 2015. http://repositorio.bc.ufg.br/tede/handle/tede/4847.
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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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Zhu, Lei. "On Visualizing Branched Surface: an Angle/Area Preserving Approach." Diss., Available online, Georgia Institute of Technology, 2004, 2004. http://etd.gatech.edu/theses/available/etd-09142004-114941/.
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Thesis (Ph. D.)--Biomedical Engineering, Georgia Institute of Technology, 2006.Anthony J. Yezzi, Committee Member ; James Gruden, Committee Member ; Allen Tannenbaum, Committee Chair ; May D. Wang, Committee Member ; Oskar Skrinjar, Committee Member. Vita. Includes bibliographical references.
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Arhab, Slimane. "Profilométrie optique par méthodes inverses de diffraction électromagnétique." Thesis, Aix-Marseille, 2012. http://www.theses.fr/2012AIXM4321/document.
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La profilométrie optique est une technique de métrologie de surface rapide et non destructive. Dans ce mémoire, nous avons abordé cette problématique par des méthodes inverses de diffraction électromagnétique et dans une configuration de type Microscopie Tomographique Optique par Diffraction (ODTM). La surface est sondée par un éclairement sous plusieurs angles d'incidences ; la mesure en amplitude et en phase du champ lointain diffracté constitue les données du problème. Des profils de surfaces ont été reconstruits en considérant différents modèles de diffraction, parmi lesquelles une méthode approchée fondée sur les approximations de diffusion simple et de paraxialité. La résolution latérale de cette méthode et des techniques classiques de profilométrie est limitée par le critère d'Abbe-Rayleigh, défini sur la base de l'ouverture numérique pour l'éclairement et la détection du champ. Afin de dépasser cette limite de résolution, nous avons développé une méthode itérative de Newton-Kantorovitch régularisée. L'opérateur de diffraction y est rigoureusement modélisé par une méthode des moments, résolution numérique des équations du formalisme intégral de frontière, et l'expression de la dérivée de Fréchet de cet opérateur est obtenue par la méthode des états adjoints, à partir du théorème de réciprocité. Pour les surfaces unidimensionnelles métalliques, notre technique permet d'inverser à partir de données synthétiques des surfaces très rugueuses avec une résolution au delà du critère d'Abbe-RayleighOptical profilometry is a nondestructive and fast noncontact surface metrology technique. In this thesis, we have tackled this issue with inverse scattering electromagnetic methods and in an Optical Digital Tomographic Microscopy (ODTM) configuration. The surface is probed with illuminations under several incidence angles; the measure of far scattered field amplitude and phase constitutes the problem data. Surface profiles have been reconstructed using different scattering models among which an approximate theory based on single scattering and paraxiality. The lateral resolution of this technique and classical profilometric approaches is limited by the so-called Abbe-Rayleigh's criterion defined out of the numerical aperture for illumination and field detection. In order to overpass this resolution limit, we have developed a regularized iterative Newton-Kantorovitch's method. The scattering operator is rigorously modelized with the method of moments, that is a numerical solution of boundary integral equations, and its Fréchet derivative adjoint states expression is deduced from the reciprocity theorem. For one-dimensional metallic surfaces, our method succeeds in inverting from synthetic data very rough surfaces with the resolutions beyond the Abbe-Rayleigh's criterion. The performance of this technique and inversion conditions clearly differ from one polarization to the other : in the TM case, interactions at longer distance than in the TE case improve yet the resolution. This work includes also an experimental validation of our inverse model on grooves in indium phosphure substrate at 633 nm
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Bernot, Marc. "Transport optimal et irrigation." Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2005. http://tel.archives-ouvertes.fr/tel-00132078.
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L'objet de cette thèse est de modéliser et d'étudier des structures d'irrigation telles les nervures des feuilles, réseau sanguin, poumons,etc. Un modèle généralisant le problème de Gilbert Steiner est introduit ; on étudie alors les propriétés d'existence, de stabilité et régularité. Des algorithmes sont alors proposés pour la simulation.
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Karlsson, Johan. "Inverse Problems in Analytic Interpolation for Robust Control and Spectral Estimation." Doctoral thesis, Stockholm : Matematik, Mathematics, Kungliga Tekniska högskolan, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-9248.
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Lavaux, Guilhem. "Reconstruction des vitesses propres des galaxies : méthodes et applications aux observations." Phd thesis, Université Paris Sud - Paris XI, 2008. http://tel.archives-ouvertes.fr/tel-00412146.
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Bien que nous ayons accès à des catalogues de galaxies très détaillés, notre compréhension de la distribution spatiale de la matière noire reste encore limitée. D'importantes informations à son sujet sont cachées dans les vitesses propres des galaxies, qui reflètent la dynamique de la matière noire à différentes échelles. Malheureusement, ces vitesses sont très difficiles à observer. Nous présentons ici une approche différente pour "mesurer" ces vitesses par l'intermédiaire de méthodes de reconstruction des champs de vitesse. Nous utilisons en particulier la reconstruction dîte de Monge-Ampère-Kantorovitch (MAK). Nous testons cette méthode sur des simulations à N-corps ainsi que sur des catalogues virtuels de galaxies. Nous vérifions sa fiabilité par la comparaison des vitesses reconstruites aux vitesses simulées et aussi à travers la mesure de la densité moyenne de matière de ces univers.Après avoir testé cette méthode, nous l'utilisons sur un vrai catalogue de galaxie: le 2MASS Redshift survey. Après l'avoir corrigé des effets observationnels connus, nous étudions l'origine de la vitesse du Groupe Local par rapport au fond diffus cosmologique. Nous montrons que plus de la moitié de notre vitesse est due à des structures situées à plus de 40 Mpc/h. Une fois étudié le mouvement d'ensemble des structures locales, nous comparons directement les vitesses reconstruites et les distances observées dans notre voisinage de 30 Mpc/h. Nous proposons une estimation indépendante du paramètre de densité. Cette estimation peut être utilisée afin de réduire les dégénérescences dans l'espace des paramètres du modèle d'univers à base de matière noire froide.
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Perrone, Paolo. "Categorical Probability and Stochastic Dominance in Metric Spaces." 2018. https://ul.qucosa.de/id/qucosa%3A32641.
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In this work we introduce some category-theoretical concepts and techniques to study probability distributions on metric spaces and ordered metric spaces.
In Chapter 1 we give an overview of the concept of a probability monad, first defined by Giry.
Probability monads can be interpreted as a categorical tool to talk about random elements of a space X. We can consider these random elements as formal convex combinations, or mixtures, of elements of X.
Spaces where the convex combinations can be actually evaluated are called algebras of the probability monad.
In Chapter 2 we define a probability monad on the category of complete metric spaces and 1-Lipschitz maps called the Kantorovich monad, extending a previous construction due to van Breugel. This monad assigns to each complete metric space X its Wasserstein space PX.
It is well-known that finitely supported probability measures with rational coefficients, or empirical distributions of finite sequences, are dense in the Wasserstein space.
This density property can be translated into categorical language as a colimit of a diagram involving certain powers of X.
The monad structure of P, and in particular the integration map, is uniquely determined by this universal property.
We prove that the algebras of the Kantorovich monad are exactly the closed convex subsets of Banach spaces.
In Chapter 3 we extend the Kantorovich monad of Chapter 2 to metric spaces equipped with a partial order. The order is inherited by the Wasserstein space, and is called the stochastic order.
Differently from most approaches in the literature, we define a compatibility condition of the order with the metric itself, rather then with the topology it induces. We call the spaces with this property L-ordered spaces.
On L-ordered spaces, the stochastic order induced on the Wasserstein spaces satisfies itself a form of Kantorovich duality.
The Kantorovich monad can be extended to the category of L-ordered metric spaces. We prove that its algebras are the closed convex subsets of ordered Banach spaces, i.e. Banach spaces equipped with a closed cone.
The category of L-ordered metric spaces can be considered a 2-category, in which we can describe concave and convex maps categorically as the lax and oplax morphisms of algebras.
In Chapter 4 we develop a new categorical formalism to describe operations evaluated partially.
We prove that partial evaluations for the Kantorovich monad, or partial expectations, define a closed partial order on the Wasserstein space PA over every algebra A, and that the resulting ordered space is itself an algebra.
We prove that, for the Kantorovich monad, these partial expectations correspond to conditional expectations in distribution.
Finally, we study the relation between these partial evaluation orders and convex functions.
We prove a general duality theorem extending the well-known duality between convex functions and conditional expectations to general ordered Banach spaces.
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Nguyen, Van thanh. "Problèmes de transport partiel optimal et d'appariement avec contrainte." Thesis, 2017. http://www.theses.fr/2017LIMO0052/document.
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Cette thèse est consacrée à l'analyse mathématique et numérique pour les problèmes de transport partiel optimal et d'appariement avec contrainte (constrained matching problem). Ces deux problèmes présentent de nouvelles quantités inconnues, appelées parties actives. Pour le transport partiel optimal avec des coûts qui sont donnés par la distance finslerienne, nous présentons des formulations équivalentes caractérisant les parties actives, le potentiel de Kantorovich et le flot optimal. En particulier, l'EDP de condition d'optimalité permet de montrer l'unicité des parties actives. Ensuite, nous étudions en détail des approximations numériques pour lesquelles la convergence de la discrétisation et des simulations numériques sont fournies. Pour les coûts lagrangiens, nous justifions rigoureusement des caractérisations de solution ainsi que des formulations équivalentes. Des exemples numériques sont également donnés. Le reste de la thèse est consacré à l'étude du problème d'appariement optimal avec des contraintes pour le coût de la distance euclidienne. Ce problème a un comportement différent du transport partiel optimal. L'unicité de solution et des formulations équivalentes sont étudiées sous une condition géométrique. La convergence de la discrétisation et des exemples numériques sont aussi établis. Les principaux outils que nous utilisons dans la thèse sont des combinaisons des techniques d'EDP, de la théorie du transport optimal et de la théorie de dualité de Fenchel--Rockafellar. Pour le calcul numérique, nous utilisons des méthodes du lagrangien augmentéThe manuscript deals with the mathematical and numerical analysis of the optimal partial transport and optimal constrained matching problems. These two problems bring out new unknown quantities, called active submeasures. For the optimal partial transport with Finsler distance costs, we introduce equivalent formulations characterizing active submeasures, Kantorovich potential and optimal flow. In particular, the PDE of optimality condition allows to show the uniqueness of active submeasures. We then study in detail numerical approximations for which the convergence of discretization and numerical simulations are provided. For Lagrangian costs, we derive and justify rigorously characterizations of solution as well as equivalent formulations. Numerical examples are also given. The rest of the thesis presents the study of the optimal constrained matching with the Euclidean distance cost. This problem has a different behaviour compared to the partial transport. The uniqueness of solution and equivalent formulations are studied under geometric condition. The convergence of discretization and numerical examples are also indicated. The main tools which we use in the thesis are some combinations of PDE techniques, optimal transport theory and Fenchel--Rockafellar dual theory. For numerical computation, we make use of augmented Lagrangian methods
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Pass, Brendan. "Structural Results on Optimal Transportation Plans." Thesis, 2011. http://hdl.handle.net/1807/31893.
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In this thesis we prove several results on the structure of solutions to optimal transportation problems.
The second chapter represents joint work with Robert McCann and Micah Warren; the main result is that, under a non-degeneracy condition on the cost function, the optimal is concentrated on a $n$-dimensional Lipschitz submanifold of the product space. As a consequence, we provide a simple, new proof that the optimal map satisfies a Jacobian equation almost everywhere. In the third chapter, we prove an analogous result for the multi-marginal optimal transportation problem; in this context, the dimension of the support of the solution depends on the signatures of a $2^{m-1}$ vertex convex polytope of semi-Riemannian metrics on the product space, induce by the cost function. In the fourth chapter, we identify sufficient conditions under which the solution to the multi-marginal problem is concentrated on the graph of a function over one of the marginals. In the fifth chapter, we investigate the regularity of the optimal map when the dimensions of the two spaces fail to coincide. We prove that a regularity theory can be developed only for very special cost functions, in which case a quotient construction can be used to reduce the problem to an optimal transport problem between spaces of equal dimension. The final chapter applies the results of chapter 5 to the principal-agent problem in mathematical economics when the space of types and the space of available goods differ. When the dimension of the space of types exceeds the dimension of the space of goods, we show if the problem can be formulated as a maximization over a convex set, a quotient procedure can reduce the problem to one where the two dimensions coincide. Analogous conditions are investigated when the dimension of the space of goods exceeds that of the space of types.
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Su, Hung-Kung, and 蘇宏恭. "Some analysis on one-dimensional Monge-Kantorovich's problem." Thesis, 2012. http://ndltd.ncl.edu.tw/handle/04317984255595970585.
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碩士國立清華大學
數學系
100
In order to better understand the Monge-Kantorovich's problem, in this thesis we study some elementary examples of cost functions for the one dimensional transportation problem. We also consider a special case of the problem with marginals , that are continuously distributed on the line with piecewise continuous cost functions c of distance, and then use doubly stochastic matrices associated to mu and nu to obtain numerical estimates of the optimal transportation cost.