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Structural Results on Optimal Transportation PlansPass, Brendan 11 January 2012 (has links)
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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On Visualizing Branched Surface: an Angle/Area Preserving ApproachZhu, Lei 12 September 2004 (has links)
The techniques of surface deformation and mapping are useful tools for the visualization of medical surfaces, especially for highly undulated or branched surfaces. In this thesis, two algorithms
are presented for flattened visualizations of multi-branched medical surfaces, such as vessels. The first algorithm is an angle preserving approach, which is based on conformal analysis. The mapping function is obtained by minimizing two Dirichlet functionals. On a triangulated representation of vessel surfaces, this algorithm can be implemented efficiently using a finite
element method. The second algorithm adjusts the result from conformal mapping to produce a flattened representation of the original surface while preserving areas. It employs the theory of
optimal mass transport via a gradient descent approach.
A new class of image morphing algorithms is also considered based on the theory of optimal mass transport. The mass moving energy functional is revised by adding an intensity penalizing term, in
order to reduce the undesired "fading" effects. It is a parameter free approach. This technique has been applied on several natural and medical images to generate in-between image sequences.
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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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