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Dissertations / Theses on the topic 'Variational theory'
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Aghassi, Michele Leslie. "Robust optimization, game theory, and variational inequalities." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/33670.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2005.Includes bibliographical references (p. 193-109).
We propose a robust optimization approach to analyzing three distinct classes of problems related to the notion of equilibrium: the nominal variational inequality (VI) problem over a polyhedron, the finite game under payoff uncertainty, and the network design problem under demand uncertainty. In the first part of the thesis, we demonstrate that the nominal VI problem is in fact a special instance of a robust constraint. Using this insight and duality-based proof techniques from robust optimization, we reformulate the VI problem over a polyhedron as a single- level (and many-times continuously differentiable) optimization problem. This reformulation applies even if the associated cost function has an asymmetric Jacobian matrix. We give sufficient conditions for the convexity of this reformulation and thereby identify a class of VIs, of which monotone affine (and possibly asymmetric) VIs are a special case, which may be solved using widely-available and commercial-grade convex optimization software. In the second part of the thesis, we propose a distribution-free model of incomplete- information games, in which the players use a robust optimization approach to contend with payoff uncertainty.
(cont.) Our "robust game" model relaxes the assumptions of Harsanyi's Bayesian game model, and provides an alternative, distribution-free equilibrium concept, for which, in contrast to ex post equilibria, existence is guaranteed. We show that computation of "robust-optimization equilibria" is analogous to that of Nash equilibria of complete- information games. Our results cover incomplete-information games either involving or not involving private information. In the third part of the thesis, we consider uncertainty on the part of a mechanism designer. Specifically, we present a novel, robust optimization model of the network design problem (NDP) under demand uncertainty and congestion effects, and under either system- optimal or user-optimal routing. We propose a corresponding branch and bound algorithm which comprises the first constructive use of the price of anarchy concept. In addition, we characterize conditions under which the robust NDP reduces to a less computationally demanding problem, either a nominal counterpart or a single-level quadratic optimization problem. Finally, we present a novel traffic "paradox," illustrating counterintuitive behavior of changes in cost relative to changes in demand.
by Michele Leslie Aghassi.
Ph.D.
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Worthing, Rodney A. (Rodney Alan). "Contributions to the variational theory of convection." Thesis, Massachusetts Institute of Technology, 1996. http://hdl.handle.net/1721.1/10577.
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Gmeineder, Franz Xaver. "Regularity theory for variational problems on BD." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:1f412087-de70-44a8-a045-8923f1e29611.
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In this thesis we provide regularity results for convex and semiconvex variational problems which are of linear growth and depend on the symmetric rather than the full gradient. By the non-availability of Korn's Inequality (known as Ornstein's Non-Inequality), usual approaches need to be modified in order to obtain higher regularity of generalised minima.
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Scott, Matthew. "Theory of electrode polarization, application of variational methods." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape3/PQDD_0015/MQ55238.pdf.
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Türköz, Ş. (Şemsettin). "Variational procedure for [phi]4-scalar field theory." Thesis, Massachusetts Institute of Technology, 1989. http://hdl.handle.net/1721.1/52913.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 1990.On t.p. "[phi]" is the original Greek letter.
Includes bibliographical references (leaves 81-83).
by Ş. Türköz.
Ph.D.
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Santambrogio, Filippo. "Variational problems in transport theory with mass concentration." Doctoral thesis, Scuola Normale Superiore, 2006. http://hdl.handle.net/11384/85701.
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Buquicchio, Luke J. "Variational Open Set Recognition." Digital WPI, 2020. https://digitalcommons.wpi.edu/etd-theses/1377.
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In traditional classification problems, all classes in the test set are assumed to also occur in the training set, also referred to as the closed-set assumption. However, in practice, new classes may occur in the test set, which reduces the performance of machine learning models trained under the closed-set assumption. Machine learning models should be able to accurately classify instances of classes known during training while concurrently recognizing instances of previously unseen classes (also called the open set assumption). This open set assumption is motivated by real world applications of classifiers wherein its improbable that sufficient data can be collected a priori on all possible classes to reliably train for them. For example, motivated by the DARPA WASH project at WPI, a disease classifier trained on data collected prior to the outbreak of COVID-19 might erroneously diagnose patients with the flu rather than the novel coronavirus. State-of-the-art open set methods based on the Extreme Value Theory (EVT) fail to adequately model class distributions with unequal variances. We propose the Variational Open-Set Recognition (VOSR) model that leverages all class-belongingness probabilities to reject unknown instances. To realize the VOSR model, we design a novel Multi-Modal Variational Autoencoder (MMVAE) that learns well-separated Gaussian Mixture distributions with equal variances in its latent representation. During training, VOSR maps instances of known classes to high-probability regions of class-specific components. By enforcing a large distance between these latent components during training, VOSR then assumes unknown data lies in the low-probability space between components and uses a multivariate form of Extreme Value Theory to reject unknown instances. Our VOSR framework outperforms state-of-the-art open set classification methods with a 15% F1 score increase on a variety of benchmark datasets.
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Black, Joshua. "Development and applications of Quasi-Variational Coupled-Cluster theory." Thesis, Cardiff University, 2017. http://orca.cf.ac.uk/105353/.
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The Quasi-Variational (QV) family of methods are a set of single-reference algorithms that can be used to investigate multireference systems with large nondynamic correlation effects. Within this current work, the Quasi-Variational Coupled Cluster Doubles (QVCCD) equations are derived and implemented into Molpro’s Integrated Tensor Framework (ITF), to produce fast and efficient code. This code, coupled with a new orbital optimisation implementation, is used to calculate potential energy curves for third-row diatomic molecules. In contrast to Traditional Coupled-Cluster methods, the QV methods are able to correctly describe the dissociation of these molecules. QV and several other single-reference methods are also applied to 5 chemical databases comprising of 88 unique reactions. From this, the activation and reaction energies are determined and contrasted. The QV methods produce larger activation energies that may correct the shortcomings of the perturbative triples correction. These results also include a new QV method with n ‘asymmetricrenormalised’ triples correction. The numerical results show there is little difference between this procedure and ‘symmetric-renormalised’ triples. Currently, only closed-shell QVCCD programs exist. Unrestricted QVCCD equations are derived and presented in the hope that this will facilitate the realisation of an open-shell QVCCD program. Finally, calculating the rate of a chemical reaction is of fundamental importance to chemistry. Knowledge of how quickly a reaction proceeds allows for an understanding of macroscopic chemical change. Rate constants are calculated with the on-the-fly Instanton method. In contrast to semi-classical Transition State Theory, the Instanton method incorporates quantum effects like atomic tunnelling into its rate constants. The effects of hydrogen tunnelling are examined for a reaction involving a Criegee intermediate. It is discovered that tunnelling does play a role in the reaction rate and may increase it by a factor of 1000. Combination of the Instanton calculations with the QV methods are discussed.
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Brown, Bruce J. L. "A variational approach to local optimality in control theory." Doctoral thesis, University of Cape Town, 2001. http://hdl.handle.net/11427/4869.
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Bibliography: leaves 203-212.A new approach to control theory is investigated in this thesis. The approach is based on a locally specified state space model of the control dynamics; together with a goal function, which defines a generalized distance from each state position to the desired equilibrium point or trajectory. A feedback control function is sought, which will result in a system response which approximates the gradient descent trajectories of the specified goal function. The approximation is chosen so that the resulting trajectories satisfy a certain local optimality criterion, involving the averaged second derivative of the goal function along the trajectories.
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Laatz, C. D. "Cosmological perturbation theory and the variational principle in gravitation." Master's thesis, University of Cape Town, 2000. http://hdl.handle.net/11427/6671.
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Summary in English.Include bibliographical references.
In this thesis firstly the theory of relativistic cosmological perturbations is studies, in the process being reviewed over the period 1960-1993. Secondly the variational principle, apropos of gravitation, is formulated and discussed. These two fields are then synthesised via a variational formulation of general relativity and cosmological perturbation theory. In the process new light is shed on Covariant Perturbation Theory via the development of generalised alternative variables, culminating in a unique variational formulation.
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Lazzaroni, Giuliano. "Some results in the variational theory of crack growth." Doctoral thesis, SISSA, 2009. http://hdl.handle.net/20.500.11767/4621.
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Allagi, Mabruk Omar F. Mehemed. "Variational processing of Monte Carlo solutions in neutron transport theory." Thesis, University of Cambridge, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.624171.
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Almoukhalalati, Adel. "Applications of variational perturbation theory in relativistic molecular quantum mechanics." Toulouse 3, 2016. http://www.theses.fr/2016TOU30172.
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Le père même de la mécanique quantique relativiste P. A. M. Dirac a prédit que la version plus réaliste de la mécanique quant ique qu'il a misen place n'offrirait pas beaucoup plus par rapport à la formulation non relativiste de la mécanique quantique lorsqu'il est appliqué à des systèmes atomiques et moléculaires ordinaires. Lorsque la théorie quantique relativiste avait environ 40 années, les gens avaient commencé à recogenize à quel point les effets relativistes peuvent être même pour l'étude des systèmes atomiques et moléculaires. Les effets relativistes se manifestent par la contraction dess atomiques et orbitales p, l'expansion des orbitales d et 1 atomiques, et le couplage spin-orbite. Un exemple classique de l'importance des effets relativistes est la structure de bande d'or métallique pour lequel les calculs non-relativistes vont conduire à une surestimation de l'écart 5d - 6p et prédire une bande d'absorption UV qui est compatible avec un métal qui ressemble à l'argent. La thèse porte sur les calculs atomiques et moléculaires dans le cadre relativiste à 4-composantes. En particulier, l'utilisation de la théorie des perturbations variationnelle dans un cadre relativiste. La théorie des perturbations dans la mécanique quantique, est basée sur le partitionnement du Hamiltonien il en l'Hamiltonien Ho a l'ordre zéro et \Î qui forme la perturbation par le biais d'un paramètre lambda. Dans la théorie des perturbations à N-corps {Rayleigh-Schrodinger), nous disposons d 'une solution exacte de l'Hamiltonien Ho. Alors que dans la théorie des perturbations variationnelle, nous supposons d'avoir une énergie optimisée pour toute valeur du paramètre À. La thèse contient deux projets principaux. Le premier projet concerne la discription de la corrélation électronique dans le cadre relativiste. Dans ce projet, nous nous sommes concentrés sur l'approche perturbative pour dériver des formules necessiry relativiste de l'énergie dans les atomes à deux électrons. L'énergie de corrélation est la différence entre la valeur propre exacte de l'hamiltonien et sa valeur d'attente dans l'approximationHartree-Fock. La valeur propre exacte ne sont pas disponibles, mais dans le domaine non-relativiste la meilleure solution est un Cl complet pour une base donnée. Notre objectif principal, dans ce projet, est de montrer que la meilleure solution de l'équation d'onde pour l' Hamiltonien DiracCoulomb, n'est pas un Cl complète, comme dans le cas non-relativis te, mais un MCSCF q ui utilise un développement Cl en orbitales énergiepositive seulement, mais qui permet la rotation entre les o rbitales d 'énerg ie positive et négative afin d 'optimiser l'opérateur de project ion. Le second projet concerne une étude sur les effets du volume nucléaire dans les spectres de vibration des molécules d iatomiques. Au début desannées 80, le groupe du professeur Eberhardt T iemann à Hanovre a util isé la spectroscopie de rotation avec une haute résolution pour étudier unesérie de molécules diatomiques contenant des atomes lo urds comme le plomb, afin d'établir des constantes spectroscopiques (Re longueur de laliaison, la fréquence vibratoire w. Etc. ) avec une grande précision. Une molécule AB a plusieurs isotopomères selon les isotopes des atomes A etB, et il était bien connu à cette époque que le spectre de chaque isotopomère est légèrement différente en raison des d ifférences de masse entrechaque isotope de l'atomes A et B. Prof. Tiemann et ses collaborateurs découvert que nous devons également ten ir compte de la différence devolume nucléa ire de chaque isotope. Nous fournissons un contrôle indépendant sur les études expérimentales et t héoriques précédentes d'effetsde volume nucléaires en spectroscopie de rotation, notamment re-calcul de la t héorie et des calculs antérieurs de référence par l'état relativiste4-composantes de l'art corrélée calculsThe father of relativistic quantum mechan ics P. A. M. Dirac predicted that, the more realistic version of quantum mechanics that he established wouId not offer much more when compared to the non-relativistic formulation of quantum mechanics when applied to ordinary atomic and molecular systems. When the relativistic quantum theory was around forty years old, people had started to recognize how important relativistic effects can beeven for the study of atomic and molecular systems. Relativistic effects are manifested via the contraction of atomics and p orbitais, the expansion of atomic d and 1 orbitais, and spin-orbit coupling. A classical example on t he importance of relativistic effects is the band struct ure of metallic gold for which non-relativistic caleulations will lead to an overestimation of the 5d-6p gap predicting a UV absorption band which is compatible with a metal that looks like silver. The thesis focuses on the atomic and molecular calculations within the 4-component relativistic framework. Ln particular, the use of the variational perturbation theory in relativistic framework. The perturbation theory in quantum mechanics is based on partitioning the Hamiltonian H into zeroth-order Hamiltonian Ho and V that forms the perturbation through a para meter lambda. Ln many-body (Rayleigh-Sch rodinger) perturbation theory, we have an exact solution of t he Hamiltonian l/0 , whereas in the variational perturbation theory, we assume to have anoptimized energy for any value of the parameter À. The thesis contains two principal projects, the first project concerns the description of the electron correlation in the relativistic framework. Ln this project , we focused on the perturbative approach to derive t he relativistic formulas nece~sary for the energy in two-electron atoms. T hecorrelation energy is the difference between the exact eigenvalue of the Ha mi ltonian and its expectation value in the Hartree-Fock approximation. The exact eigenvalue is not avail able, but in the non- relativistic domain t he best solution is a full Cl for a given basis. Our main goal, in this project , will be to show that the best solution of the wave equation for the embedded Dirac-Coulomb Hamil tonian, is not a Full Cl, as in thenon- relativistic case, but a MCSCF which uses a Cl development in positive-energy orbitais only, but which keeps rotations between the positive and negative energy orbitais to optimize the projection operator. The second project concerns a study of the effects of t he nuclear volume in the vibrational spectra of diatomic molecules. Ln the early 80s, Theg roup of Professor Eberhardt Tiemann in Hanover used the rotational spectroscopy with high resolution to study a series of diatomic molecules containing heavy a toms like lead in order to establish spectroscopie constants (R. Bond length, vibrational frequency W c etc. ) with a great precision. A molecule AB has several isotopomers according to isotopes atoms A and B and it was weil known at that t ime only the spectrum of eachisotopomer is slightly d iffe rent because of the mass differences between each isotope of the atoms A and B. Prof. Tiemann and his collaborators discovered that we must also take into account the difference in nuclear volume of each isotope. We provide an independent check on previous experimental and t heoretical studies of nuclear volume effects in rotational spectroscopy, notably re-derivation of theory and benchmark previous calculations by 4-component relativistic state of the art correlated calculations
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Ding, Bingfeng. "Variational particle-antiparticle bound states in the scalar Yukawa model." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ59128.pdf.
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Mosher, Scott William. "A Variational Transport Theory Method for Two-Dimensional Reactor Core Calculations." Diss., Georgia Institute of Technology, 2004. http://hdl.handle.net/1853/5070.
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A Variational Transport Theory Method for
Two-Dimensional Reactor Core Calculations
Scott W. Mosher
110 Pages
Directed by Dr. Farzad Rahnema
It seems very likely that the next generation of reactor analysis methods will be based largely on neutron transport theory, at both the assembly and core levels. Signifi-cant progress has been made in recent years toward the goal of developing a transport method that is applicable to large, heterogeneous coarse-meshes. Unfortunately, the ma-jor obstacle hindering a more widespread application of transport theory to large-scale calculations is still the computational cost.
In this dissertation, a variational heterogeneous coarse-mesh transport method has been extended from one to two-dimensional Cartesian geometry in a practical fashion. A generalization of the angular flux expansion within a coarse-mesh was developed. This allows a far more efficient class of response functions (or basis functions) to be employed within the framework of the original variational principle. New finite element equations were derived that can be used to compute the expansion coefficients for an individual coarse-mesh given the incident fluxes on the boundary. In addition, the non-variational method previously used to converge the expansion coefficients was developed in a new and more thorough manner by considering the implications of the fission source treat-ment imposed by the response expansion.
The new coarse-mesh method was implemented for both one and two-dimensional (2-D) problems in the finite-difference, multigroup, discrete ordinates approximation. An efficient set of response functions was generated using orthogonal boundary conditions constructed from the discrete Legendre polynomials. Several one and two-dimensional heterogeneous light water reactor benchmark problems were studied. Relatively low-order response expansions were used to generate highly accurate results using both the variational and non-variational methods. The expansion order was found to have a far more significant impact on the accuracy of the results than the type of method. The varia-tional techniques provide better accuracy, but at substantially higher computational costs. The non-variational method is extremely robust and was shown to achieve accurate re-sults in the 2-D problems, as long as the expansion order was not very low.
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Ilas, Danut. "Coarse mesh transport theory model for heterogeneous systems." Diss., Georgia Institute of Technology, 2001. http://hdl.handle.net/1853/16089.
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Franklin, C. P. "Scattering theory with applications to muon catalysed fusion and positron H2+ collisions." Thesis, University of Nottingham, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.281636.
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Kovvali, Ravi Kumar. "A nonlinear theory of Cosserat elastic plates using the variational-asymptotic method." Diss., Georgia Institute of Technology, 2015. http://hdl.handle.net/1853/54342.
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One of the most important branches of applied mechanics is the theory of plates - defined to be plane structural elements whose thickness is very small when compared to the two planar dimensions. There is an abundance of plate theories in the literature modeling classical elastic solids that fit this description. Recently, however, there has been a steady growth of interest in modeling materials with microstructures that exhibit length-scale dependent behavior, generally known as Cosserat elastic materials. Concurrently, there has also been an increased interest in the construction of reduced dimensional models of such materials owing to advantages like reduced computational effort and a simpler, yet elegant, resulting mathematical formulation.
The objective of this work is the formulation and implementation of a theory of elastic plates with microstructure. The mathematical underpinning of the approach used is the Variational Asymptotic Method (VAM), a powerful tool used to construct asymptotically correct plate models. Unlike existing Cosserat plate models in the literature, the VAM allows for a plate formulation that is free of a priori assumptions regarding the kinematics. The result is a systematic derivation of the two-dimensional constitutive relations and a set of geometrically-exact, fully intrinsic equations gov- erning the motion of a plate. An important consequence is the extraction of the drilling degree of freedom and the associated stiffness. Finally, a Galerkin approach for the solution of the fully-intrinsic formulation will be developed for a Cosserat sur- face analysis which will also be compatible with more traditional plate solvers based on the classical theory of elasticity. Results and validation are presented from linear static and dynamic analyses, along with a discussion on some challenges and solution techniques for nonlinear problems.One of the most important branches of applied mechanics is the theory of plates - defined to be plane structural elements whose thickness is very small when compared to the two planar dimensions. There is an abundance of plate theories in the literature modeling classical elastic solids that fit this description. Recently, however, there has been a steady growth of interest in modeling materials with microstructures that exhibit length-scale dependent behavior, generally known as Cosserat elastic materials. Concurrently, there has also been an increased interest in the construction of reduced dimensional models of such materials owing to advantages like reduced computational effort and a simpler, yet elegant, resulting mathematical formulation.
The objective of this work is the formulation and implementation of a theory of elastic plates with microstructure. The mathematical underpinning of the approach used is the Variational Asymptotic Method (VAM), a powerful tool used to construct asymptotically correct plate models. Unlike existing Cosserat plate models in the literature, the VAM allows for a plate formulation that is free of a priori assumptions regarding the kinematics. The result is a systematic derivation of the two-dimensional constitutive relations and a set of geometrically-exact, fully intrinsic equations gov- erning the motion of a plate. An important consequence is the extraction of the drilling degree of freedom and the associated stiffness. Finally, a Galerkin approach for the solution of the fully-intrinsic formulation will be developed for a Cosserat sur- face analysis which will also be compatible with more traditional plate solvers based on the classical theory of elasticity. Results and validation are presented from linear static and dynamic analyses, along with a discussion on some challenges and solution techniques for nonlinear problems.
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SADE, MARTIN CHARLES. "VARIATIONAL PRINCIPLES FOR FIELD VARIABLES SUBJECT TO GROUP ACTIONS (GAUGE)." Diss., The University of Arizona, 1985. http://hdl.handle.net/10150/188075.
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This dissertation is concerned with variational problems whose field variables are functions on a product manifold M x G of two manifolds M and G. These field variables transform as type (0,1) tensor fields on M and are denoted by ψ(h)ᵅ (h = 1, ..., n = dim M, α = 1, ..., r = dim G). The dependence of ψ(h)ᵅ on the coordinates of G is given by a generalized gauge transformation that depends on a local map h:M → G. The requirement that a Lagrangian that is defined in terms of these field variables be independent of the coordinates of G and the choice of the map h endows G with a local Lie group structure. The class of Lagrangians that exhibits this type of invariance may be characterized by three invariance identities. These identities, together with an arbitrary solution of a system of partial differential equations, may be used to define field strengths associated with the ψ(h)ᵅ as well as connection and curvature forms on M. The former may be used to express the Euler-Lagrange equations in a particularly simple form. An energy-momentum tensor may also be defined in the usual manner; however additional conditions must be imposed in order to guarantee the existance of conservation laws resulting from this tensor. The above analysis may be repeated for the case that the field variables behave as type (0,2) tensor fields under coordinate transformations on M. For these field variables, the Euler-Lagrange expressions may be expressed as a product of a covariant divergence with the components λʰ of a type (1,0) vector field on M. An unexpected consequence of this construction is the fact that the Euler-Lagrange equations that result for the vector field λʰ are satisfied whenever the Euler-Lagrange equations associated with the field variables are satisfied.
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Zhang, Chen-Song. "Adaptive finite element methods for variational inequalities theory and applications in finance /." College Park, Md. : University of Maryland, 2007. http://hdl.handle.net/1903/7167.
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Thesis (Ph. D.) -- University of Maryland, College Park, 2007.Thesis research directed by: Applied Mathematics and Scientific Computation Program. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.
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Doole, Stuart Harvey. "Steady gravity waves on flows with vorticity : bifurcation theory and variational principles." Thesis, University of Oxford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.357537.
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Chyou, Hui-Huang Abel. "Variational formulation and finite element implementation of Pagano's theory of laminated plates /." The Ohio State University, 1989. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487671108308444.
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Galbraith, Grant N. "Applications of variational analysis to optimal trajectories and nonsmooth Hamilton-Jacobi theory /." Thesis, Connect to this title online; UW restricted, 1999. http://hdl.handle.net/1773/5766.
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Kim, Yunho. "Variational methods theory and its applications to image deblurring and denoising problems /." Diss., Restricted to subscribing institutions, 2009. http://proquest.umi.com/pqdweb?did=1872146111&sid=1&Fmt=2&clientId=1564&RQT=309&VName=PQD.
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Mongiardo, Mauro. "Variational analysis of waveguide discontinuities by integral equation including the edge condition." Thesis, University of Bath, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.305044.
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Ponsiglione, Marcello. "Stability of Neumann problems and applications to the variational theory of crack propagation." Doctoral thesis, SISSA, 2004. http://hdl.handle.net/20.500.11767/4175.
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Chi, Xuguang. "A non-variational approach to the quantum three-body coulomb problem /." View abstract or full-text, 2004. http://library.ust.hk/cgi/db/thesis.pl?PHYS%202004%20CHI.
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Wang, Jiabin. "Variational Bayes inference based segmentation algorithms for brain PET-CT images." Thesis, The University of Sydney, 2012. https://hdl.handle.net/2123/29251.
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Dual modality PET-CT imaging can provide aligned anatomical (CT) and functional (PET) images in a single scanning session, and has nowadays steadily replaced single modality PET imaging in clinical practice. The enormous number of PET-CT images produced in hospitals are currently analysed almost entirely through visual inspection on a slice-by-slice basis, which requires a high degree of skill and concentration, and is time-consuming, expensive, prone to operator bias, and unsuitable for the processing large-scale studies. Computer-aided diagnosis, where image segmentation is an essential step, would enable doctors and researchers to bypass these issues. However, most medical image segmentation methods are designed for single modality images. In this thesis, the automated segmentation of dual-modality brain PET-CT images has been comprehensively investigated by using variational learning techniques. Two novel statistical segmentation algorithms, namely the DE-VEM algorithm and PA-VEM algorithm, have been proposed to delineate brain PET-CT images into grey matter (GM), white matter (WM) and cerebrospinal fluid (CSF).
In statistical image segmentation, voxel values are usually characterised by probabilistic models, whose parameters can be estimated by using the maximum likelihood estimation, and the optimal segmentation result is regarded as the one that maximises the posterior probability. Despite of their simplicity, statistical approaches intrinsically suffer from overfitting and local convergence. In variational Bayes inference, statistical model parameters are further assumed to be random variables to improve the model's flexibility. Instead of directly estimating the posterior probability, variational learning techniques use a variational distribution to approximate the posterior probability, and thus are able to overcome the drawback of overfitting. The most widely used variational learning technique is the variational expectation maximisation (VEM) algorithm. As a natural extension of the traditional expectation maximisation (EM) algorithm, the VEM algorithm is also a two-step iterative process and still faces the risk of being trapped in a local maximum and the difficulty of incorporating prior knowledge.
Inspired by the fact that global optimisation techniques, such as the genetic algorithm, have been successfully applied to replace the EM algorithm in the maximum-likelihood estimation of probabilistic models, this research combines the differential evolution (DE) algorithm and VEM algorithm to solve the optimisation problem involved in the variational Bayes inference, and thus proposes the DE-VEM algorithm for brain PET -CT image segmentation. In this algorithm, the DE scheme is introduced to search a global solution and the VEM scheme is employed to perform a local search. Since DE is population-based global optimisation technique and has proven itself in a variety of applications with good, the DEYEM algorithm has the potential to avoid local convergence. The proposed algorithm has been compared with the YEM algorithm and the segmentation function in the statistical parametric mapping (SPM, Version 2008) package in 21 clinical brain PET -CT images. My results show that the DE-YEM algorithm outperforms the other two algorithms and can produce accurate segmentation of brain PET-CT images.
Meanwhile, to incorporate the prior anatomical information into the variational learning based brain image segmentation process, the probabilistic brain atlas is generated and used to guide the search of an optimal segmentation result through performing the YEM iteration. As a result, the probabilistic atlas based YEM (PA-YEM) algorithm is developed to allow each voxel to have an adaptable prior probability of belonging to each class. This algorithm has been compared to the segmentation functions in the SPM8 package and the EMS package, the DE-YEM algorithm, and the DEV algorithm in 21 clinical brain PET-CT images. My results demonstrate that the proposed PA-YEM algorithm can substantially improve the accuracy of segmenting brain PET -CT images.
Although this research uses the brain PET -CT images as case studies, the theoretical outcomes are generic and can be extended to the segmentation of other dual-modality medical images. The future work in this area should be focused mainly on improving the computational efficiency of variational learning based image segmentation approaches.
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Simo, Tao Lee Walter Cédric. "On the variational approach to mollification in the theory of ill-posed problems and applications." Thesis, Toulouse 3, 2020. http://www.theses.fr/2020TOU30130.
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Les problèmes inverses constituent un domaine en pleine expansion en mathématiques appliquées qui a suscité une grande attention au cours des dernières décennies en raison de son omniprésence dans plusieurs domaines des sciences et technologies. Le plus souvent, les problèmes inverses donnent lieu à des équations mathématiques instables. Autrement dit, les solutions ne dépendent pas continument des données. En effet, de très petites perturbations sur les données peuvent causer des erreurs arbitrairement grandes sur les solutions. Étant donné que le bruit est généralement inévitable, inverser l'équation mal-posée échoue à résoudre le problème. Il est alors nécessaire d'appliquer une méthode de régularisation afin de récupérer des approximations stables des solutions. À cet égard, plusieurs techniques de régularisation ont été développées dans la littérature. Globalement, ces méthodes de régularisation peuvent être divisées en deux classes : Une classe de méthodes qui tentent de reconstruire les solutions inconnues initiales et une classe de méthodes qui tentent de reconstruire des versions lisses des solutions inconnues. L'objectif de cette thèse est de contribuer à la promotion de la deuxième classe de méthode de régularisation à travers l'étude et l'application de la formulation variationnelle de la mollification. Dans ce manuscrit, nous montrons que l'approche variationnelle de la mollification peut être étendue à la régularisation de problèmes mal-posés impliquant des opérateurs non compacts. À cet égard, nous étudions et appliquons avec succès la méthode à la régression instrumentale non-paramétrique. Une contribution supplémentaire de cette thèse est la conception et l'étude d'une nouvelle méthode de régularisation adaptée aux problèmes linéaires exponentiellement mal-posés. Une comparaison numérique de cette nouvelle méthode aux méthodes classiques de régularisation telles que Tikhonov, la spectral cut-off, la régularisation asymptotique et la méthode des gradients conjugués est effectuée sur trois problèmes test tirés de la littérature. L'aspect pratique de la sélection du paramètre de régularisation avec un niveau de bruit inconnu est également considéré. Outre l'étude et l'application des méthodes de régularisation, cette thèse traite également de l'application d'une règle de sélection de paramètres de régularisation très populaire connue sous le nom du principe de Morozov. En utilisant la dualité de Lagrange, nous fournissons un algorithme simple et rapide pour le calcul du paramètre de régularisation correspondant à cette règle pour les méthodes de régularisation du type Tikhonov. L'intérêt de cette étude est qu'elle met en avant une méthode de régularisation mal connue qui pourtant a un grand potentiel et est capable de fournir des solutions approchées comparativement meilleures que certaines techniques de régularisation classiques bien connues. Un autre apport de cette thèse est la conception d'une nouvelle méthode de régularisation qui, selon nous, est prometteuse dans la régularisation de problèmes exponentiellement mal-posés, en particulier pour les problèmes inverses de conduction thermiqueInverse problems is a fast growing area in applied mathematics which has gained a great attention in the last decades due to its ubiquity in several fields of sciences and technology. Yet, most often, inverse problems result in mathematical equation which are unstable. That is, the solutions do not continuously depend on the data. As a matter of fact, very little perturbations on the data might cause arbitrary large errors on the solutions. Therefore, given that the noise is generally unavoidable in the data, direct attempts to solve the problem fail and one needs to apply a regularization method in order to recover stable approximates of the unknown solutions. In this respect, several regularization techniques have been developed in the literature. Globally, all these regularization methods can be split into two classes: A class of methods which attempt to reconstruct the unknown solutions and a class of methods which try to recover smooth versions of the unknown solutions. The aim of this thesis is to contribute to the promotion of the second class of regularization method via the study and application of the variational formulation of mollification. In this work, we show that the variational approach can be extended to the regularization of ill-posed problems involving non-compact operators. In this respect, we study and successfully apply the method to a problem coming from statistics namely the nonparametric instrumental regression. An additional contribution of this thesis is the design and study of a novel regularization method suitable for linear exponentially ill-posed problems. A numerical comparison of the new method to classical regularization methods such as Tikhonov, spectral cut-off, asymptotic regularization and conjugate gradient is carried out on three test problems from literature. The practical aspect of selection of the regularization parameter without knowledge of the noise level is also considered. Apart from the study and application of regularization methods, this thesis also focuses on the application of a very popular parameter selection rule known as the Morozov principle. Using Lagrange duality, we provide a simple and rapid algorithm for the computation of the regularization parameter corresponding to this rule for Tikhonov-like regularization methods. A relevance of this study is that it highlights a poorly known regularization method which yet has a great potential and is able to provide comparatively better approximate solutions compared to well-known classical regularization techniques. Another benefit of this thesis is the design of a new regularization method which, we believe, is promising in the regularization of exponentially ill-posed problems, especially for inverse heat conduction problems
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Borghi, Giovanni. "Gutzwiller approximation applied to inhomogeneous lattice models and solid-state systems." Doctoral thesis, SISSA, 2011. http://hdl.handle.net/20.500.11767/4290.
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The plan of this thesis is as follows. Chapt. 1 is devoted to the explanation of the main theoretical tool of our work, namely the GVM and GA. After introducing their earliest formulation by Martin C. Gutzwiller, we discuss their effectiveness in describing the physics of strongly correlated conductors, emphasizing the improvements they bring in comparison with mean-field, independent-electron approximations such as HF, and their limitations with respect to more refined, though computationally more costly, methods like DMFT and VQMC. We mention how the GA was initially exploited as an approximate tool for analytical calculation of expectation values on the GVW, and how later studies proved its exactness in the limit of infinite lattice coordination. After that, we discuss its more recent multi-band formulation which, together with the mixed-basis parametrization of Gutzwiller parameter matrix, is particularly important for combining the GVM with DFT.
In Chapt. 2 we present our results for the strongly correlated Hubbard lattice with broken translational invariance due to the presence of a surface (panel (a) in Fig. 1), a metal-metal or metal-insulator junction (panel (b)), or a “sandwich” of Mott insulator or strongly correlated metal between metallic leads (panel (c)). For all geometries, we show the layer dependence of the quasi-particle weight and provide approximate analytical fits for the data, together with a comparison with DMFT calculations on similar systems.
In Chapt. 3, we introduce the formalism of DFT, the Kohn-Sham self-consistent
equations for the functional minimization and the LDA for exchange and correlation functionals. We further discuss the performance and limitations of LDA and present the LDA+U method as a way to correct the self-interaction error of LDA. We explain the details of the GDF in Chapt. 4, and underline its similarities and differences with respect to the LDA+U functional. In the same chapter we present our data for paramagnetic and ferromagnetic bcc iron obtained through our implementation of LDA+G in the Siesta code. We show energy differences between spin-polarized and unpolarized Iron computed within LDA, GGA and LDA+G and with different basis sets. We compare the band structure, lattice parameters and magnetic moments (some sample data is shown in Table 1) obtained with these functionals, and discuss the implications of our results on the understanding of the origin of magnetism in
transition metals.
In the appendices we list some important results that we believed too detailed or too marginal to be presented in the main body of the thesis. Appendix A is devoted to some proofs and detailed explanations related to the GVM. In Appendix B we include all details related to the calculations on the layered geometries of Chapt. 2.
In Appendix C we explain how to implement spin and orbital symmetries in the
parametrization of the Gutzwiller projector, while in Appendix D we give the details of the minimization algorithm we implemented for optimizing the variational energy of the LDA+G calculation with respect to Gutzwiller parameters. Finally, Appendix E contains various topics of DFT and LDA+U that are important for the understanding of the GDF we implemented and discussed in Chapt. 4.
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Bui, Hoa. "Extremality and stationarity of collections of sets : metric, slope and normal cone characterisations." Thesis, Federation University of Australia, 2019. http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/178600.
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Variational analysis, a relatively new area of research in mathematics, has become one of the most powerful tools in nonsmooth optimisation and neighbouring areas. The extremal principle, a tool to substitute the conventional separation theorem in the general nonconvex environment, is a fundamental result in variational analysis. There have seen many attempts to generalise the conventional extremal principle in order to tackle certain optimisation models. Models involving collections of sets, initiated by the extremal principle, have proved their usefulness in analysis and optimisation, with non-intersection properties (or their absence) being at the core of many applications: recall the ubiquitous convex separation theorem, extremal principle, Dubovitskii Milyutin formalism and various transversality/regularity properties. We study elementary nonintersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish nonlinear primal (slope) and linear/nonlinear dual (generalised separation) characterisations of these non-intersection properties. We establish a series of consequences of our main results covering all known formulations of extremality/ stationarity and generalised separability properties. This research develops a universal theory, unifying all the current extensions of the extremal principle, providing new results and better understanding for the exquisite theory of variational analysis. This new study also results in direct solutions for many open questions and new future research directions in the fields of variational analysis and optimisation. Some new nonlinear characterisations of the conventional extremality/stationarity properties are obtained. For the first time, the intrinsic transversality property is characterised in primal space without involving normal cones. This characterisation brings a new perspective on intrinsic transversality. In the process, we thoroughly expose and classify all quantitative geometric and metric characterisations of transversality properties of collections of sets and regularity properties of set-valued mappings.Doctor of Philosophy
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Spencer, Paul. "Variational problems arising in classical mechanics and nonlinear elasticity." Thesis, University of Bath, 1999. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.323570.
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Chen, Guang. "General Variational Principles : theory and applications to the approximate solutions of nonlinear and/or nonconservative oscillations." Diss., Georgia Institute of Technology, 1986. http://hdl.handle.net/1853/14996.
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Caruso, Valeria. "Variational inequalities and networks for organ transplants and for humanitarian organizations." Doctoral thesis, Università di Catania, 2018. http://hdl.handle.net/10761/3788.
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In this thesis we focus our attention on two mathematical models applied to two real situations, both studied with the theory of variational inequalities. The first network-based model describes the organ transplant system with the aim of minimizing the total costs associated with this process. We find the related optimality conditions and the variational inequality formulation. Some existence and uniqueness results as well as the Lagrange formulation are stated and some numerical examples are studied. The second mathematical model presented is a Generalized Nash Equilibrium model for postdisaster
humanitarian relief. We identify the network structure of the problem, with logistical and financial ows, and propose a variational equilibrium framework, which allows us to then formulate, analyze, and solve the model using the theory of variational inequalities. We then utilize Lagrange analysis and finally we illustrate the game theory model through a case study.
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Warschkow, Oliver. "A divide-and-conquer implementation of the discrete variational DFT method for large molecular and solid systems." Thesis, University of Southampton, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.284652.
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Singh, Kumaresh. "Efficient Computational Tools for Variational Data Assimilation and Information Content Estimation." Diss., Virginia Tech, 2010. http://hdl.handle.net/10919/39125.
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The overall goals of this dissertation are to advance the field of chemical data assimilation, and to develop efficient computational tools that allow the atmospheric science community benefit from state of the art assimilation methodologies. Data assimilation is the procedure to combine data from observations with model predictions to obtain a more accurate representation of the state of the atmosphere.
As models become more complex, determining the relationships between pollutants and their sources and sinks becomes computationally more challenging. The construction of an adjoint model ( capable of efficiently computing sensitivities of a few model outputs with respect to many input parameters ) is a difficult, labor intensive, and error prone task. This work develops adjoint systems for two of the most widely used chemical transport models: Harvardâ s GEOS-Chem global model and for Environmental Protection Agencyâ s regional CMAQ regional air quality model. Both GEOS-Chem and CMAQ adjoint models are now used by the atmospheric science community to perform sensitivity analysis and data assimilation studies.
Despite the continuous increase in capabilities, models remain imperfect and models alone cannot provide accurate long term forecasts. Observations of the atmospheric composition are now routinely taken from sondes, ground stations, aircraft, and satellites, etc. This work develops three and four dimensional variational data assimilation capabilities for GEOS-Chem and CMAQ which allow to estimate chemical states that best fit the observed reality.
Most data assimilation systems to date use diagonal approximations of the background covariance matrix which ignore error correlations and may lead to inaccurate estimates. This dissertation develops computationally efficient representations of covariance matrices that allow to capture spatial error correlations in data assimilation.
Not all observations used in data assimilation are of equal importance. Erroneous and redundant observations not only affect the quality of an estimate but also add unnecessary computational expense to the assimilation system. This work proposes techniques to quantify the information content of observations used in assimilation; information-theoretic metrics are used.
The four dimensional variational approach to data assimilation provides accurate estimates but requires an adjoint construction, and uses considerable computational resources. This work studies versions of the four dimensional variational methods (Quasi 4D-Var) that use approximate gradients and are less expensive to develop and run.
Variational and Kalman filter approaches are both used in data assimilation, but their relative merits and disadvantages in the context of chemical data assimilation have not been assessed. This work provides a careful comparison on a chemical assimilation problem with real data sets. The assimilation experiments performed here demonstrate for the first time the benefit of using satellite data to improve estimates of tropospheric ozone.Ph. D.
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Pontaza, Juan Pablo. "Least-squares variational principles and the finite element method: theory, formulations, and models for solid and fluid mechanics." Diss., Texas A&M University, 2003. http://hdl.handle.net/1969.1/288.
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We consider the application of least-squares variational principles and the finite element method to the numerical solution of boundary value problems arising in the fields of solidand fluidmechanics.For manyof these problems least-squares principles offer many theoretical and computational advantages in the implementation of the corresponding finite element model that are not present in the traditional weak form Galerkin finite element model.Most notably, the use of least-squares principles leads to a variational unconstrained minimization problem where stability conditions such as inf-sup conditions (typically arising in mixed methods using weak form Galerkin finite element formulations) never arise. In addition, the least-squares based finite elementmodelalways yields a discrete system ofequations witha symmetric positive definite coeffcientmatrix.These attributes, amongst manyothers highlightedand detailed in this work, allow the developmentofrobust andeffcient finite elementmodels for problems of practical importance. The research documented herein encompasses least-squares based formulations for incompressible and compressible viscous fluid flow, the bending of thin and thick plates, and for the analysis of shear-deformable shell structures.
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Vierling, Morten [Verfasser], and Michael [Akademischer Betreuer] Hinze. "Control-constrained parabolic optimal control problems on evolving surfaces : theory and variational discretization / Morten Vierling. Betreuer: Michael Hinze." Hamburg : Staats- und Universitätsbibliothek Hamburg, 2013. http://d-nb.info/1042753873/34.
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Haro, Àlex. "The Primitive Function of an Exact Symplectomorphism. Variational principles, Converse KAM Theory and the problems of determination and interpolation." Doctoral thesis, Universitat de Barcelona, 1998. http://hdl.handle.net/10803/2116.
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We have divided this thesis in four parts:a) PART I: Exact symplectic geometry (introduction of the problems). This part contains the basic tools of symplectic geometry and outlines the four subjects that we have study along the thesis: the determination problem, the interpolation problem, the variational problem and the breakdown problem.
b) PART II: On the standard symplectic manifold (analytical part). We recall the necessary tools to work on R(d) x R(d). That is we perform a coordinate treatment of the results. First of all we relate different kinds of generating functions to the primitive function and later we solve formally the determination problem. Then we introduce different variational principles: for fixed points, periodic orbits and orbital segments. Their invariance under certain kind of transformations of phase space is proved, and we interpret physically such results. Finally we give the basic properties of invariant exact Lagrangian graphs obtaining at last that if our graph is minimizing then its orbits are minimizing.
c) PART III: On the cotangent bundle (geometrical part). The first three chapters are similar to the three previous ones with the difference that we do an intrinsic treatment of the results by considering any cotangent bundle. The fourth chapter in this part deals with the solution of the interpolation problem given in analytic set up.
d) PART IV: Converse KAM theory (numerical part). The last part deals with the applications to converse Kolmogorv-Arnold-Moser (KAM) theory. First of all we give a small list of different examples that we shall study later. Then we generalize converse KAM theory and we related it to the Lipschitz theory by Birkhoff and Herman. Then we perform our variational Greene method and apply it to different examples. Also we study numerically the Aubry-Mather sets in higher dimensions. After this we apply our methods to the rotational standard map that is a symplectic skew product. Then we give some ideas about the geometrical obstructions for existence of invariant tori showing them with a simple example. We also find some known Birkhoff normal forms using our methods. Finally we explain briefly how our theory can be used for arbitrary Lagrangian foliations.
La present memòria es troba dividida en quatre parts ben diferenciades. La primera conté les eines bàsiques de la geometria simplèctica i planteja els quatre problemes que tractarem al llarg de la memòria: el problema de determinació, el problema d'interpolació, el problema variacional i el problema del trencament de tors invariants.
La segona part tracta sobre la varietat simpléctica estàndard, i vindria a ser la part analítica. Aquí hem treballat a R(d) x R(d), és a dir hem fet un tractament coordenat dels resultats. Primer relacionem les funcions generatrius amb la funció primitiva i després resolem formalment el problema de determinación. Tot seguit tractem diferents principis variacionals per als punts fixos per a les òrbites periòdiques i per als segments orbitals. La seva invariància respecte a certs tipus de transformacions de l'espai de fase és demostrada donant una interpretació física. Finalment donem les propietats bàsiques dels grafs Lagrangians invariants, especialment aquella que diu que les òrbites sobre un graf minimitzant són minimitzants.
La tercera part abraça el tema del fibrat cotangent, la part geométrica de l'obra. Els tres primers capítols segueixen més o menys la línia dels tres precedents amb la diferéncia fonamental que aquí considerem qualsevol fibrat cotangent. Fem llavors un tractament intrínsec. El quart capítol d'aquesta part està dedicat a resoldre el problema d'interpolació en el cas analític.
La quarta i darrera part (que vindria a ser la secció numèrica de la tesi), tracta de les aplicacions a la teoria Kolmogorv, Arnold i Moser (KAM) inversa o del trencament dels tors invariants. Primer donem una llista d'exemples que utilitzarem més endavant. Després generalitzem la teoria KAM inversa i la relacionem amb la teoria Lipschitziana de Birkhoff i Herman. Llavors implementem el nostre criteri de Greene variacional i l'apliquem a diferents exemples. També estudiem els equivalents dels conjunts d'Aubry-Mather en dimensió alta (bé = 4). Després apliquem aquesta metodologia a l'aplicació estàndard rotacional (3D), indicant abans la teoria necessària. Llavors donem algunes idees de com generalitzar els criteris obstruccionals a dimensions altes hi ho mostrem amb un petit exemple. Finalment retrobem algunes formes normals de Birkhoff utilitzant la nostra metodologia basada en la funcióprimitiva i expliquem una mica com es podria considerar la nostra teoria tenint en compte foliacions Lagrangianes arbitràries.
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Goßler, Timo, Tina Wakolbinger, Anna Nagurney, and Patrizia Daniele. "How to increase the impact of disaster relief: a study of transportation rates, framework agreements and product distribution." Elsevier, 2019. http://dx.doi.org/10.1016/j.ejor.2018.09.045.
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Due to restricted budgets of relief organizations, costs of hiring transportation service providers steer
distribution decisions and limit the impact of disaster relief. To improve the success of future humanitarian
operations, it is of paramount importance to understand this relationship in detail and to identify
mitigation actions, always considering the interdependencies between multiple independent actors in humanitarian
logistics. In this paper, we develop a game-theoretic model in order to investigate the influence
of transportation costs on distribution decisions in long-term relief operations and to evaluate measures
for improving the fulfillment of beneficiary needs. The equilibrium of the model is a Generalized Nash
Equilibrium, which has had few applications in the supply chain context to date. We formulate it, utilizing
the construct of a Variational Equilibrium, as a Variational Inequality and perform numerical simulations
in order to study the effects of three interventions: an increase in carrier competition, a reduction of
transportation costs and an extension of framework agreements. The results yield important implications
for policy makers and humanitarian organizations (HOs). Increasing the number of preselected carriers
strengthens the bargaining power of HOs and improves impact up to a certain limit. The limit is reached
when carriers set framework rates equal to transportation unit costs. Reductions of transportation costs
have a consistently positive, but decreasing marginal benefit without any upper bound. They provide
the highest benefit when the bargaining power of HOs is weak. On the contrary, extending framework
agreements enables most improvements when the bargaining power of HOs is strong.
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Barbagallo, Mathias. "Statistical energy analysis and variational principles for the prediction of sound transmission in multilayered structures." Doctoral thesis, KTH, MWL Marcus Wallenberg Laboratoriet, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-118427.
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Multilayered structures have many application in industry and society: they have peculiar properties and serve a variety of purposes, like structural support, thermal insulation, vibrational and acoustic isolation. This thesis concerns the prediction of sound transmission in multilayered structures. Two problems are herein investigated: the transmission of energy through structures and the transmission of energy along structures. The focus of the analysis is on the mid to high frequency range. To predict sound transmission in these structures, statistical energy analysis (SEA) is used.SEA models are devised for the prediction of the sound reduction index for two kinds of multilayered structures, double-walls used in buildings and trim-panels in vehicles; the double-walls comprise an air cavity in between flat plasterboard or glass plates, whereas the trim-panels a porous layer in between curved aluminium and rubber layers. The SEA models are based upon the wave-types carrying energy. The novelty in these SEAs is an element describing the waves in the air cavity, or in the porous layer, fully coupled to the mass-impeded external layers. Compared to measurements, the proposed SEA performs well: for double-walls, it performs better than previous models; for trim-panels, it is an original result. The parameters of the new SEA element, such as modal density, are derived from the coupling equations describing the fully coupled waves. For double-walls, these equations are derived via Newton's laws. For trim-panels, a variational approach based upon a modified Hamilton's principle valid for non-conservative systems is preferred, because it is a powerful machinery for deriving equations of motion and coupling conditions of a medium as complex as the porous layer. The modified Hamilton's principle for non-conservative systems is based upon a self-adjoint functional analogous to the Lagrangian, inspired by Morse and Feshbach's construction. A self-adjoint variational principle for Biot's equations in the displacement formulation is devised. An equivalent mixed formulation is obtained changing the coordinates of the displacement formulation via Lagrange multipliers. From this mixed formulation, the Lagrangian for a porous material with a limp frame is derived, which yields the continuity of the total displacement of the porous layer. Lagrange multipliers help to obtain the correct coupling functionals between a porous material and a solid. The Lagrange multipliers introducing the continuity of the frame and the solid displacements equal the traction of the in-vacuo frame, thus disappearing if the latter is limp. Measurements to gather material parameters for a Biot model of the porous layer have been conducted.The effects of spatial energy decay in the transmission along structures predicted by SEA is studied: a major effect is the increased relevance of indirect coupling loss factors between SEA elements. This may jeopardize the usefulness of SEA at higher frequencies.QC 20130218
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Maad, Sara. "Critical point theory with applications to semilinear problems without compactness." Doctoral thesis, Uppsala : Matematiska institutionen, Univ. [distributör], 2002. http://publications.uu.se/theses/91-506-1557-2/.
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Smith, Todd Blanton. "Variational embedded solitons, and traveling wavetrains generated by generalized Hopf bifurcations, in some NLPDE systems." Doctoral diss., University of Central Florida, 2011. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/5042.
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In this Ph.D. thesis, we study regular and embedded solitons and generalized and degenerate Hopf bifurcations. These two areas of work are seperate and independent from each other. First, variational methods are employed to generate families of both regular and embedded solitary wave solutions for a generalized Pochhammer PDE and a generalized microstructure PDE that are currently of great interest. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the family of the trial functions). Thus, the residual is calculated, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that only the parameter regimes for the existence of solitary waves had previously been analyzed for the microstructure PDE considered here, the results obtained here are both new and timely. Second, we consider generalized and degenerate Hopf bifurcations in three different models: i. a predator-prey model with general predator death rate and prey birth rate terms, ii. a laser-diode system, and iii. traveling-wave solutions of twospecies predator-prey/reaction-diffusion equations with arbitrary nonlinear/reaction terms. For specific choices of the nonlinear terms, the quasi-periodic orbit in the post-bifurcation regime is constructed for each system using the method of multiple scales, and its stability is analyzed via the corresponding normal form obtained by reducing the system down to the center manifold. The resulting predictions for the post-bifurcation dynamics provide an organizing framework for the variety of possible behaviors.; These predictions are verified and supplemented by numerical simulations, including the computation of power spectra, autocorrelation functions, and fractal dimensions as appropriate for the periodic and quasiperiodic attractors, attractors at infinity, as well as bounded chaotic attractors obtained in various cases. The dynamics obtained in the three systems is contrasted and explained on the basis of the bifurcations occurring in each. For instance, while the two predator-prey models yield a variety of behaviors in the post-bifurcation regime, the laser-diode evinces extremely stable quasiperiodic solutions over a wide range of parameters, which is very desirable for robust operation of the system in oscillator mode.ID: 029809927; System requirements: World Wide Web browser and PDF reader.; Mode of access: World Wide Web.; Thesis (Ph.D.)--University of Central Florida, 2011.; Includes bibliographical references (p. 121-129).
Ph.D.
Doctorate
Mathematics
Sciences
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Yu, Haofeng. "A Numerical Investigation Of The Canonical Duality Method For Non-Convex Variational Problems." Diss., Virginia Tech, 2011. http://hdl.handle.net/10919/29095.
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This thesis represents a theoretical and numerical investigation of the canonical duality theory, which has been recently proposed as an alternative to the classic and direct methods for non-convex variational problems. These non-convex variational problems arise in a wide range of scientific and engineering applications, such as phase transitions, post-buckling of large deformed beam models, nonlinear field theory, and superconductivity. The numerical discretization of these non-convex variational problems leads to global minimization problems in a finite dimensional space.
The primary goal of this thesis is to apply the newly developed canonical duality theory to two non-convex variational problems: a modified version of Ericksen's bar and a problem of Landau-Ginzburg type. The canonical duality theory is investigated numerically and compared with classic methods of numerical nature. Both advantages and shortcomings of the canonical duality theory are discussed. A major component of this critical numerical investigation is a careful sensitivity study of the various approaches with respect to changes in parameters, boundary conditions and initial conditions.Ph. D.
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Aragão, Cristiane Moura Lima de. "Cálculo de integrais de trajetória em mecânica estatística e teoria de campos através de técnicas variacionais." Universidade de São Paulo, 2002. http://www.teses.usp.br/teses/disponiveis/43/43134/tde-18022014-140626/.
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Estendemos para a teria de campos o método variacional de Kleinert. Este método foi primeiramente usado na mecânica quântica e fornece uma expansão em cumulantes convergente. Sua extensão para a teoria de campos não é trivial devido às divergências ultravioletas que aparecem quando a dimensão do espaço é maior que 2. Devido a estas divergências, a teoria deve ser regularizada e normalizada. Além das dificuldades usuais associadas com a renormalização, devemos decidir se calculamos o valor ótimo do parâmetro variacional antes ou depois da renormalização. Nesta tese abordamos o problema da renormalização do potencial efetivo variacional. Primeiramente, mostramos que o potencial efetivo variacional em temperatura zero coincide com o \"potencial efetivo pós-gaussiano\" introduzido por Stancu e Stevenson. Em seguida, apresentamos um esquema de renormalização que permite que renormalizemos a teoria antes de calcular o parâmetro variacional ótimo. Usando este esquema mostramos que o potencial efetivo usual, calculado em ordem 1-loop, pode ser obtido a partir do esquema variacional de Kleinert inteirando uma única vez a equação que determina o parâmetro variacional. Para o potencial efetivo em ordem 2-loops esta aproximação não é tão boa. A renormalização da teoria antes do cálculo do parâmetro variacional permite que estudemos o potencial efetivo variacional numericamente e de forma não-perturbativa, como foi feito por Kleinert para a mecânica quântica.We have extended the Kleinert variational technique to field theory. This method was first used in quantum mechanics and provides a convergent cumulate expansion that is extremely accurate. Its extension to field theory is non-trivial because of the ultraviolet divergences that appear when the space dimension is greater than 2. Due to these divergences the theory has to be regularized and renormalized. In addition to the usual difficulties associated with renormalization, one has to decide whether one calculates the optimum value of the variational parameter before or after renormalization. In this thesis we deal with the renormalization of the variational effective potential. Firstly, we show that the zero temperature regularized variational potential coincides with the post-Gaussian effective potential introduced by Stancu and Stenvenson. Secondly, we present a renormalization scheme that enables one to renormalize the theory before calculating the optimum variational parameter. Using this scheme we show that the usual 1-loop effective potential can be obtained from the Kleinert variational scheme by interacting only once the equation that determines the variational parameter. In this sense, the 1-loop expansion is contained within the variational scheme. For the 2-loop effective potential the same approximation is not so good. The renormalization of the theory before the calculation of the variational parameter allows one to study the variational effective potential numerically and in a non-pertubative way, as it was done in quantum mechanics by Kleinert.
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Maslovaric, Marcel [Verfasser], Henrik [Akademischer Betreuer] Seppänen, Henrik [Gutachter] Seppänen, Viktor [Gutachter] Pidstrygach, and Daniel [Gutachter] Greb. "Variational Geometric Invariant Theory and Moduli of Quiver Sheaves / Marcel Maslovaric ; Gutachter: Henrik Seppänen, Viktor Pidstrygach, Daniel Greb ; Betreuer: Henrik Seppänen." Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2018. http://d-nb.info/1161942149/34.
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Hu, Xu. "Towards efficient learning of graphical models and neural networks with variational techniques." Thesis, Paris Est, 2019. http://www.theses.fr/2019PESC1037.
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Dans cette thèse, je me concentrerai principalement sur l’inférence variationnelle et les modèles probabilistes. En particulier, je couvrirai plusieurs projets sur lesquels j'ai travaillé pendant ma thèse sur l'amélioration de l'efficacité des systèmes AI / ML avec des techniques variationnelles. La thèse comprend deux parties. Dans la première partie, l’efficacité des modèles probabilistes graphiques est étudiée. Dans la deuxième partie, plusieurs problèmes d’apprentissage des réseaux de neurones profonds sont examinés, qui sont liés à l’efficacité énergétique ou à l’efficacité des échantillonsIn this thesis, I will mainly focus on variational inference and probabilistic models. In particular, I will cover several projects I have been working on during my PhD about improving the efficiency of AI/ML systems with variational techniques. The thesis consists of two parts. In the first part, the computational efficiency of probabilistic graphical models is studied. In the second part, several problems of learning deep neural networks are investigated, which are related to either energy efficiency or sample efficiency
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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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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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Eskandari, Aram. "VAE-clustering of neural signals and their association to cytokines." Thesis, KTH, Matematisk statistik, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-273627.
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In this thesis we start by reproducing previous experiments by Zanos et al., where they have shown that it is possible to associate neural signals with specific cytokines. One future aim of this project is to send synthetic neural signals through the efferent arc of the vagus nerve and observe reactions without the corresponding catalyst of the symptoms. We use a variational autoencoder (VAE) in our experiment to create a model able to generate new neural signals, and we introduce a novel clustering technique called VAE-clustering, which will be used to cluster neural signals with their associated cytokines. The focus of this paper is the implementation of this method and applying it on the neural signals. Running VAE-clustering on the MNIST dataset shows it to be viable for finding detailed properties of a dataset. We also find that using a VAE as a generative model for neural signals is a good way for recreating detailed waveforms.I detta examensarbete börjar vi med att reproducera tidigare experiment av Zanos et al., där dom visat att det är möjligt att associera nervsignaler med specifika cytokiner. Ett framtida mål med detta projekt är att skicka syntetiska nervsignaler till kroppen för att observera reaktioner utan motsvarande katalysator av symptomen. Vi använder en variational autoencoder (VAE) i våra experiment för att skapa en modell kapabel till att generera nya nervsignaler, och vi introducerar en ny klusterings-teknik kallad VAE-klustring, vilken kommer att användas för att klustra nervsignaler med dess associerade cytokiner. Fokuset i detta arbete ligger i implementationen av denna metod och applicerandet på nervsignaler. Efter att ha kört VAE-klustring på MNIST dataset fann vi att det det är användbart för att hitta detaljerade egenskaper hos ett dataset. Vi har även funnit att användningen av en VAE som en generativ modell för nervsignaler är ett bra sätt att återskapa detaljerade vågformer.