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Mazure, Marie-Laurence. "Analyse varationnelle des formes quadratiques convexes." Toulouse 3, 1986. http://www.theses.fr/1986TOU30109.
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Ce travail a pour motivation le principe variationnel de maxwell: lorsque deux réseaux électriques sont disposés en parallèle, la distribution du courant, soumise à la loi de Kirchhoff, se fait de façon a minimiser la puissance totale dissipée. Ce principe se présente donc comme une manifestation de l'inf-convolution des fonctions puissances associées aux deux réseaux. Si les résistances généralisées de ces réseaux sont les matrices symétriques semi-définies positives a et b, cette constation conduit au fait que la résistance généralisée équivalente, appelée somme parallèle de a et b, est la matrice associée a la forme quadratique convexe résultant de l'inf-convolution des formes quadratiques convexes associées à a et b. Cette définition variationnelle de l'addition parallèle permet d'utiliser, d'une manière systématique, les techniques de l'analyse convexe pour en développer les propriétés de manière élégante. Toujours à partir de formulations variationnelles, il est également possible d'interpréter en termes d'analyse convexe d'une part la notion électrique de court-circuit, et d'autre part celle de différence parallèle de deux operateurs symétriques semi-définis positifs. Cette dernière opération est définie a partir d'une notion nouvelle en analyse convexe: la déconvolution d'une fonction convexe par une autre, opération qui, grâce a un résultat récent sur la conjuguée de la différence de deux fonctions, peut être, dans une certaine mesure, considérée comme l'opération inverse de l'inf-convolution. Les résultats concernant l'addition parallèle, l'opération de court-circuit et la différence parallèle sont d'abord énoncés en dimension finie puis prolonges au cadre hilbertien. L’extension possible des diverses opérations étudiées aux sous-espaces hilbertiens d'un espace vectoriel topologique est suggérée en annexe.
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Hillyard, Cinnamon. "A Maple Package for the Variational Calculus." DigitalCommons@USU, 1992. https://digitalcommons.usu.edu/etd/7124.
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The HELMHOLTZ package, written in Maple V, is a collection of commands to support research in the variational calculus. These commands include the standard operators on differential forms, Euler-Lagrange operators, homotopy operators, Lie bracket, Lie derivatives, and the prolongation of a vector field. We give a brief introduction to the variational calculus. We describe each of the commands in the HELMHOLTZ package completely along with numerous examples of each. Applications of the package include verification of symmetry groups for differential equations, solving the inverse problem of the calculus of variations, computing generalized symmetries, and finding variational integrating factors. A complete listing of the Maple code for HELMHOLTZ is found in an appendix.
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Iqbal, Zamin. "Variational methods in solid mechanics." Thesis, University of Oxford, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.301901.
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Botelho, Fabio Silva. "Variational Convex Analysis." Diss., Virginia Tech, 2009. http://hdl.handle.net/10919/28351.
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This work develops theoretical and applied results for variational convex analysis. First we present the basic tools of analysis necessary to develop the core theory and applications.
New results concerning duality principles for systems originally modeled by non-linear differential equations are shown in chapters 9 to 17. A key aspect of this work is that although the original problems are non-linear with corresponding non-convex variational formulations, the dual formulations obtained are almost always concave and amenable to numerical computations. When the primal problem has no solution in the classical sense, the solution of dual problem is a weak limit of minimizing sequences, and the evaluation of such average behavior is important in many practical applications. Among the results we highlight the dual formulations for micro-magnetism, phase transition models, composites in elasticity and conductivity and others. To summarize, in the present work we introduce convex analysis as an interesting alternative approach for the understanding and computation of some important problems in the modern calculus of variations.Ph. D.
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Muller, Stefan. "Variational problems in mechanics and analysis." Thesis, Heriot-Watt University, 1989. http://hdl.handle.net/10399/925.
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Hanisch, Florian. "Variational problems on supermanifolds." Phd thesis, Universität Potsdam, 2011. http://opus.kobv.de/ubp/volltexte/2012/5975/.
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In this thesis, we discuss the formulation of variational problems on supermanifolds. Supermanifolds incorporate bosonic as well as fermionic degrees of freedom. Fermionic fields take values in the odd part of an appropriate Grassmann algebra and are thus showing an anticommutative behaviour. However, a systematic treatment of these Grassmann parameters requires a description of spaces as functors, e.g. from the category of Grassmann algberas into the category of sets (or topological spaces, manifolds). After an introduction to the general ideas of this approach, we use it to give a description of the resulting supermanifolds of fields/maps. We show that each map is uniquely characterized by a family of differential operators of appropriate order. Moreover, we demonstrate that each of this maps is uniquely characterized by its component fields, i.e. by the coefficients in a Taylor expansion w.r.t. the odd coordinates. In general, the component fields are only locally defined. We present a way how to circumvent this limitation. In fact, by enlarging the supermanifold in question, we show that it is possible to work with globally defined components.
We eventually use this formalism to study variational problems. More precisely, we study a super version of the geodesic and a generalization of harmonic maps to supermanifolds. Equations of motion are derived from an energy functional and we show how to decompose them into components. Finally, in special cases, we can prove the existence of critical points by reducing the problem to equations from ordinary geometric analysis. After solving these component equations, it is possible to show that their solutions give rise to critical points in the functor spaces of fields.In dieser Dissertation wird die Formulierung von Variationsproblemen auf Supermannigfaltigkeiten diskutiert. Supermannigfaltigkeiten enthalten sowohl bosonische als auch fermionische Freiheitsgrade. Fermionische Felder nehmen Werte im ungeraden Teil einer Grassmannalgebra an, sie antikommutieren deshalb untereinander. Eine systematische Behandlung dieser Grassmann-Parameter erfordert jedoch die Beschreibung von Räumen durch Funktoren, z.B. von der Kategorie der Grassmannalgebren in diejenige der Mengen (der topologischen Räume, Mannigfaltigkeiten, ...). Nach einer Einführung in das allgemeine Konzept dieses Zugangs verwenden wir es um eine Beschreibung der resultierenden Supermannigfaltigkeit der Felder bzw. Abbildungen anzugeben. Wir zeigen, dass jede Abbildung eindeutig durch eine Familie von Differentialoperatoren geeigneter Ordnung charakterisiert wird. Darüber hinaus beweisen wir, dass jede solche Abbildung eineindeutig durch ihre Komponentenfelder, d.h. durch die Koeffizienten einer Taylorentwickelung bzgl. von ungeraden Koordinaten bestimmt ist. Im Allgemeinen sind Komponentenfelder nur lokal definiert. Wir stellen einen Weg vor, der diese Einschränkung umgeht: Durch das Vergrößern der betreffenden Supermannigfaltigkeit ist es immer möglich, mit globalen Koordinaten zu arbeiten. Schließlich wenden wir diesen Formalismus an, um Variationsprobleme zu untersuchen, genauer betrachten wir eine super-Version der Geodäte und eine Verallgemeinerung von harmonischen Abbildungen auf Supermannigfaltigkeiten. Bewegungsgleichungen werden von Energiefunktionalen abgeleitet und wir zeigen, wie sie sich in Komponenten zerlegen lassen. Schließlich kann in Spezialfällen die Existenz von kritischen Punkten gezeigt werden, indem das Problem auf Gleichungen der gewöhnlichen geometrischen Analysis reduziert wird. Es kann dann gezeigt werden, dass die Lösungen dieser Gleichungen sich zu kritischen Punkten im betreffenden Funktor-Raum der Felder zusammensetzt.
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黃志榮 and Chi-wing Wong. "On Cartan form and equivalence of variational problems." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1997. http://hub.hku.hk/bib/B31220071.
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Wong, Chi-wing. "On Cartan form and equivalence of variational problems /." Hong Kong : University of Hong Kong, 1997. http://sunzi.lib.hku.hk/hkuto/record.jsp?B19472602.
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OKASSA, EUGENE. "Geometrie des points proches et applications." Toulouse 3, 1989. http://www.theses.fr/1989TOU30003.
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On donne une caracterisation du prolongement des champs de vecteurs sur une variete des points proches. On determine le type d'algebres locales pour lesquel certaines relevees de formes symplectiques (respectivement pseudo-riemanniennes) sont symplectiques (respectivement pseudo-riemanniennes). On definit les prolongements d'algebres locales: ce qui permet de definir le complexe differentiel de lagrange sur une variete de points proches
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Forclaz, A. "Variational methods in materials science." Thesis, University of Oxford, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.249532.
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Three problems are being investigated in this thesis. The first two relate to the modelling and analysis of martensitic phase transitions, while the third is concerned with some mathematical tools used in this setting. After a short introduction (Chapter 1) and overviews of the calculus of variations and martensitic phase transformations (Chapter 2), the research part of this thesis is divided into three chapters. We show in Chapter 3 that for the two wells $\mathrm{SO}(3)U$ and $\mathrm{SO}(3)V$ to be rank-one connected, where the $3\times 3$ symmetric positive definite $U$ and $V$ have the same eigenvalues, it is necessary and sufficient that $\mathrm{det}(U-V)=0$, a result that does not hold in higher dimensions. Using this criterion and a result of Gurtin, formulae for the twinning plane and the shearing vector are obtained, which yield an extremely simple condition for the occurrence of so-called compound twins. Our results also provide a simple classification of the twinning mode of the two wells by looking at the crystallographic properties of the eigenvectors of the difference $U-V$. As an illustration, we apply our results to cubic-to-tetra gonal,tetragonal-to-monoclinic and cubic-to-monoclinic transitions. Chapter 4 focuses on the mathematical analysis of biaxial loading experiments in martensite, more particularly on how hysteresis relates to metastability. These experiments were carried out by Chu and James and their mathematical treatment was initiated by Ball, Chu and James. Experimentally it is observed that a homogeneous deformation $y_1(x)= U_1x$ is the stable state for `small' loads while $y_2(x)=U_2x$ is stable for `large' loads. A model was proposed by Ball, Chu and James which, for a certain intermediate range of loads, predicts crucially that $y_1(x)=U_1x$ remains metastable i.e., a local - as opposed to global - minimiser of the energy). This result explains convincingly the hysteresis that is observed experimentally. It is easy to get an upper bound for when metastability finishes. However, it was also noticed that this bound (the Schmid Law) may not be sharp, though this required some geometric conditions on the sample. In this chapter, we rigorously justify the Ball-Chu-James model by means of De Giorgi's $\Gamma$-convergence, establish some properties of local minimisers of the (limiting) energy and prove the metastability result mentioned above. An important part of the chapter is then devoted to establishing which geometric conditions are necessary and sufficient for the counter-example to the Schmid Law to apply. Finally, Chapter 5 investigates the structure of the solutions to the two-well problem. Restricting ourselves to the subset $K=\{H\}\cup \mathrm{SO}(2)V \subset\mathrm{SO}(2)U\cup\mathrm{SO}(2)V$ and assuming the two wells to be compatible, we let $T_1$ and $T_2$ denote the two (not necessarily distinct) twins of $H$ on $\mathrm{SO}(2)V$ and ask the following question: if $\nu_x$ is a non-trivial gradient Young measure almost everywhere supported on $K$, does its support necessarily contain a pair of rank-one connected matrices on a set of positive measure? Although we do not provide a solution for the general case, we show that this is true whenever (a) $\nu_x\equiv \nu$ is homogeneous and $\mathrm{supp}\nu\cap \mathrm{SO}(2)V$ is connected, (b) $\nu_x\equiv \nu$ is homogeneous and $T_1=T_2$ i.e., when the two wells are trivially rank-one connected) or (c) $\mathrm{supp}\nu_x \subset F$ a.e., for some finite set $F$. We also establish a more general case provided a strong `rigidity' conjecture holds.
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Santos, Simão Pedro Silva. "Calculus of variations of Herglotz type." Doctoral thesis, Universidade de Aveiro, 2017. http://hdl.handle.net/10773/22503.
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Doutoramento em MatemáticaWe consider several problems based on Herglotz’s generalized variational problem. We dedicate two chapters to extensions on Herglotz’s generalized variational problem to higher-order and first-order problems with time delay, using a variational approach. In the last four chapters, we rewrite Herglotz's type problems in the optimal control form and use an optimal control approach. We prove generalized higher- order Euler-Lagrange equations, first without and then with time delay; higher-order natural boundary conditions; Noether's first theorem for the first-order problem of Herglotz with time delay; Noether's first theorem for higher-order problems of Herglotz without and with time delay; and existence of Noether currents as a version of Noether's second theorem of optimal control.
Consideramos vários problemas com base no problema variacional generalizado de Herglotz. Dois capítulos são dedicados à extensão do problema variacional generalizado de Herglotz para ordem superior e para problemas de primeira ordem com atraso no tempo, utilizando uma abordagem variacional. Nos últimos quatro capítulos, reescrevemos os problemas de Herglotz na forma do controlo ótimo e usamos essa abordagem. Demonstramos equações generalizadas de Euler-Lagrange de ordem superior, inicialmente sem e depois com atraso no tempo; condições de fronteira de ordem superior; o primeiro teorema de Noether para o problema de Herglotz de primeira ordem com atraso no tempo; o primeiro teorema de Noether para problemas de ordem superior de Herglotz sem e com atraso no tempo; e a existência de leis de conservação de Noether numa versão do segundo teorema de Noether do controlo ótimo.
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Hartmann, Kevin. "Calcul variationnel sur l'espace de Wiener." Electronic Thesis or Diss., Paris, ENST, 2016. http://www.theses.fr/2016ENST0049.
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Ce travail vise à étendre la représentation variationnelle classique du logarithme de l’espérance de e−f par rapport à la mesure de Wiener à des mesures plus générales. Nous donnons d’abord une condition suffisante de différentiabilité forte sur l’espace de Cameron-Martin. Dans un second temps nous étendons la formulation variationnelle à la mesure image d’une diffusion, puis nous utilisons cet exemple pour généraliser la représentation à un large ensemble de mesure. Nous diminuons aussi les hypothèses d’intégrabilité sur f et prouvons de nouveaux résultats sur l’inversibilité stochastique et l’existence de solutions fortes pour certaines équations différentielles stochastiques. Finalement, nous étendons encore une fois la représentationThis work aims at extending the classical variational formulation of the logarithm of the expectation of e −f with respect to the Wiener measure to more general measures. First we give a sufficient criteria for functions to be strongly differentiable over the Cameron-Martin space. Then we extend the variational formulation to the case of the image measure of a diffusion, and we use this example to generalize the variational formulation to a wide set of measures, while reducing the integrability hypothesis over f and obtaining new results concerning stochastic invertibility and existence of strong solutions of stochastic differential equations. Finally, we extend once more this formulation by considering conditional expectations with respect to the same set of measures
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Hopper, Christopher Peter. "On the regularity of holonomically constrained minimisers in the calculus of variations." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:d8bde7a2-7dae-44d2-919d-48b9f2543789.
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This thesis concerns the regularity of holonomic minimisers of variational integrals in the context of direct methods in the calculus of variations. Specifically, we consider Sobolev mappings from a bounded domain into a connected compact Riemannian manifold without boundary, to which such mappings are said to be holonomically constrained. For a general class of strictly quasiconvex integral functionals, we give a direct proof of local C1,α-Hölder continuity, for some 0 < α < 1, of holonomic minimisers off a relatively closed 'singular set' of Lebesgue measure zero. Crucially, the proof constructs comparison maps using the universal covering of the target manifold, the lifting of Sobolev mappings to the covering space and the connectedness of the covering space. A certain tangential A-harmonic approximation lemma obtained directly using a Lipschitz approximation argument is also given. In the context of holonomic minimisers of regular variational integrals, we also provide bounds on the Hausdorff dimension of the singular set by generalising a variational difference quotient method to the holonomically constrained case with critical growth. The results are analogous to energy-minimising harmonic maps into compact manifolds, however in this case the proof does not use a monotonicity formula. We discuss several applications to variational problems in condensed matter physics, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals. In these problems, the class of mappings are constrained to an orbit of 'broken symmetries' or 'manifold of internal states', which correspond to a sub-group of residual symmetries.
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Cahill, Nathan D. "Constructing and solving variational image registration problems." Thesis, University of Oxford, 2009. http://ora.ox.ac.uk/objects/uuid:ed43a6f4-216f-45b5-88c5-2baaba1e684a.
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Nonrigid image registration has received much attention in the medical imaging and computer vision research communities, because it enables a wide variety of applications. Feature tracking, segmentation, classification, temporal image differencing, tumour growth estimation, and pharmacokinetic modeling are examples of the many tasks that are enhanced by the use of aligned imagery. Over the years, the medical imaging and computer vision communties have developed and refined image registration techniques in parallel, often based on similar assumptions or underlying paradigms. This thesis focuses on variational registration, which comprises a subset of nonrigid image registration. It is divided into chapters that are based on fundamental aspects of the variational registration problem: image dissimilarity measures, changing overlap regions, regularizers, and computational solution strategies. Key contributions include the development of local versions of standard dissimilarity measures, the handling of changing overlap regions in a manner that is insensitive to the amount of non-interesting background information, the combination of two standard taxonomies of regularizers, and the generalization of solution techniques based on Fourier methods and the Demons algorithm for use with many regularizers. To illustrate and validate the various contributions, two sets of example imagery are used: 3D CT, MR, and PET images of the brain as well as 3D CT images of lung cancer patients.
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Pérez, Aros Pedro Antonio. "Subdifferential calculus in the framework of Epi-pointed variational analysis, integral functions, and applications." Tesis, Universidad de Chile, 2018. http://repositorio.uchile.cl/handle/2250/150762.
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Tesis para optar al grado de Doctor en Ciencias de la Ingeniería, Mención Modelación MatemáticaLa investigación de esta tesis es presentada en seis capítulos, desde el Capítulo 2 al Capítulo 7. El capítulo 2 proporciona una demostración directa de una caracterización reciente de convexidad dada en el marco de los espacios de Banach en [J. Saint Raymond, J. Convexo no lineal Anal., 14 (2013), pp. 253-262]. Estos resultados también extienden esta caracterización a espacios localmente convexos bajo condiciones más débiles y se basa en la definición de una función epi-puntada. El Capítulo 3 proporciona una extensión del Teorema Br{\o}ndsted-Rockafellar, y algunas de sus importantes consecuencias, a las funciones convexas semicontinuas inferiores definidas en espacios localmente convexos. Este resulado es demostrado usando un nuevo enfoque basado en un principio variacional simple, que también permite recuperar los resultados clásicos de una manera natural. El Capítulo 4 continúa el estudio de la epi-puntadas no convexas, bajo una definición general de subdiferencial. Este trabajo proporciona una generalización del teorema del valor medio de Zagrodny. Posteriormente este resultado es aplicado a los problemas relacionados con la integración de subdiferenciales y caracterización de la convexidad en términos de la monotonicidad del subdiferencial. El Capítulo 5 proporciona una fórmula general para $\epsilon$-subdiferencial de una función integral convexa en términos de $\epsilon$-subdiferenciales de la funcion integrante. Bajo condiciones de calificación, esta fórmula recupera los resultados clásicos en la literatura. Además, este trabajo investiga caracterizaciones del subdiferencial en términos de selecciones medibles que convergen al punto de interés. El Capítulo 6 proporciona fórmulas secuenciales para subdiferenciales bornológicos de un funcional integral no convexo. También son presentadas fórmulas exactas para el subiferencial Limiting/Mordukhovich, el subdiferencial Geometrico de Ioffe y el subdiferencial de Clarke-Rockafellar. El Capítulo 7 proporciona fórmulas para el subdiferencial de funciones de probabilidad bajo distribuciones Gaussianas. En este trabajo la variables de decisión esta tomada en un espacio infinito dimensional. Estas fórmulas se basan en la descomposición esférico-radial de vectores aleatorios Gaussianos.
CONICYT-PCHA/doctorado Nacional / 2014-21140621 y CMM - Conicyt PIA AFB170001
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Chen, Pei-Tai. "Axisymmetric vibration, acoustic radiation, and the influence of eigenvalue veering phenomena in prolate spheroidal shells using variational principles." Diss., Georgia Institute of Technology, 1992. http://hdl.handle.net/1853/19407.
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Tang, Wenbo. "What can variational calculus tell us about ocean turbulence : rigorous bounds on mixing and dissipation in geophysical flows /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2005. http://wwwlib.umi.com/cr/ucsd/fullcit?p3189211.
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Couprie, Camille. "Graph-based variational optimization and applications in computer vision." Phd thesis, Université Paris-Est, 2011. http://tel.archives-ouvertes.fr/tel-00666878.
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Many computer vision applications such as image filtering, segmentation and stereovision can be formulated as optimization problems. Recently discrete, convex, globally optimal methods have received a lot of attention. Many graph-based methods suffer from metrication artefacts, segmented contours are blocky in areas where contour information is lacking. In the first part of this work, we develop a discrete yet isotropic energy minimization formulation for the continuous maximum flow problem that prevents metrication errors. This new convex formulation leads us to a provably globally optimal solution. The employed interior point method can optimize the problem faster than the existing continuous methods. The energy formulation is then adapted and extended to multi-label problems, and shows improvements over existing methods. Fast parallel proximal optimization tools have been tested and adapted for the optimization of this problem. In the second part of this work, we introduce a framework that generalizes several state-of-the-art graph-based segmentation algorithms, namely graph cuts, random walker, shortest paths, and watershed. This generalization allowed us to exhibit a new case, for which we developed a globally optimal optimization method, named "Power watershed''. Our proposed power watershed algorithm computes a unique global solution to multi labeling problems, and is very fast. We further generalize and extend the framework to applications beyond image segmentation, for example image filtering optimizing an L0 norm energy, stereovision and fast and smooth surface reconstruction from a noisy cloud of 3D points
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Bourdin, Loïc. "Contributions au calcul des variations et au principe du maximum de Pontryagin en calculs time scale et fractionnaire." Thesis, Pau, 2013. http://www.theses.fr/2013PAUU3009/document.
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Cette thèse est une contribution au calcul des variations et à la théorie du contrôle optimal dans les cadres discret, plus généralement time scale, et fractionnaire. Ces deux domaines ont récemment connu un développement considérable dû pour l’un à son application en informatique et pour l’autre à son essor dans des problèmes physiques de diffusion anormale. Que ce soit dans le cadre time scale ou dans le cadre fractionnaire, nos objectifs sont de : a) développer un calcul des variations et étendre quelques résultats classiques (voir plus bas); b) établir un principe du maximum de Pontryagin (PMP en abrégé) pour des problèmes de contrôle optimal. Dans ce but, nous généralisons plusieurs méthodes variationnelles usuelles, allant du simple calcul des variations au principe variationnel d’Ekeland (couplé avec la technique des variations-aiguilles), en passant par l’étude d’invariances variationnelles par des groupes de transformations. Les démonstrations des PMPs nous amènent également à employer des théorèmes de point fixe et à prendre en considération la technique des multiplicateurs de Lagrange ou encore une méthode basée sur un théorème d’inversion locale conique. Ce manuscrit est donc composé de deux parties : la Partie 1 traite de problèmes variationnels posés sur time scale et la Partie 2 est consacrée à leurs pendants fractionnaires. Dans chacune de ces deux parties, nous suivons l’organisation suivante : 1. détermination de l’équation d’Euler-Lagrange caractérisant les points critiques d’une fonctionnelle Lagrangienne ; 2. énoncé d’un théorème de type Noether assurant l’existence d’une constante de mouvement pour les équations d’Euler-Lagrange admettant une symétrie ; 3. énoncé d’un théorème de type Tonelli assurant l’existence d’un minimiseur pour une fonctionnelle Lagrangienne et donc, par la même occasion, d’une solution pour l’équation d’Euler-Lagrange associée (uniquement en Partie 2) ; 4. énoncé d’un PMP (version forte en Partie 1, version faible en Partie 2) donnant une condition nécessaire pour les trajectoires qui sont solutions de problèmes de contrôle optimal généraux non-linéaires ; 5. détermination d’une condition de type Helmholtz caractérisant les équations provenant d’un calcul des variations (uniquement en Partie 1 et uniquement dans les cas purement continu et purement discret). Des théorèmes de type Cauchy-Lipschitz nécessaires à l’étude de problèmes de contrôle optimal sont démontrés en AnnexeThis dissertation deals with the mathematical fields called calculus of variations and optimal control theory. More precisely, we develop some aspects of these two domains in discrete, more generally time scale, and fractional frameworks. Indeed, these two settings have recently experience a significant development due to its applications in computing for the first one and to its emergence in physical contexts of anomalous diffusion for the second one. In both frameworks, our goals are: a) to develop a calculus of variations and extend some classical results (see below); b) to state a Pontryagin maximum principle (denoted in short PMP) for optimal control problems. Towards these purposes, we generalize several classical variational methods, including the Ekeland’s variational principle (combined with needle-like variations) as well as variational invariances via the action of groups of transformations. Furthermore, the investigations for PMPs lead us to use fixed point theorems and to consider the Lagrange multiplier technique and a method based on a conic implicit function theorem. This manuscript is made up of two parts : Part A deals with variational problems on time scale and Part B is devoted to their fractional analogues. In each of these parts, we follow (with minor differences) the following organization: 1. obtaining of an Euler-Lagrange equation characterizing the critical points of a Lagrangian functional; 2. statement of a Noether-type theorem ensuring the existence of a constant of motion for Euler-Lagrange equations admitting a symmetry;3. statement of a Tonelli-type theorem ensuring the existence of a minimizer for a Lagrangian functional and, consequently, of a solution for the corresponding Euler-Lagrange equation (only in Part B); 4. statement of a PMP (strong version in Part A and weak version in Part B) giving a necessary condition for the solutions of general nonlinear optimal control problems; 5. obtaining of a Helmholtz condition characterizing the equations deriving from a calculus of variations (only in Part A and only in the purely continuous and purely discrete cases). Some Picard-Lindelöf type theorems necessary for the analysis of optimal control problems are obtained in Appendices
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Sousa, Júnior José Ribamar Alves de. "O cálculo variacional e o problema da braquistócrona /." Rio Claro : [s.n.], 2010. http://hdl.handle.net/11449/94359.
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Orientador: Suzinei Aparecida Siqueira MarconatoBanca: Renata Zotin Gomes de Oliveira
Banca: Sueli Mieko Tanaki Aki
Resumo: Neste trabalho estudamos o problema da Braquistócrona de duas formas distintas: através da teoria do Cálculo Variacional para problemas com fronteiras xas e também através das considerações feitas por Johann Bernoulli, utilizando conceitos de Óptica e Geometria. Apresentamos também uma simulação computacional dos resultados obtidos
Abstract: In this work we study the Brachistochrone Problem of two di erent ways; by theory of Variational Calculus for problems with xed boundary and by considerations of Johann Bernoulli, with concepts of Optics and Geometry. A computational simulation of the obtained results, is presented too
Mestre
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Tolaba, Angel Gustavo. "Otimização de bioprocessos baseada em modelos matemáticos e cálculo variacional /." Araraquara, 2019. http://hdl.handle.net/11449/190747.
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Orientador: Samuel Conceição de OliveiraResumo: A determinação da estratégia de controle adequada para bioprocessos batelada e batelada alimentada é uma questão prática importante devido ao alto valor agregado de alguns bioprodutos. Desde que é altamente desejável otimizar a produção de bioprodutos, vários métodos têm sido propostos para esse objetivo. Uma vez dispondo de um modelo matemático adequado para o bioprocesso, o problema de otimização pode ser formulado no âmbito do princípio do máximo de Pontryagin e da teoria de controle ótimo para determinar a melhor trajetória de controle para certas variáveis manipuladas, como temperatura, pH e taxa de alimentação do substrato. Neste estudo, duas aplicações dessas técnicas baseadas em modelos matemáticos para otimizar e controlar bioprocessos de produção de antibióticos são revisadas e novos aspectos são enfatizados. Os casos analisados incluem a otimização da taxa de alimentação de substrato em um reator batelada alimentada e da temperatura em um reator batelada durante fermentações penicilínicas. Os principais resultados obtidos neste estudo foram: (i) a constatação de que métodos numéricos simples (Runge-Kutta, Newton-Raphson) podem ser aplicados para resolver satisfatoriamente os problemas de valor no contorno propostos; (ii) a demonstração de que a operação não isotérmica é mais produtiva em antibiótico do que a operação sob temperatura constante; (iii) a necessidade de adoção de um modelo matemático apropriado para o bioprocesso visando à resolução do problema de cont... (Resumo completo, clicar acesso eletrônico abaixo)
Mestre
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Aubert, Gilles. "Contribution aux problèmes du calcul des variations et application à l'élasticité non linéaire." Paris 6, 1986. http://www.theses.fr/1986PA066011.
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Théorèmes d'existence pour un type de problèmes du calcul des variations, et en optimisation non convexe. Minimisation d'une fonctionnelle non convexe, non différentiable en dimension 1. Remarques sur un problème d'élasticité non linéaire. Faible fermeture de certains ensembles de contraintes en élasticité non linéaire plane. Conditions nécessaires de faible fermeture et de 1-rang convexité en dimension 3. Théorèmes de caractérisation de la polyconvexité et de la 1-rang convexité en dimensions 2 et 3.
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Bouchitte, Guy. "Calcul des variations en cadre non reflexif : representation et relaxation de fonctionnelles integrales sur un espace de mesures, applications en plasticite et homogeneisation." Perpignan, 1987. http://www.theses.fr/1987PERP0033.
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Etude variationnelle de fonctionnelles convexes du type somomega f(x,u(x),du)dx lorsque la croissance de f necessite du travail dans un banach non reflexif. On abordera en particulier les questions relatives a la semi-continuite inferieure, la relaxation, la convergence variationnelle et l'approximation, l'equation d'euler. . . Ayant en vue certaines applications dans le dommaine de la mecanique (plasticite, milieux fissures), on s'interessera principalement au cas ou l'integrande f presente une croissance lineaire par rapport au gradient. Suivant une demarche desormais classique, cela nous amenera a etendre la definition de la fonctionnelle a des fonctions dont le gradient est une mesure bornee (espaces de type bv(omega ) ou bd(omega ). Dans cette optique, une contribution substanstielle est apportee (chapitres ii et iii notamment) au probleme de la relaxation sur un espace de mesures d'une fonctionnelle integrale dependant du parametre. Les 3 derniers chapitres ont une connotation beaucoup plus "appliquee" et sont consacres a quelques problemes d'homogeneisation et d'analyse limite issus des equations de la plasticite (chapitre v) ou de l'etude des phenomenes de diffraction en electromagnetisme (chapitre vi et vii)
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Rocha, Kelvin Raymond. "A variational approach for viewpoint-based visibility maximization." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/24816.
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Thesis (Ph.D.)--Electrical and Computer Engineering, Georgia Institute of Technology, 2008.Committee Chair: Allen R. Tannenbaum; Committee Member: Anthony J. Yezzi; Committee Member: Gregory Turk; Committee Member: Joel R. Jackson; Committee Member: Patricio A. Vela
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Henao, Manrique Duvan Alberto. "Variational modelling of cavitation and fracture in nonlinear elasticity." Thesis, University of Oxford, 2009. http://ora.ox.ac.uk/objects/uuid:82b6fdc7-4b66-4853-86ad-7fa4488a32ea.
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Motivated by experiments on titanium alloys of Petrinic et al. (2006), which show the formation of cracks through the growth and coalescence of voids in ductile fracture, we consider the problem of formulating a variational model in nonlinear elasticity compatible both with cavitation and the appearance of discontinuities across two-dimensional surfaces. As in the model for cavitation of Müller and Spector (1995) we address this problem, which is connected to the sequential weak continuity of the determinant of the deformation gradient in spaces of functions having low regularity, by means of adding an appropriate surface energy term to the elastic energy. Based upon considerations of invertibility, we derive an expression for the surface energy that admits a physical and a geometrical interpretation, and that allows for the formulation of a model with better analytical properties. We obtain, in particular, important regularity results for the inverses of deformations, as well as the weak continuity of the determinants and the existence of minimizers. We show, further, that the creation of surface can be modeled by carefully analyzing the jump set of the inverses, and we point out some connections between the analysis of cavitation and fracture, the theory of SBV functions, and the theory of Cartesian currents of Giaquinta, Modica, and Soucek. In addition to the above, we extend previous work of Sivaloganathan, Spector and Tilakraj (2006) on the approximation of minimizers for the problem of cavitation with a constraint in the number of flaw points, and present some numerical results for this problem.
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Heinz, Sebastian. "Preservation of quasiconvexity and quasimonotonicity in polynomial approximation of variational problems." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2008. http://dx.doi.org/10.18452/15808.
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Die vorliegende Arbeit beschäftigt sich mit drei Klassen ausgewählter nichtlinearer Probleme, die Forschungsgegenstand der angewandten Mathematik sind. Diese Probleme behandeln die Minimierung von Integralen in der Variationsrechnung (Kapitel 3), das Lösen partieller Differentialgleichungen (Kapitel 4) und das Lösen nichtlinearer Optimierungsaufgaben (Kapitel 5). Mit deren Hilfe lassen sich unterschiedlichste Phänomene der Natur- und Ingenieurwissenschaften sowie der Ökonomie mathematisch modellieren. Als konkretes Beispiel werden mathematische Modelle der Theorie elastischer Festkörper betrachtet. Das Ziel der vorliegenden Arbeit besteht darin, ein gegebenes nichtlineares Problem durch polynomiale Probleme zu approximieren. Um dieses Ziel zu erreichen, beschäftigt sich ein großer Teil der vorliegenden Arbeit mit der polynomialen Approximation von nichtlinearen Funktionen. Den Ausgangspunkt dafür bildet der Weierstraßsche Approximationssatz. Auf der Basis dieses bekannten Satzes und eigener Sätze wird als Hauptresultat der vorliegenden Arbeit gezeigt, dass im Übergang von einer gegebenen Funktion zum approximierenden Polynom wesentliche Eigenschaften der gegebenen Funktion erhalten werden können. Die wichtigsten Eigenschaften, für die dies bisher nicht bekannt war, sind: Quasikonvexität im Sinne der Variationsrechnung, Quasimonotonie im Zusammenhang mit partiellen Differentialgleichungen sowie Quasikonvexität im Sinne der nichtlinearen Optimierung (Theoreme 3.16, 4.10 und 5.5). Schließlich wird gezeigt, dass die zu den untersuchten Klassen gehörenden nichtlinearen Probleme durch polynomiale Probleme approximiert werden können (Theoreme 3.26, 4.16 und 5.8). Die dieser Approximation zugrunde liegende Konvergenz garantiert sowohl eine Approximation im Parameterraum als auch eine Approximation im Lösungsraum. Für letztere werden die Konzepte der Gamma-Konvergenz (Epi-Konvergenz) und der G-Konvergenz verwendet.In this thesis, we are concerned with three classes of non-linear problems that appear naturally in various fields of science, engineering and economics. In order to cover many different applications, we study problems in the calculus of variation (Chapter 3), partial differential equations (Chapter 4) as well as non-linear programming problems (Chapter 5). As an example of possible applications, we consider models of non-linear elasticity theory. The aim of this thesis is to approximate a given non-linear problem by polynomial problems. In order to achieve the desired polynomial approximation of problems, a large part of this thesis is dedicated to the polynomial approximation of non-linear functions. The Weierstraß approximation theorem forms the starting point. Based on this well-known theorem, we prove theorems that eventually lead to our main result: A given non-linear function can be approximated by polynomials so that essential properties of the function are preserved. This result is new for three properties that are important in the context of the considered non-linear problems. These properties are: quasiconvexity in the sense of the calculus of variation, quasimonotonicity in the context of partial differential equations and quasiconvexity in the sense of non-linear programming (Theorems 3.16, 4.10 and 5.5). Finally, we show the following: Every non-linear problem that belongs to one of the three considered classes of problems can be approximated by polynomial problems (Theorems 3.26, 4.16 and 5.8). The underlying convergence guarantees both the approximation in the parameter space and the approximation in the solution space. In this context, we use the concepts of Gamma-convergence (epi-convergence) and of G-convergence.
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Silva, Edcarlos Domingos da. "Multiplicidade de soluções para sistemas gradientes semilineares ressonantes." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306981.
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Orientadores: Djairo Guedes de Figueiredo, Francisco Odair Vieira de PaivaTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Nesta tese lidamos com três classes de sistemas gradientes ressonantes. A primeira classe é um sistema com ressonância do tipo Landesman-Lazer. A segunda classe é um sistema fortemente ressonante enquanto a terceira classe é um sistema com ressonância no infinito e na origem. Analisamos as questões de existência e multiplicidade de soluções em cada uma das classes mencionadas. Para obtermos os nossos principais resultados aplicamos alguns métodos variacionais, tais como, teoremas Min-Max e minimização. Além disso, usamos a Teoria de Morse para distinguirmos soluções dados por métodos variacionais distintos.
Abstract: In this thesis we deal with three classes of gradient elliptic systems with resonance. The first class is a resonant system of Landesman-Lazer type. The second class is a system of strong resonance type while the third class is a system with resonance at infinity and at origin. We are concerned about the questions of existence and multiplicity of solutions in each of the classes mentioned. To obtain our main results we apply variational methods, such as, Min-max theorems and minimization. Moreover, we use Morse Theory to distinguish the solutions given by different variational methods.
Doutorado
Doutor em Matemática
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Sousa, Júnior José Ribamar Alves de [UNESP]. "O cálculo variacional e o problema da braquistócrona." Universidade Estadual Paulista (UNESP), 2010. http://hdl.handle.net/11449/94359.
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sousajunior_jra_me_rcla.pdf: 1174734 bytes, checksum: cbdf2669884098c54b72817cfc625edd (MD5)Neste trabalho estudamos o problema da Braquistócrona de duas formas distintas: através da teoria do Cálculo Variacional para problemas com fronteiras xas e também através das considerações feitas por Johann Bernoulli, utilizando conceitos de Óptica e Geometria. Apresentamos também uma simulação computacional dos resultados obtidos
In this work we study the Brachistochrone Problem of two di erent ways; by theory of Variational Calculus for problems with xed boundary and by considerations of Johann Bernoulli, with concepts of Optics and Geometry. A computational simulation of the obtained results, is presented too
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Muehlemann, Anton. "Variational models in martensitic phase transformations with applications to steels." Thesis, University of Oxford, 2016. https://ora.ox.ac.uk/objects/uuid:bb7f4ff4-0911-4dad-bb23-ada904839d73.
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This thesis concerns the mathematical modelling of phase transformations with a special emphasis on martensitic phase transformations and their application to the modelling of steels. In Chapter 1, we develop a framework that determines the optimal transformation strain between any two Bravais lattices and use it to give a rigorous proof of a conjecture by E.C. Bain in 1924 on the optimality of the so-called Bain strain. In Chapter 2, we review the Ball-James model and related concepts. We present some simplification of existing results. In Chapter 3, we pose a conjecture for the explicit form of the quasiconvex hull of the three tetragonal wells, known as the three-well problem. We present a new approach to finding inner and outer bounds. In Chapter 4, we focus on highly compatible, so called self-accommodating, martensitic structures and present new results on their fine properties such as estimates on their minimum complexity and bounds on the relative proportion of each martensitic variant in them. In Chapter 5, we investigate the contrary situation when self-accommodating microstructures do not exist. We determine, whether in this situation, it is still energetically favourable to nucleate martensite within austenite. By constructing different types of inclusions, we find that the optimal shape of an inclusion is flat and thin which is in agreement with experimental observation. In Chapter 6, we introduce a mechanism that identifies transformation strains with orientation relationships. This mechanism allows us to develop a simpler, strain-based approach to phase transformation models in steels. One novelty of this approach is the derivation of an explicit dependence of the orientation relationships on the ratio of tetragonality of the product phase. In Chapter 7, we establish a correspondence between common phenomenological models for steels and the Ball-James model. This correspondence is then used to develop a new theory for the (5 5 7) lath transformation in low-carbon steels. Compared to existing theories, this new approach requires a significantly smaller number of input parameters. Furthermore, it predicts a microstructure morphology which differs from what is conventionally believed.
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Gairing, Jan Martin. "Variational and Ergodic Methods for Stochastic Differential Equations Driven by Lévy Processes." Doctoral thesis, Humboldt-Universität zu Berlin, 2018. http://dx.doi.org/10.18452/18984.
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Diese Dissertation untersucht Aspekte des Zusammenspiels von ergodischem Langzeitver-
halten und der Glättungseigenschaft dynamischer Systeme, die von stochastischen Differen-
tialgleichungen (SDEs) mit Sprüngen erzeugt sind. Im Speziellen werden SDEs getrieben
von Lévy-Prozessen und der Marcusschen kanonischen Gleichung untersucht. Ein vari-
ationeller Ansatz für den Malliavin-Kalkül liefert eine partielle Integration, sodass eine
Variation im Raum in eine Variation im Wahrscheinlichkeitsmaß überführt werden kann.
Damit lässt sich die starke Feller-Eigenschaft und die Existenz glatter Dichten der zuge-
hörigen Markov-Halbgruppe aus einer nichtstandard Elliptizitätsbedingung an eine Kom-
bination aus Gaußscher und Sprung-Kovarianz ableiten. Resultate für Sprungdiffusionen
auf Untermannigfaltigkeiten werden aus dem umgebenden Euklidischen Raum hergeleitet.
Diese Resultate werden dann auf zufällige dynamische Systeme angewandt, die von lin-
earen stochastischen Differentialgleichungen erzeugt sind. Ruelles Integrierbarkeitsbedin-
gung entspricht einer Integrierbarkeitsbedingung an das Lévy-Maß und gewährleistet die
Gültigkeit von Oseledets multiplikativem Ergodentheorem. Damit folgt die Existenz eines
Lyapunov-Spektrums. Schließlich wird der top Lyapunov-Exponent über eine Formel der
Art von Furstenberg–Khasminsikii als ein ergodisches Mittel der infinitesimalen Wachs-
tumsrate über die Einheitssphäre dargestellt.The present thesis investigates certain aspects of the interplay between the ergodic long time behavior and the smoothing property of dynamical systems generated by stochastic differential equations (SDEs) with jumps, in particular SDEs driven by Lévy processes and the Marcus’ canonical equation. A variational approach to the Malliavin calculus generates an integration-by-parts formula that allows to transfer spatial variation to variation in the probability measure. The strong Feller property of the associated Markov semigroup and the existence of smooth transition densities are deduced from a non-standard ellipticity condition on a combination of the Gaussian and a jump covariance. Similar results on submanifolds are inferred from the ambient Euclidean space. These results are then applied to random dynamical systems generated by linear stochas- tic differential equations. Ruelle’s integrability condition translates into an integrability condition for the Lévy measure and ensures the validity of the multiplicative ergodic theo- rem (MET) of Oseledets. Hence the exponential growth rate is governed by the Lyapunov spectrum. Finally the top Lyapunov exponent is represented by a formula of Furstenberg– Khasminskii–type as an ergodic average of the infinitesimal growth rate over the unit sphere.
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Крючкова, А. С., Сергій Вікторович Соколов, Сергей Викторович Соколов, and Serhii Viktorovych Sokolov. "Оптимізація системи керування ходового візка прольотних кранів." Thesis, Сумський державний університет, 2017. http://essuir.sumdu.edu.ua/handle/123456789/65127.
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Saddek, Lhassane. "Solutions d'un problème aux limites non linéaire discontinu à l'infini." Paris 9, 1988. https://portail.bu.dauphine.fr/fileviewer/index.php?doc=1988PA090010.
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On recherche par une méthode variationnelle les solutions t-périodiques d'un système dynamique comportant un potentiel convexe sous quadratique ou super quadratique. On démontre des théorèmes d'existence d'une solution non triviale
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Kwa, Kiam Heong. "Laser-Driven Charged Particles as a Dynamical System." The Ohio State University, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=osu1250103994.
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Foare, Marion. "Analyse d'images par des méthodes variationnelles et géométriques." Thesis, Université Grenoble Alpes (ComUE), 2017. http://www.theses.fr/2017GREAM043/document.
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Dans cette thèse, nous nous intéressons à la fois aux aspects théoriques et à la résolution numérique du problème de Mumford-Shah avec anisotropie pour la restauration et la segmentation d'image. Cette fonctionnelle possède en effet la particularité de reconstruire une image dégradée tout en extrayant l'ensemble des contours des régions d'intérêt au sein de l'image. Numériquement, on utilise l'approximation d'Ambrosio-Tortorelli pour approcher un minimiseur de la fonctionnelle de Mumford-Shah. Elle Gamma-converge vers cette dernière et permet elle aussi d'extraire les contours. Les implémentations avec des schémas aux différences finies ou aux éléments finis sont toutefois peu adaptées pour l'optimisation de la fonctionnelle d'Ambrosio-Tortorelli. On présente ainsi deux nouvelles formulations discrètes de la fonctionnelle d'Ambrosio-Tortorelli à l'aide des opérateurs et du formalisme du calcul discret. Ces approches sont utilisées pour la restauration d'images ainsi que pour le lissage du champ de normales et la détection de saillances des surfaces digitales de l'espace. Nous étudions aussi un second problème d'optimisation de forme similaire avec conditions aux bords de Robin. Nous démontrons dans un premier temps l'existence et la régularité partielle des solutions, et dans un second temps deux approximations par Gamma-convergence pour la résolution numérique du problème. L'analyse numérique montre une nouvelle fois les difficultés rencontrées pour la minimisation d'approximations par Gamma-convergenceIn this work, we study both theoretical and numerical aspects of an anisotropic Mumford-Shah problem for image restoration and segmentation. The Mumford-Shah functional allows to both reconstruct a degraded image and extract the contours of the region of interest. Numerically, we use the Amborsio-Tortorelli approximation to approach a minimizer of the Mumford-Shah functional. It Gamma-converges to the Mumford-Shah functional and allows also to extract the contours. However, the minimization of the Ambrosio-Tortorelli functional using standard discretization schemes such as finite differences or finite elements leads to difficulties. We thus present two new discrete formulations of the Ambrosio-Tortorelli functional using the framework of discrete calculus. We use these approaches for image restoration and for the reconstruction of normal vector field and feature extraction on digital data. We finally study another similar shape optimization problem with Robin boundary conditions. We first prove existence and partial regularity of solutions and then construct and demonstrate the Gamma-convergence of two approximations. Numerical analysis shows once again the difficulties dealing with Gamma-convergent approximations
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Tavares, Dina dos Santos. "Fractional calculus of variations." Doctoral thesis, Universidade de Aveiro, 2017. http://hdl.handle.net/10773/22184.
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Doutoramento em Matemática e AplicaçõesO cálculo de ordem não inteira, mais conhecido por cálculo fracionário, consiste numa generalização do cálculo integral e diferencial de ordem inteira. Esta tese é dedicada ao estudo de operadores fracionários com ordem variável e problemas variacionais específicos, envolvendo também operadores de ordem variável. Apresentamos uma nova ferramenta numérica para resolver equações diferenciais envolvendo derivadas de Caputo de ordem fracionária variável. Consideram- -se três operadores fracionários do tipo Caputo, e para cada um deles é apresentada uma aproximação dependendo apenas de derivadas de ordem inteira. São ainda apresentadas estimativas para os erros de cada aproximação. Além disso, consideramos alguns problemas variacionais, sujeitos ou não a uma ou mais restrições, onde o funcional depende da derivada combinada de Caputo de ordem fracionária variável. Em particular, obtemos condições de otimalidade necessárias de Euler–Lagrange e sendo o ponto terminal do integral, bem como o seu correspondente valor, livres, foram ainda obtidas as condições de transversalidade para o problema fracionário.
The calculus of non–integer order, usual known as fractional calculus, consists in a generalization of integral and differential integer-order calculus. This thesis is devoted to the study of fractional operators with variable order and specific variational problems involving also variable order operators. We present a new numerical tool to solve differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them, an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided. Furthermore, we consider variational problems subject or not to one or more constraints, where the functional depends on a combined Caputo derivative of variable fractional order. In particular, we establish necessary optimality conditions of Euler–Lagrange. As the terminal point in the cost integral, as well the terminal state, are free, thus transversality conditions are obtained.
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Ferreira, Rui Alexandre Cardoso. "Calculus of variations on time scales and discrete fractional calculus." Doctoral thesis, Universidade de Aveiro, 2010. http://hdl.handle.net/10773/2921.
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Doutoramento em MatemáticaEstudamos problemas do cálculo das variações e controlo óptimo no contexto das escalas temporais. Especificamente, obtemos condições necessárias de optimalidade do tipo de Euler–Lagrange tanto para lagrangianos dependendo de derivadas delta de ordem superior como para problemas isoperimétricos. Desenvolvemos também alguns métodos directos que permitem resolver determinadas classes de problemas variacionais através de desigualdades em escalas temporais. No último capítulo apresentamos operadores de diferença fraccionários e propomos um novo cálculo das variações fraccionário em tempo discreto. Obtemos as correspondentes condições necessárias de Euler– Lagrange e Legendre, ilustrando depois a teoria com alguns exemplos.
We study problems of the calculus of variations and optimal control within the framework of time scales. Specifically, we obtain Euler–Lagrange type equations for both Lagrangians depending on higher order delta derivatives and isoperimetric problems. We also develop some direct methods to solve certain classes of variational problems via dynamic inequalities. In the last chapter we introduce fractional difference operators and propose a new discrete-time fractional calculus of variations. Corresponding Euler–Lagrange and Legendre necessary optimality conditions are derived and some illustrative examples provided.
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Bedford, Stephen James. "Calculus of variations and its application to liquid crystals." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:a2004679-5644-485c-bd35-544448f53f6a.
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The thesis concerns the mathematical study of the calculus of variations and its application to liquid crystals. In the first chapter we examine vectorial problems in the calculus of variations with an additional pointwise constraint so that any admissible function n ε W1,1(ΩM), and M is a manifold of suitable regularity. We formulate necessary and sufficient conditions for any given state n to be a strong or weak local minimiser of I. This is achieved using a nearest point projection mapping in order to use the more classical results which apply in the absence of a constraint. In the subsequent chapters we study various static continuum theories of liquid crystals. More specifically we look to explain a particular cholesteric fingerprint pattern observed by HP Labs. We begin in Chapter 2 by focusing on a specific cholesteric liquid crystal problem using the theory originally derived by Oseen and Frank. We find the global minimisers for general elastic constants amongst admissible functions which only depend on a single variable. Using the one-constant approximation for the Oseen-Frank free energy, we then show that these states are global minimisers of the three-dimensional problem if the pitch of the cholesteric liquid crystal is sufficiently long. Chapter 3 concerns the application of the results from the first chapter to the situations investigated in the second. The local stability of the one-dimensional states are quantified, analytically and numerically, and in doing so we unearth potential shortcomings of the classical Oseen-Frank theory. In Chapter 4, we ascertain some equivalence results between the continuum theories of Oseen and Frank, Ericksen, and Landau and de Gennes. We do so by proving lifting results, building on the work of Ball and Zarnescu, which relate the regularity of line and vector fields. The results prove to be interesting as they show that for a director theory to respect the head to tail symmetry of the liquid crystal molecules, the appropriate function space for the director field is S BV2 (Ω,S2,/sup>). We take this idea and in the final chapter we propose a mathematical model of liquid crystals based upon the Oseen-Frank free energy but using special functions of bounded variation. We establish the existence of a minimiser, forms of the Euler-Lagrange equation, and find solutions of the Euler-Lagrange equation in some simple cases. Finally we use our proposed model to re-examine the same problems from Chapter 2. By doing so we extend the analysis we were able to achieve using Sobolev spaces and predict the existence of multi-dimensional minimisers consistent with the known experimental properties of high-chirality cholesteric liquid crystals.
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Zhang, Chengdian. "Calculus of variations with multiple integration." Bonn : [s.n.], 1989. http://catalog.hathitrust.org/api/volumes/oclc/20436929.html.
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Capet, SteÌphane. "Calculus of variations in quantum mechanics." Thesis, University of Warwick, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.444831.
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Gratwick, Richard. "Singular minimizers in the calculus of variations." Thesis, University of Warwick, 2011. http://wrap.warwick.ac.uk/47653/.
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This thesis examines the possible failure of regularity for minimizers of onedimensional variational problems. The direct method of the calculus of variations gives rigorous assurance that minimizers exist, but necessarily admits the possibility that minimizers might not be smooth. Regularity theory seeks to assert some extra smoothness of minimizers. Tonelli's partial regularity theorem states that any absolutely continuous minimizer has a (possibly infinite) classical derivative everywhere, and this derivative is continuous as a function into the extended real line. We examine the limits of this theorem. We find an example of a reasonable problem where partial regularity fails, and examples where partial regularity holds, but the infinite derivatives of minimizers permitted by the theorem occur very often, in precise senses. We construct continuous Lagrangians, strictly convex and superlinear in the third variable, such that the associated variational problems have minimizers nondifferentiable on dense second category sets. Thus mere continuity is an insufficient smoothness assumption for Tonelli's partial regularity theorem. Davie showed that any compact null set can occur as the singular set of a minimizer to a problem given via a smooth Lagrangian with quadratic growth. The proof relies on enforcing the occurrence of the Lavrentiev phenomenon. We give a new proof of the result, but constructing also a Lagrangian with arbitrary superlinear growth, and in which the Lavrentiev phenomenon does not occur in the problem. Universal singular sets record how often a given Lagrangian can have minimizers with infinite derivative. Despite being negligible in terms of both topology and category, they can have dimension two: any compact purely unrectifiable set can lie inside the universal singular set of a Lagrangian with arbitrary superlinearity. We show this also to be true of Fσ purely unrectifiable sets, suggesting a possible characterization of universal singular sets.
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Taheri, Ali. "Local minimizers in the calculus of variations." Thesis, Heriot-Watt University, 1997. http://hdl.handle.net/10399/656.
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Chan, Ka-bo, and 陳家寶. "On Griffiths' formalism of the calculus of variations." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2004. http://hub.hku.hk/bib/B30456630.
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Arora, Raman. "Analysis of Economic Models Through Calculus of Variations." TopSCHOLAR®, 2005. http://digitalcommons.wku.edu/theses/453.
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This thesis is a combination of two science fields: Mathematics and Economics. Mathematics is often used to formulate a clear and concise solution to economic problems. In my observation calculus of variation has often been used in various macroeconomic problems. This mathematical method deals with maximizing or minimizing of various objective functions given a set of constraints. This topic brings out one of the best ways to show the relationship between mathematics and economics. My thesis consists of three parts: The first chapter contains a review of the calculus of variations. Basic definitions and important conditions have been stated. The aim of this chapter was to set the groundwork for understanding calculus of variations so that it can be used in solving various economics models. In the second chapter we study an economic model from which calculus of variations has been used to solve it. The macroeconomic model deals with optimizing the social welfare function. The entire working of the model has been discussed and documented in the thesis report. The third chapter deals with the analysis of the Lucas model which concentrated on how the accumulation of human capital impacts the growth rate of the economy. Lucas assumes that the growth rate of the human capital is linearly related to its level. If we abandon this assumption, will the optimal value of the time devoted to education in the steady state exist? If it exists, will it be same or different? So we introduced a new model in which the only modification we made to the Lucas model was in the equation that describes the process of human accumulation by introducing a nonlinear component. On investigation of this new model we have shown that it is possible that optimal behavior for an individual can be not to educate himself.
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Campos, Cordero Judith. "Regularity and uniqueness in the calculus of variations." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:81e69dac-5ba2-4dc3-85bc-5d9017286f13.
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This thesis is about regularity and uniqueness of minimizers of integral functionals of the form F(u) := ∫Ω F(∇u(x)) dx; where F∈C2(RNn) is a strongly quasiconvex integrand with p-growth, Ω⊆RnRn is an open bounded domain and u∈W1,pg(Ω,RN) for some boundary datum g∈C1,α(‾Ω, RN). The first contribution of this work is a full regularity result, up to the boundary, for global minimizers of F provided that the boundary condition g satisfies that ΙΙ∇gΙΙLP < ε for some ε > 0 depending only on n;N, the parameters given by the strong quasiconvexity and p-growth conditions and, most importantly, on an arbitrary but fixed constant M > 0 for which we require that ΙΙ∇gΙΙO,α < M. Furthermore, when the domain Ω is star-shaped, we extend the regularity result to the case of W1,p-local minimizers. On the other hand, for the case of global minimizers we exploit the compactness provided by the aforementioned regularity result to establish the main contribution of this thesis: we prove that, under essentially the same smallness assumptions over the boundary condition g that we mentioned above, the minimizer of F in W1,pg is unique. This result appears in contrast to the non-uniqueness examples previously given by Spadaro [Spa09], for which the boundary conditions are required to be suitably large.
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Chen, Chuei Yee. "Quasiminimality and coercivity in the calculus of variations." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:d6daadfd-92fb-4fa6-9d5b-fd8403955079.
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Cordesse, Pierre. "Contribution to the study of combustion instabilities in cryotechnic rocket engines : coupling diffuse interface models with kinetic-based moment methods for primary atomization simulations." Thesis, université Paris-Saclay, 2020. http://www.theses.fr/2020UPASC016.
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Gardiens de l’espace, les lanceurs de fusée font l’objet d’une amélioration continue et concurrentielle, grâce à des campagnes de tests expérimentaux et numériques. Les simulations prédictives sont devenues indispensables pour accroître notre compréhension de la physique. Ajustables, elles se prêtent parfaitement à la conception et l’optimisation, en particuliers de la chambre de combustion, pour garantir la sureté et maximiser l’efficacité. L’atomisation primaire est l’un des phénomènes déterminants de la combustion du combustible et de l’oxydant, pilotant à la fois la distribution de gouttes et les potentielles instabilités hautes-fréquences en conditions sous-critiques. Elle couvre un large spectre de topologies d’écoulement diphasique, depuis ceux de type phases séparées jusqu’à la phase dispersée, en passant par une région mixte caractérisée par la complexité de la physique à petites échelles et de la topologie de l’écoulement. Les modèles d’ordre réduit constituent de bons candidats pour réaliser des simulations numériques prédictives et relativement peu coûteuses en ressource de calcul sur des configurations industrielles. Cependant, jusqu’à présent ils ne décrivent correctement que la dynamique des grandes échelles et doivent donc être couplés à des modèles de phase dispersée nécessitant un réglage minutieux de paramètres pour prédire la formation du spray. Afin de décrire à la fois les régions mixte et dispersée, l’amélioration de la hiérarchie de modèles d’ordre réduit repose sur quelques principes clefs au cœur de la thèse ci-présente et fournit des problèmes interdisciplinaires faisant appel tant à l’analyse mathématique et la modélisation physique de ces systèmes d’EDP qu’à leur discrétisation numérique et leur implémentation dans des codes de CFD à des fins industriels. Grâce d’une part à l’extension de la théorie des équations de conservation supplémentaires à des systèmes impliquant des termes non-conservatifs et d’autre part à un formalisme de thermodynamique multi-fluide tenant compte des effets non-idéaux, nous proposons de nouvelles pistes pour définir une entropie de mélange strictement convexe et consistante avec le système d’équation et les lois de pression, dans le but de permettre la symmétrisation entropique des modèles diphasiques, de prouver leur hyperbolicité et d’obtenir des termes sources généraux. De plus, en rompant avec la vision géométrique de l’interface, nous proposons une description multi-échelle de l’interface pour décrire un mélange multi-fluide comportant une dynamique interfaciale complexe. Le Principe de Moindre Action a permis de dériver un modèle bifluide à une vitesse couplant grandes et petites échelles de l’écoulement. Nous avons ensuite développé une stratégie de séparation d’opérateurs basée sur la discrétisation par Volumes Finis, et nous avons implémenté le nouveau modèle dans le logiciel industriel multiphysique de CFD, CEDRE, de l’ONERA afin d’évaluer numériquement ce dernier. Enfin, nous avons construit et analysé les fondations d’une hiérarchie de cas tests accessibles à la DNS tout en étant au plus proche de configurations industrielles, dans le but d’évaluer les résultats de simulations du nouveau modèle ou de tout autre modèle à venirGatekeepers to the open space, launchers are subject to intense and competitive enhancements, through experimental and numerical test campaigns. Predictive numerical simulations have become mandatory to increase our understanding of the physics. Adjustable, they provide early-stage optimization processes, in particular of the combustion chamber, to guaranty safety and maximize efficiency. One of the major physical phenomenon involved in the combustion of the fuel and oxidizer is the jet atomization, which pilotes both the droplet distributions and the potential high-frequency instabilities in subcritical conditions. It encompasses a large sprectrum of two-phase flow topologies, from separated phases to disperse phase, with a mixed region where the small scale physics and topology of the flow are very complex. Reduced-order models are good candidates to perform predictive but low CPU demanding simulations on industrial configurations but have only been able so far to capture large scale dynamics and have to be coupled to disperse phase models through adjustable and weakly reliable parameters in order to predict spray formation. Improving the hierarchy of reduced order models in order to better describe both the mixed region and the disperse region requires a series of building blocks at the heart of the present work and give on to complex problems in the mathematical analysis and physical modelling of these systems of PDE as well as their numerical discretization and implementation in CFD codes for industrial uses. Thanks to the extension of the theory on supplementary conservative equations to system of non-conservation laws and the formalism of the multi-fluid thermodynamics accounting for non-ideal effects, we give some new leads to define a strictly convex mixture entropy consistent with the system of equations and the pressure laws, which would allow to recover the entropic symmetrization of two-phase flow models, prove their hyperbolicity and obtain generalized source terms. Furthermore, we have departed from a geometric approach of the interface and proposed a multi-scale rendering of the interface to describe multi-fluid flow with complex interface dynamics. The Stationary Action Principle has returned a single velocity two-phase flow model coupling large and small scales of the flow. We then have developed a splitting strategy based on a Finite Volume discretization and have implemented the new model in the industrial CFD software CEDRE of ONERA to proceed to a numerical verification. Finally, we have constituted and investigated a first building block of a hierarchy of test-cases designed to be amenable to DNS while close enough to industrial configurations in order to assess the simulation results of the new model but also to any up-coming models
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Chow, Hong-Yu, and 周康宇. "Griffiths' formalism of the calculus of variations and applications toinvariants." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2005. http://hub.hku.hk/bib/B35812503.
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Onofrei, Daniel T. "Homogenization of an elastic-plastic problem." Link to electronic thesis, 2003. http://www.wpi.edu/Pubs/ETD/Available/etd-0430103-121632.
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Seghir, Driss. "Calcul des variations et contrôle optimal dans des espaces de fonctions à variation bornée." Toulouse 3, 1993. http://www.theses.fr/1993TOU30102.
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L'auteur etudie des problemes vectoriels de calcul des variations et de controle optimal non coercifs dans les espaces de sobolev. Leur extension a des espaces de fonctions a variation bornee (bv en abrege) s'obtient par regularisation semi-continue inferieure pour la topologie faible etoile des espaces bv. Des resultats de representation integrale sont donnes pour les fonctionnelles regularisees lorsque les fonctions de recession des integrandes sont partout finies. D'autres resultats de semi-continuite sont obtenus pour des fonctionnelles explicitement definies sur l'espace bv lorsque cette condition sur les fonctions de recession n'est pas satisfaite. L'existence des solutions est alors obtenue par la methode directe du calcul des variations et les courbes-bv solutions des problemes etudies sont caracterisees par les solutions lipschitziennes d'un probleme parametrique auxiliaire. Ces methodes sont ensuite developpees pour des problemes de controle optimal gouvernes par des equations differentielles ordinaires avec donnees mesures. Certains resultats d'existence sont obtenus en l'absence de toute hypothese de convexite sur les integrandes des fonctionnelles etudiees
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Chow, Hong-Yu. "Griffiths' formalism of the calculus of variations and applications to invariants." Click to view the E-thesis via HKUTO, 2005. http://sunzi.lib.hku.hk/hkuto/record/B35812503.
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