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Dissertations / Theses on the topic 'Variational inequalities (Mathematics)'
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Riaz, Samia. "Domain decomposition method for variational inequalities." Thesis, University of Birmingham, 2014. http://etheses.bham.ac.uk//id/eprint/4815/.
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Variational inequalities have found many applications in applied science. A partial list includes obstacles problems, fluid flow in porous media, management science, traffic network, and financial equilibrium problems. However, solving variational inequalities remain a challenging task as they are often subject to some set of complex constraints, for example the obstacle problem. Domain decomposition methods provide great flexibility to handle these types of problems. In our thesis we consider a general variational inequality, its finite element formulation and its equivalence with linear and quadratic programming. We will then present a non-overlapping domain decomposition formulation for variational inequalities. In our formulation, the original problem is reformulated into two subproblems such that the first problem is a variational inequality in subdomain Ω\(^i\) and the other is a variational equality in the complementary subdomain Ω\(^e\). This new formulation will reduce the computational cost as the variational inequality is solved on a smaller region. However one of the main challenges here is to obtain the global solution of the problem, which is to be coupled through an interface problem. Finally, we validate our method on a two dimensional obstacle problem using quadratic programming.
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Zhang, Guohan. "Complementarity problems, variational inequalities and extended Lorentz cones." Thesis, University of Birmingham, 2017. http://etheses.bham.ac.uk//id/eprint/8003/.
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In this thesis, we introduced the concept of extended Lorentz cones. We discussed the solvability of variational inequalities and complementarity problems associated with an unrelated closed convex cone. This cone does not have to be an isotone projection cone. We showed that the solution of variational inequalities and complementarity problems can be reached as a limit of a sequence defined in an ordered space which is ordered by extended Lorentz cone. Moreover, we applied our results in game theory and conic optimization problems. We also discussed the positive operators. We showed necessary and sufficient conditions under which a linear operator is a positive operator of extended Lorentz cone. We also showed sufficient and necessary conditions under which a linear operator in a specific form is a positive operator.
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Denault, M. (Michel). "Variational inequalities with the analytic center cutting plane method." Thesis, McGill University, 1998. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=34945.
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This thesis concerns the solution of variational inequalities (VIs) with analytic center cutting plane methods (ACCPMs). A convex feasibility problem reformulation of the variational inequality is used; this reformulation applies to VIs defined with pseudo-monotone, single-valued mappings or with maximal monotone, multi-valued mappings.Two cutting plane methods are presented: the first is based on linear cuts while the second uses quadratic cuts. The first method, ACCPM-VI (linear cuts), requires mapping evaluations but no Jacobian evaluations; in fact, no differentiability assumption is needed. The cuts are placed at approximate analytic centers that are tracked with infeasible primal-dual Newton steps. Linear equality constraints may be present in the definition of the VI's set of reference, and are treated explicitly. The set of reference is assumed to be polyhedral, or is convex and iteratively approximated by polyhedra. Alongside of the sequence of analytic centers, another sequence of points is generated, based on convex combinations of the analytic centers. This latter sequence is observed to converge to a solution much faster than the former sequence.
The second method, ACCPM-VI (quadratic cuts), has cuts based on both mapping evaluations and Jacobian evaluations. The use of such a richer information set allows cuts that guide more accurately the sequence of analytic centers towards a solution. Mappings are assumed to be strongly monotone. However, Jacobian approximations, relying only on mapping evaluations, are observed to work very well in practice, so that differentiability of the mappings may not be required. There are two versions of the ACCPM-VI (quadratic cuts), that differ in the way a new analytic center is reached after the introduction of a cut. One version uses a curvilinear search followed by dual Newton centering steps. The search entails a full eigenvector-eigenvalue decomposition of a dense matrix of the order of the number of variables. The other version uses two line searches, primal-dual Newton steps, but no eigenvector-eigenvalue decomposition.
The algorithms described in this thesis were implemented in the M ATLAB environment. Numerical tests were performed on a variety of problems, some new and some traditional applications of variational inequalities.
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Liu, Wenbin. "Optimal shape design for systems governed by variational inequalities." Thesis, University of Leeds, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.293800.
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Sarbu, Lavinia. "Primal-dual active set methods for Allen-Cahn variational inequalities." Thesis, University of Sussex, 2010. http://sro.sussex.ac.uk/id/eprint/6267/.
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This thesis aims to introduce and analyse a primal-dual active set strategy for solving Allen-Cahn variational inequalities. We consider the standard Allen-Cahn equation with non-local constraints and a vector-valued Allen-Cahn equation with and without non-local constraints. Existence and uniqueness results are derived in a formulation involving Lagrange multipliers for local and non-local constraints. Local Convergence is shown by interpreting the primal-dual active set approach as a semi-smooth Newton method. Properties of the method are discussed and several numerical simulations in two and three space dimensions demonstrate its efficiency. In the second part of the thesis various applications of the Allen-Cahn equation are discussed. The non-local Allen-Cahn equation can be coupled with an elasticity equation to solve problems in structural topology optimisation. The model can be extended to handle multiple structures by using the vector-valued Allen-Cahn variational inequality with non-local constraints. Since many applications of the Allen-Cahn equation involve evolution of interfaces in materials an important extension of the standard Allen-Cahn model is to allow materials to exhibit anisotropic behaviour. We introduce an anisotropic version of the Allen-Cahn variational inequality and we show that it is possible to apply the primal-dual active set strategy efficiently to this model. Finally, the Allen-Cahn model is applied to problems in image processing, such as segmentation, denoising and inpainting. The primal-dual active set method proves exible and reliable for all the applications considered in this thesis.
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Swaters, Gordon Edwin. "On the stability and propagation of barotropic modons in slowly varying media." Thesis, [S.l. : s.n.], 1985. http://catalog.hathitrust.org/api/volumes/oclc/13002210.html.
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Chengalur-Smith, Indushobha Narayanan. "Variable sampling in multiparameter Shewhart charts." Diss., Virginia Polytechnic Institute and State University, 1989. http://hdl.handle.net/10919/54782.
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This dissertation deals with the use of Shewhart control charts, modified to have variable sampling intervals, to simultaneously monitor a set of parameters. Fixed sampling interval control charts are modified to utilize sampling intervals that vary depending on what is being observed from the data. Two problems are emphasized, namely, the simultaneous monitoring of the mean and the variance and the simultaneous monitoring of several means. For each problem, two basic strategies are investigated. One strategy uses separate control charts for each parameter. A second strategy uses a single statistic which combines the information in the entire sample and is sensitive to shifts in any of the parameters. Several variations on these two basic strategies are studied. Numerical studies investigate the optimal number of sampling intervals and the length of the sampling intervals to be used. Each procedure is compared to corresponding fixed interval procedures in terms of time and the number of samples taken to signal. The effect of correlation on multiple means charts is studied through simulation. For both problems, it is seen that the variable sampling interval approach is substantially more efficient than fixed interval procedures, no matter which strategy is used.Ph. D.
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Detournay, Sylvie. "Méthodes multigrilles pour les jeux stochastiques à deux joueurs et somme nulle, en horizon infini." Phd thesis, Ecole Polytechnique X, 2012. http://pastel.archives-ouvertes.fr/pastel-00762010.
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Dans cette thèse, nous proposons des algorithmes et présentons des résultats numériques pour la résolution de jeux répétés stochastiques, à deux joueurs et somme nulle dont l'espace d'état est de grande taille. En particulier, nous considérons la classe de jeux en information complète et en horizon infini. Dans cette classe, nous distinguons d'une part le cas des jeux avec gain actualisé et d'autre part le cas des jeux avec gain moyen. Nos algorithmes, implémentés en C, sont principalement basés sur des algorithmes de type itérations sur les politiques et des méthodes multigrilles. Ces algorithmes sont appliqués soit à des équations de la programmation dynamique provenant de problèmes de jeux à deux joueurs à espace d'états fini, soit à des discrétisations d'équations de type Isaacs associées à des jeux stochastiques différentiels. Dans la première partie de cette thèse, nous proposons un algorithme qui combine l'algorithme des itérations sur les politiques pour les jeux avec gain actualisé à des méthodes de multigrilles algébriques utilisées pour la résolution des systèmes linéaires. Nous présentons des résultats numériques pour des équations d'Isaacs et des inéquations variationnelles. Nous présentons également un algorithme d'itérations sur les politiques avec raffinement de grilles dans le style de la méthode FMG. Des exemples sur des inéquations variationnelles montrent que cet algorithme améliore de façon non négligeable le temps de résolution de ces inéquations. Pour le cas des jeux avec gain moyen, nous proposons un algorithme d'itération sur les politiques pour les jeux à deux joueurs avec espaces d'états et d'actions finis, dans le cas général multichaine (c'est-à-dire sans hypothèse d'irréductibilité sur les chaînes de Markov associées aux stratégies des deux joueurs). Cet algorithme utilise une idée développée dans Cochet-Terrasson et Gaubert (2006). Cet algorithme est basé sur la notion de projecteur spectral non-linéaire d'opérateurs de la programmation dynamique de jeux à un joueur (lequel est monotone et convexe). Nous montrons que la suite des valeurs et valeurs relatives satisfont une propriété de monotonie lexicographique qui implique que l'algorithme termine en temps fini. Nous présentons des résultats numériques pour des jeux discrets provenant d'une variante des jeux de Richman et sur des problèmes de jeux de poursuite. Finalement, nous présentons de nouveaux algorithmes de multigrilles algébriques pour la résolution de systèmes linéaires singuliers particuliers. Ceux-ci apparaissent, par exemple, dans l'algorithme d'itérations sur les politiques pour les jeux stochastiques à deux joueurs et somme nulle avec gain moyen, décrit ci-dessus. Nous introduisons également une nouvelle méthode pour la recherche de mesures invariantes de chaînes de Markov irréductibles basée sur une approche de contrôle stochastique. Nous présentons un algorithme qui combine les itérations sur les politiques d'Howard et des itérations de multigrilles algébriques pour les systèmes linéaires singuliers.
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Klein, Nicole [Verfasser]. "Consistent FE-analysis of elliptic variational inequalities / Nicole Klein." Siegen : Universitätsbibliothek der Universität Siegen, 2013. http://d-nb.info/1034425897/34.
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Kulieva, Gulchehra. "Some special problems in elliptic and parabolic variational inequalities." Licentiate thesis, Luleå : Department of Mathematics, Luleå University of Technology, 2006. http://epubl.ltu.se/1402-1757/2006/77/.
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Butz, Martin [Verfasser], and Harald [Akademischer Betreuer] Garcke. "Computational methods for Cahn-Hilliard variational inequalities / Martin Butz. Betreuer: Harald Garcke." Regensburg : Universitätsbibliothek Regensburg, 2012. http://d-nb.info/1023362139/34.
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Glas, Silke [Verfasser]. "Noncoercive and parabolic variational inequalities : analysis, applications and model reduction / Silke Glas." Ulm : Universität Ulm, 2018. http://d-nb.info/1166756882/34.
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Tietz, Christoph [Verfasser], Siegfried Gutachter] Carl, and Dumitru [Gutachter] [Motreanu. "Variational inequalities with multivalued bifunctions / Christoph Tietz ; Gutachter: Siegfried Carl, Dumitru Motreanu." Halle (Saale) : Universitäts- und Landesbibliothek Sachsen-Anhalt, 2019. http://d-nb.info/1210731800/34.
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Tietz, Christoph [Verfasser], Siegfried [Gutachter] Carl, and Dumitru [Gutachter] Motreanu. "Variational inequalities with multivalued bifunctions / Christoph Tietz ; Gutachter: Siegfried Carl, Dumitru Motreanu." Halle (Saale) : Universitäts- und Landesbibliothek Sachsen-Anhalt, 2019. http://d-nb.info/1210731800/34.
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Köhler, Karoline Sophie. "On efficient a posteriori error analysis for variational inequalities." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2016. http://dx.doi.org/10.18452/17635.
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Effiziente und zuverlässige a posteriori Fehlerabschätzungen sind eine Hauptzutat für die effiziente numerische Berechnung von Lösungen zu Variationsungleichungen durch die Finite-Elemente-Methode. Die vorliegende Arbeit untersucht zuverlässige und effiziente Fehlerabschätzungen für beliebige Finite-Elemente-Methoden und drei Variationsungleichungen, nämlich dem Hindernisproblem, dem Signorini Problem und dem Bingham Problem in zwei Raumdimensionen. Die Fehlerabschätzungen hängen vom zum Problem gehörenden Lagrange Multiplikator ab, der eine Verbindung zwischen der Variationsungleichung und dem zugehörigen linearen Problem darstellt. Effizienz und Zuverlässigkeit werden bezüglich eines totalen Fehlers gezeigt. Die Fehleranschätzungen fordern minimale Regularität. Die Approximation der exakten Lösung erfüllt die Dirichlet Randbedingungen und die Approximation des Lagrange Multiplikators ist nicht-positiv im Falle des Hindernis- und Signoriniproblems, und hat Betrag kleiner gleich 1 für das Bingham Problem. Dieses allgemeine Vorgehen ermöglicht das Einbinden nicht-exakter diskreter Lösungen, welche im Kontext dieser Ungleichungen auftreten. Aus dem Blickwinkel der Anwendungen ist Effizienz und Zuverlässigkeit im Bezug auf den Fehler der primalen Variablen in der Energienorm von großem Interesse. Solche Abschätzungen hängen von der Wahl eines effizienten diskreten Lagrange Multiplikators ab. Im Falle des Hindernis- und Signorini Problems werden postive Beispiele für drei Finite-Elemente Methoden, der konformen Courant Methode, der nicht-konformen Crouzeix-Raviart Methode und der gemischten Raviart-Thomas Methode niedrigster Ordnung hergeleitet. Partielle Resultate liegen im Fall des Bingham Problems vor. Numerischer Experimente heben die theoretischen Ergebnisse hervor und zeigen Effizienz und Zuverlässigkeit. Die numerischen Tests legen nahe, dass der aus den Abschätzungen resultierende adaptive Algorithmus mit optimaler Konvergenzrate konvergiert.Efficient and reliable a posteriori error estimates are a key ingredient for the efficient numerical computation of solutions for variational inequalities by the finite element method. This thesis studies such reliable and efficient error estimates for arbitrary finite element methods and three representative variational inequalities, namely the obstacle problem, the Signorini problem, and the Bingham problem in two space dimensions. The error estimates rely on a problem connected Lagrange multiplier, which presents a connection between the variational inequality and the corresponding linear problem. Reliability and efficiency are shown with respect to some total error. Reliability and efficiency are shown under minimal regularity assumptions. The approximation to the exact solution satisfies the Dirichlet boundary conditions, and an approximation of the Lagrange multiplier is non-positive in the case of the obstacle and Signorini problem and has an absolute value smaller than 1 for the Bingham flow problem. These general assumptions allow for reliable and efficient a posteriori error analysis even in the presence of inexact solve, which naturally occurs in the context of variational inequalities. From the point of view of the applications, reliability and efficiency with respect to the error of the primal variable in the energy norm is of great interest. Such estimates depend on the efficient design of a discrete Lagrange multiplier. Affirmative examples of discrete Lagrange multipliers are presented for the obstacle and Signorini problem and three different first-order finite element methods, namely the conforming Courant, the non-conforming Crouzeix-Raviart, and the mixed Raviart-Thomas FEM. Partial results exist for the Bingham flow problem. Numerical experiments highlight the theoretical results, and show efficiency and reliability. The numerical tests suggest that the resulting adaptive algorithms converge with optimal convergence rates.
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Zhang, Zhenying Verfasser], Karen [Akademischer Betreuer] [Veroy, and Michael [Akademischer Betreuer] Herty. "Certified reduced basis method for variational inequalities / Zhenying Zhang ; Karen Paula Veroy-Grepl, Michael Matthias Herty." Aachen : Universitätsbibliothek der RWTH Aachen, 2016. http://d-nb.info/1156922216/34.
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Zhang, Zhenying [Verfasser], Karen [Akademischer Betreuer] Veroy, and Michael [Akademischer Betreuer] Herty. "Certified reduced basis method for variational inequalities / Zhenying Zhang ; Karen Paula Veroy-Grepl, Michael Matthias Herty." Aachen : Universitätsbibliothek der RWTH Aachen, 2016. http://d-nb.info/1156922216/34.
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Kober, Bernhard [Verfasser], and Gerhard [Akademischer Betreuer] Starke. "Stress-based finite element methods for variational inequalities in contact mechanics / Bernhard Kober ; Betreuer: Gerhard Starke." Duisburg, 2020. http://d-nb.info/120400417X/34.
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Betz, Thomas [Verfasser], Christian [Akademischer Betreuer] Meyer, and Roland [Gutachter] Herzog. "Optimal control of two variational inequalities arising in solid mechanics / Thomas Betz. Betreuer: Christian Meyer. Gutachter: Roland Herzog." Dortmund : Universitätsbibliothek Dortmund, 2015. http://d-nb.info/1102159395/34.
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Betz, Thomas [Verfasser], Christian Akademischer Betreuer] Meyer, and Roland [Gutachter] [Herzog. "Optimal control of two variational inequalities arising in solid mechanics / Thomas Betz. Betreuer: Christian Meyer. Gutachter: Roland Herzog." Dortmund : Universitätsbibliothek Dortmund, 2015. http://nbn-resolving.de/urn:nbn:de:101:1-201605302603.
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Rösel, Simon. "Approximation of nonsmooth optimization problems and elliptic variational inequalities with applications to elasto-plasticity." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2017. http://dx.doi.org/10.18452/17778.
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Optimierungsprobleme und Variationsungleichungen über Banach-Räumen stellen Themen von substantiellem Interesse dar, da beide Problemklassen einen abstrakten Rahmen für zahlreiche Anwendungen aus verschiedenen Fachgebieten stellen. Nach einer Einführung in Teil I werden im zweiten Teil allgemeine Approximationsmethoden, einschließlich verschiedener Diskretisierungs- und Regularisierungsansätze, zur Lösung von nichtglatten Variationsungleichungen und Optimierungsproblemen unter konvexen Restriktionen vorgestellt. In diesem allgemeinen Rahmen stellen sich gewisse Dichtheitseigenschaften der konvexen zulässigen Menge als wichtige Voraussetzungen für die Konsistenz einer abstrakten Klasse von Störungen heraus. Im Folgenden behandeln wir vor allem Restriktionsmengen in Sobolev-Räumen, die durch eine punktweise Beschränkung an den Funktionswert definiert werden. Für diesen Restriktionstyp werden verschiedene Dichtheitsresultate bewiesen. In Teil III widmen wir uns einem quasi-statischen Kontaktproblem der Elastoplastizität mit Härtung. Das entsprechende zeit-diskretisierte Problem kann als nichtglattes, restringiertes Minimierungsproblem betrachtet werden. Zur Lösung wird eine Pfadverfolgungsmethode auf Basis des verallgemeinerten Newton-Verfahrens entwickelt, dessen Teilprobleme lokal superlinear und gitterunabhängig lösbar sind. Teil III schließt mit verschiedenen numerischen Beispielen. Der letzte Teil der Arbeit ist der quasi-statischen, perfekten Plastizität gewidmet. Auf Basis des primalen Problems der perfekten Plastizität leiten wir eine reduzierte Formulierung her, die es erlaubt, das primale Problem als Fenchel-dualisierte Form des klassischen zeit-diskretisierten Spannungsproblems zu verstehen. Auf diese Weise werden auch neue Optimalitätsbedingungen hergeleitet. Zur Lösung des Problems stellen wir eine modifizierte Form der viskoplastischen Regularisierung vor und beweisen die Konvergenz dieses neuen Regularisierungsverfahrens.Optimization problems and variational inequalities over Banach spaces are subjects of paramount interest since these mathematical problem classes serve as abstract frameworks for numerous applications. Solutions to these problems usually cannot be determined directly. Following an introduction, part II presents several approximation methods for convex-constrained nonsmooth variational inequality and optimization problems, including discretization and regularization approaches. We prove the consistency of a general class of perturbations under certain density requirements with respect to the convex constraint set. We proceed with the study of pointwise constraint sets in Sobolev spaces, and several density results are proven. The quasi-static contact problem of associative elasto-plasticity with hardening at small strains is considered in part III. The corresponding time-incremental problem can be equivalently formulated as a nonsmooth, constrained minimization problem, or, as a mixed variational inequality problem over the convex constraint. We propose an infinite-dimensional path-following semismooth Newton method for the solution of the time-discrete plastic contact problem, where each path-problem can be solved locally at a superlinear rate of convergence with contraction rates independent of the discretization. Several numerical examples support the theoretical results. The last part is devoted to the quasi-static problem of perfect (Prandtl-Reuss) plasticity. Building upon recent developments in the study of the (incremental) primal problem, we establish a reduced formulation which is shown to be a Fenchel predual problem of the corresponding stress problem. This allows to derive new primal-dual optimality conditions. In order to solve the time-discrete problem, a modified visco-plastic regularization is proposed, and we prove the convergence of this new approximation scheme.
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Gaevskaya, Alexandra [Verfasser], and Ronald H. W. [Akademischer Betreuer] Hoppe. "Adaptive finite elements for optimally controlled elliptic variational inequalities of obstacle type / Alexandra Gaevskaya. Betreuer: Ronald H. W. Hoppe." Augsburg : Universität Augsburg, 2013. http://d-nb.info/1077703309/34.
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Köhler, Karoline Sophie [Verfasser], Carsten [Gutachter] Carstensen, Andreas [Gutachter] Schröder, and Neela [Gutachter] Nataraj. "On efficient a posteriori error analysis for variational inequalities / Karoline Sophie Köhler ; Gutachter: Carsten Carstensen, Andreas Schröder, Neela Nataraj." Berlin : Mathematisch-Naturwissenschaftliche Fakultät, 2016. http://d-nb.info/1119861543/34.
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Harms, Nadja [Verfasser], and Christian [Gutachter] Kanzow. "Primal and Dual Gap Functions for Generalized Nash Equilibrium Problems and Quasi-Variational Inequalities / Nadja Harms. Gutachter: Christian Kanzow." Würzburg : Universität Würzburg, 2014. http://d-nb.info/1102828769/34.
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Abbas, Lamia. "Inégalités de Landau-Kolmogorov dans des espaces de Sobolev." Phd thesis, INSA de Rouen, 2012. http://tel.archives-ouvertes.fr/tel-00776349.
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Ce travail est dédié à l'étude des inégalités de type Landau-Kolmogorov en normes L2. Les mesures utilisées sont celles d'Hermite, de Laguerre-Sonin et de Jacobi. Ces inégalités sont obtenues en utilisant une méthode variationnelle. Elles font intervenir la norme d'un polynômes p et celles de ces dérivées. Dans un premier temps, on s'intéresse aux inégalités en une variable réelle qui font intervenir un nombre quelconque de normes. Les constantes correspondantes sont prises dans le domaine où une certaine forme bilinéaire est définie positive. Ensuite, on généralise ces résultats aux polynômes à plusieurs variables réelles en utilisant le produit tensoriel dans L2 et en faisant intervenir au plus les dérivées partielles secondes. Pour les mesures d'Hermite et de Laguerre-Sonin, ces inégalités sont étendues à toutes les fonctions d'un espace de Sobolev. Pour la mesure de Jacobi on donne des inégalités uniquement pour les polynômes d'un degré fixé par rapport à chaque variable.
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Christof, Constantin [Verfasser], Christian [Akademischer Betreuer] Meyer, and Martin [Gutachter] Brokate. "Sensitivity analysis of elliptic variational inequalities of the first and the second kind / Constantin Christof ; Gutachter: Martin Brokate ; Betreuer: Christian Meyer." Dortmund : Universitätsbibliothek Dortmund, 2018. http://d-nb.info/1163452335/34.
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Rösel, Simon [Verfasser], Michael [Gutachter] Hintermüller, Roland [Gutachter] Glowinski, and Batmanathan Dayanand [Gutachter] Reddy. "Approximation of nonsmooth optimization problems and elliptic variational inequalities with applications to elasto-plasticity / Simon Rösel ; Gutachter: Michael Hintermüller, Roland Glowinski, Batmanathan Dayanand Reddy." Berlin : Mathematisch-Naturwissenschaftliche Fakultät, 2017. http://d-nb.info/1135241651/34.
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Sigstam, Kibret Negussie. "Optimization and estimation of solutions of Riccati equations /." Uppsala : Matematiska institutionen, Univ. [distributör], 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4288.
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Strogies, Nikolai. "Optimization of nonsmooth first order hyperbolic systems." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2016. http://dx.doi.org/10.18452/17633.
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Wir betrachten Optimalsteuerungsprobleme, die von partiellen Differentialgleichungen beziehungsweise Variationsungleichungen mit Differentialoperatoren erster Ordnung abhängen. Wir führen die Reformulierung eines Tagebauplanungsproblems, das auf stetigen Funktionen beruht, ein. Das Resultat ist ein Optimalsteuerungsproblem für Viskositätslösungen einer Eikonalgleichung. Die Existenz von Lösungen dieses und bestimmter Hilfsprobleme, die von semilinearen PDG‘s mit künstlicher Viskosität abhängen, wird bewiesen, Stationaritätsbedingungen hergeleitet und ein schwaches Konsistenzresultat für stationäre Punkte präsentiert. Des Weiteren betrachten wir Optimalsteuerungsprobleme, die von stationären Variationsungleichungen erster Art mit linearen Differentialoperatoren erster Ordnung abhängen. Wir diskutieren Lösbarkeit und Stationaritätskonzepte für diese Probleme. Für letzteres vergleichen wir Ergebnisse, die entweder durch die Anwendung von Penalisierungs- und Regularisierungsansätzen direkt auf Ebene von Differentialoperatoren erster Ordnung oder als Grenzwertprozess von Stationaritätssystemen für viskositätsregularisierte Optimalsteuerungsprobleme unter passenden Annahmen erhalten werden. Um die Konsistenz von ursprünglichem und regularisierten Problemen zu sichern, wird ein bekanntes Ergebnis für Lösungen von VU’s mit degeneriertem Differentialoperator erweitert. In beiden Fällen ist die erhaltene Stationarität schwächer als W-stationarität. Die theoretischen Ergebnisse werden anhand numerischer Beispiele verifiziert. Wir erweitern diese Ergebnisse auf Optimalsteuerungsprobleme bezüglich zeitabhängiger VU’s mit Differentialoperatoren erster Ordnung. Hierfür wird die Existenz von Lösungen bewiesen und erneut ein Stationaritätssystem mit Hilfe verschwindender Viskositäten unter bestimmten Beschränktheitsannahmen hergeleitet. Die erhaltenen Ergebnisse werden anhand von numerischen Beispielen verifiziert.We consider problems of optimal control subject to partial differential equations and variational inequality problems with first order differential operators. We introduce a reformulation of an open pit mine planning problem that is based on continuous functions. The resulting formulation is a problem of optimal control subject to viscosity solutions of a partial differential equation of Eikonal Type. The existence of solutions to this problem and auxiliary problems of optimal control subject to regularized, semilinear PDE’s with artificial viscosity is proven. For the latter a first order optimality condition is established and a mild consistency result for the stationary points is proven. Further we study certain problems of optimal control subject to time-independent variational inequalities of the first kind with linear first order differential operators. We discuss solvability and stationarity concepts for such problems. In the latter case, we compare the results obtained by either utilizing penalization-regularization strategies directly on the first order level or considering the limit of systems for viscosity-regularized problems under suitable assumptions. To guarantee the consistency of the original and viscosity-regularized problems of optimal control, we extend known results for solutions to variational inequalities with degenerated differential operators. In both cases, the resulting stationarity concepts are weaker than W-stationarity. We validate the theoretical findings by numerical experiments for several examples. Finally, we extend the results from the time-independent to the case of problems of optimal control subject to VI’s with linear first order differential operators that are time-dependent. After establishing the existence of solutions to the problem of optimal control, a stationarity system is derived by a vanishing viscosity approach under certain boundedness assumptions and the theoretical findings are validated by numerical experiments.
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Merdon, Christian. "Aspects of guaranteed error control in computations for partial differential equations." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2013. http://dx.doi.org/10.18452/16818.
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Diese Arbeit behandelt garantierte Fehlerkontrolle für elliptische partielle Differentialgleichungen anhand des Poisson-Modellproblems, des Stokes-Problems und des Hindernisproblems. Hierzu werden garantierte obere Schranken für den Energiefehler zwischen exakter Lösung und diskreten Finite-Elemente-Approximationen erster Ordnung entwickelt. Ein verallgemeinerter Ansatz drückt den Energiefehler durch Dualnormen eines oder mehrerer Residuen aus. Hinzu kommen berechenbare Zusatzterme, wie Oszillationen der gegebenen Daten, mit expliziten Konstanten. Für die Abschätzung der Dualnormen der Residuen existieren viele verschiedene Techniken. Diese Arbeit beschäftigt sich vorrangig mit Equilibrierungsschätzern, basierend auf Raviart-Thomas-Elementen, welche effiziente garantierte obere Schranken ermöglichen. Diese Schätzer werden mit einem Postprocessing-Verfahren kombiniert, das deren Effizienz mit geringem zusätzlichen Rechenaufwand deutlich verbessert. Nichtkonforme Finite-Elemente-Methoden erzeugen zusätzlich ein Inkonsistenzresiduum, dessen Dualnorm mit Hilfe diverser konformer Approximationen abgeschätzt wird. Ein Nebenaspekt der Arbeit betrifft den expliziten residuen-basierten Fehlerschätzer, der für gewöhnlich optimale und leicht zu berechnende Verfeinerungsindikatoren für das adaptive Netzdesign liefert, aber nur schlechte garantierte obere Schranken. Eine neue Variante, die auf den equilibrierten Flüssen des Luce-Wohlmuth-Fehlerschätzers basiert, führt zu stark verbesserten Zuverlässigkeitskonstanten. Eine Vielzahl numerischer Experimente vergleicht alle implementierten Fehlerschätzer und zeigt, dass effiziente und garantierte Fehlerkontrolle in allen vorliegenden Modellproblemen möglich ist. Insbesondere zeigt ein Modellproblem, wie die Fehlerschätzer erweitert werden können, um auch auf Gebieten mit gekrümmten Rändern garantierte obere Schranken zu liefern.This thesis studies guaranteed error control for elliptic partial differential equations on the basis of the Poisson model problem, the Stokes equations and the obstacle problem. The error control derives guaranteed upper bounds for the energy error between the exact solution and different finite element discretisations, namely conforming and nonconforming first-order approximations. The unified approach expresses the energy error by dual norms of one or more residuals plus computable extra terms, such as oscillations of the given data, with explicit constants. There exist various techniques for the estimation of the dual norms of such residuals. This thesis focuses on equilibration error estimators based on Raviart-Thomas finite elements, which permit efficient guaranteed upper bounds. The proposed postprocessing in this thesis considerably increases their efficiency at almost no additional computational costs. Nonconforming finite element methods also give rise to a nonconsistency residual that permits alternative treatment by conforming interpolations. A side aspect concerns the explicit residual-based error estimator that usually yields cheap and optimal refinement indicators for adaptive mesh refinement but not very sharp guaranteed upper bounds. A novel variant of the residual-based error estimator, based on the Luce-Wohlmuth equilibration design, leads to highly improved reliability constants. A large number of numerical experiments compares all implemented error estimators and provides evidence that efficient and guaranteed error control in the energy norm is indeed possible in all model problems under consideration. Particularly, one model problem demonstrates how to extend the error estimators for guaranteed error control on domains with curved boundary.
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Wang, Hao. "Incremental sheet forming process : control and modelling." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:a80370f5-2287-4c6b-b7a4-44f06211564f.
Full textAbstract:
Incremental Sheet Forming (ISF) is a progressive metal forming process, where the deformation occurs locally around the point of contact between a tool and the metal sheet. The final work-piece is formed cumulatively by the movements of the tool, which is usually attached to a CNC milling machine. The ISF process is dieless in nature and capable of producing different parts of geometries with a universal tool. The tooling cost of ISF can be as low as 5–10% compared to the conventional sheet metal forming processes. On the laboratory scale, the accuracy of the parts created by ISF is between ±1.5 mm and ±3mm. However, in order for ISF to be competitive with a stamping process, an accuracy of below ±1.0 mm and more realistically below ±0.2 mm would be needed. In this work, we first studied the ISF deformation process by a simplified phenomenal linear model and employed a predictive controller to obtain an optimised tool trajectory in the sense of minimising the geometrical deviations between the targeted shape and the shape made by the ISF process. The algorithm is implemented at a rig in Cambridge University and the experimental results demonstrate the ability of the model predictive controller (MPC) strategy. We can achieve the deviation errors around ±0.2 mm for a number of simple geometrical shapes with our controller. The limitations of the underlying linear model for a highly nonlinear problem lead us to study the ISF process by a physics based model. We use the elastoplastic constitutive relation to model the material law and the contact mechanics with Signorini’s type of boundary conditions to model the process, resulting in an infinite dimensional system described by a partial differential equation. We further developed the computational method to solve the proposed mathematical model by using an augmented Lagrangian method in function space and discretising by finite element method. The preliminary results demonstrate the possibility of using this model for optimal controller design.
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Eriksson, Jonatan. "On the pricing equations of some path-dependent options." Doctoral thesis, Uppsala : Department of Mathematics, Univ. [distributör], 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-6329.
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Ponomarenko, Andrej. "Lösungsmethoden für Variationsungleichungen." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2003. http://dx.doi.org/10.18452/14841.
Full textAbstract:
Zusammenfassung Diese Arbeit ist ein Versuch, verschiedene klassische und neuere Methodender glatten bzw. nichtglatten Optimierung zu verallgemeinern und in ihrem Zusammenhang darzustellen. Als Hauptinstrument erweist sich dabei die sogenannte verallgemeinerte Kojima-Funktion. Neben reichlichen Beispielen setzen wir einen besonderen Akzent auf die Betrachtung von Variationsungleichungen, Komplementaritaetsaufgaben und der Standartaufgabeder mathematischen Programmierung. Unter natuerlichen Voraussetzungen an diese Probleme kann man u.a. Barriere-, Straf- und SQP-Typ-Methoden, die auf Newton-Verfahrenbasieren, aber auch Modelle, die sogenannte NCP-Funktionen benutzen, mittelsspezieller Stoerungen der Kojima-Funktion exakt modellieren. Daneben werdendurch explizite und natuerliche Wahl der Stoerungsparameter auch neue Methoden dieser Arten vorgeschlagen. Die Vorteile solcher Modellierungsind ueberzeugend vor allem wegen der direkt moeglichen (auf Stabilitaetseigenschaften der Kojima-Gleichung beruhendenden)Loesungsabschaetzungen und weil die entsprechenden Nullstellen ziemlich einfach als Loesungen bekannter Ersatzprobleme interpretiert werden koennen. Ein weiterer Aspekt der Arbeit besteht in der genaueren Untersuchungder "nichtglatten Faelle". Hier wird die Theorie von verschiedenen verallgemeinerten Ableitungen und dadurch entstehenden verallgemeinerten Newton-Verfahren, die im Buch "Nonsmooth Equations in Optimization" von B. Kummer und D. Klatte vorgeschlagen und untersucht wurde, intensiv benutzt. Entscheidend ist dabei, dass die benutzten verallgemeinerten Ableitungen auch praktisch angewandt werden koennen, da man sie exakt ausrechnen kann.This work attempts to generalize various classical and new methods of smooth or nonsmooth optimization and to show them in their interrelation. The main tool for doing this is the so-called generalized Kojima-function. In addition to numerous examples we specialy emphasize the consideration of variational inequalities, complementarity problems and the standard problem of mathematical programming. Under natural assumptions on these problems we can model e.g. barrier-, penalty-, and SQP-Type-methods basing on Newton methods, and also methods using the so-called NCP-function exactly by means of special perturbations of the Kojima-function. Furthermore, by the explicit and natural choice of the perturbation parameters new methods of these kinds are introduced. The benefit of such a modelling is obvious, first of all due to the direct solution estimation (basing on stability properties of the Kojima-equation) and because the corresponding zeros can easily be interpreted as solutions of known subproblems. A further aspect considered in this paper is the detailed investigation of "nonsmooth cases". The theory of various generalized derivatives and resulting generalized Newton methods, which is introduced and investigated in the book "Nonsmooth Equations in Optimization" of B. Kummer and D. Klatte, is intensely used here. The crucial point is the applicability of the used generalized derivatives in practice, since they can be calculated exactly.
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"On merit functions and error bounds for variational inequality problem." 2004. http://library.cuhk.edu.hk/record=b5892104.
Full textAbstract:
Li Guo-Yin.Thesis (M.Phil.)--Chinese University of Hong Kong, 2004.
Includes bibliographical references (leaves 105-107).
Abstracts in English and Chinese.
Chapter 1 --- Introduction --- p.1
Chapter 1.1 --- Examples for the variational inequality problem --- p.2
Chapter 1.2 --- Approaches for variational inequality problem --- p.7
Chapter 1.3 --- Error bounds results for variational inequality problem --- p.8
Chapter 1.4 --- Organization --- p.9
Chapter 2 --- Solution Theory --- p.11
Chapter 2.1 --- "Elementary Convex Analysis, Nonsmooth Analysis and Degree theory" --- p.11
Chapter 2.1.1 --- Elementary Convex Analysis --- p.11
Chapter 2.1.2 --- Elementary Nonsmooth Analysis --- p.16
Chapter 2.1.3 --- Degree Theory --- p.18
Chapter 2.2 --- Existence and Uniqueness Theory --- p.24
Chapter 3 --- Merit Functions for variational inequalities problem --- p.36
Chapter 3.1 --- Regularized gap function --- p.38
Chapter 3.2 --- D-gap function --- p.44
Chapter 3.3 --- Generalized Regularize gap function and Generalized D-gap function --- p.61
Chapter 4 --- Error bound results for the merit functions --- p.74
Chapter 4.1 --- Error bound results for Regularized gap function --- p.77
Chapter 4.2 --- Error bound results for D-gap function --- p.78
Chapter 4.3 --- Error bound results for Generalized Regularized gap function --- p.92
Chapter 4.4 --- Error bound results for Generalized D-gap function --- p.93
Bibliography --- p.105
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"Solving variational inequalities and related problems using recurrent neural networks." Thesis, 2007. http://library.cuhk.edu.hk/record=b6074418.
Full textAbstract:
During the past two decades, numerous recurrent neural networks (RNNs) have been proposed for solving VIs and related problems. However, first, the theories of many emerging RNNs have not been well founded yet; and their capabilities have been underestimated. Second, these RNNs have limitations in handling some types of problems. Third, it is certainly not true that these RNNs are best choices for solving all problems, and new network models with more favorable characteristics could be devised for solving specific problems.In the research, the above issues are extensively explored from dynamic system perspective, which leads to the following major contributions. On one hand, many new capabilities of some existing RNNs have been revealed for solving VIs and related problems. On the other hand, several new RNNs have been invented for solving some types of these problems. The contributions are established on the following facts. First, two existing RNNs, called TLPNN and PNN, are found to be capable of solving pseudomonotone VIs and related problems with simple bound constraints. Second, many more stability results are revealed for an existing RNN, called GPNN, for solving GVIs with simple bound constraints, and it is then extended to solve linear VIs (LVIs) and generalized linear VIs (GLVIs) with polyhedron constraints. Third, a new RNN, called IDNN, is proposed for solving a special class of quadratic programming problems which features lower structural complexity compared with existing RNNs. Fourth, some local convergence results of an existing RNN, called EPNN, for nonconvex optimization are obtained, and two variants of the network by incorporating two augmented Lagrangian function techniques are proposed for seeking Karush-Kuhn-Tucker (KKT) points, especially local optima, of the problems.
Variational inequality (VI) can be viewed as a natural framework for unifying the treatment of equilibrium problems, and hence has applications across many disciplines. In addition, many typical problems are closely related to VI, including general VI (GVI), complementarity problem (CP), generalized CP (GCP) and optimization problem (OP).
Hu, Xiaolin.
"July 2007."
Adviser: Jun Wang.
Source: Dissertation Abstracts International, Volume: 69-02, Section: B, page: 1102.
Thesis (Ph.D.)--Chinese University of Hong Kong, 2007.
Includes bibliographical references (p. 193-207).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Abstract in English and Chinese.
School code: 1307.
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"On merit functions, error bounds, minimizing and stationary sequences for nonsmooth variational inequality problems." Thesis, 2005. http://library.cuhk.edu.hk/record=b6074106.
Full textAbstract:
First, we study the associated regularized gap functions and the D-gap functions and compute their Clarke-Rockafellar directional derivatives and the Clarke generalized gradients. Second, using these tools and extending the works of Fukushima and Pang (who studied the case when F is smooth), we present results on the relationship between minimizing sequences and stationary sequences of the D-gap functions, regardless the existence of solutions of (VIP). Finally, as another application, we show that, under the strongly monotonicity assumption, the regularized gap functions have fractional exponent error bounds, and thereby we provide an algorithm of Armijo type to solve the (VIP).In this thesis, we investigate a nonsmooth variational inequality problem (VIP) defined by a locally Lipschitz function F which is not necessarily differentiable or monotone on its domain which is a closed convex set in an Euclidean space.
Tan Lulin.
"December 2005."
Adviser: Kung Fu Ng.
Source: Dissertation Abstracts International, Volume: 67-11, Section: B, page: 6444.
Thesis (Ph.D.)--Chinese University of Hong Kong, 2005.
Includes bibliographical references (p. 79-84) and index.
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Abstracts in English and Chinese.
School code: 1307.
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"On asymptotic analysis and error bounds in optimization." 2001. http://library.cuhk.edu.hk/record=b6073326.
Full textAbstract:
He Yiran.Includes index.
Thesis (Ph.D.)--Chinese University of Hong Kong, 2001.
Includes bibliographical references (p. 74-80) and index..
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Mode of access: World Wide Web.
Abstracts in English and Chinese.
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Pan, Jie. "Variational inequalities in the modelling and computation of spatial economic equilibria: Structural reformulations and the method of multipliers." 1992. https://scholarworks.umass.edu/dissertations/AAI9233126.
Full textAbstract:
Variational inequalities have been used to study problems involving partial differential equations with unilateral constraints, such as free-boundary problems. They have also gained much recent interest in the field of operations research, particularly in the study of competitive equilibrium problems. The main focus of this work is to develop efficient algorithms for the computation of large-scale economic equilibria under weaker conditions than those considered previously. The prototype that we use in the analysis is the spatial market equilibrium system with direct price functions. We take advantage of the special structure of the variational inequalities, hence reformulate the problems, via a dual approach of Mosco and a linear algebra argument, as multivalued equations involving two maximal monotone operators. We then apply a relaxed proximal point method with variable parameters to the new formulation. In finite dimensions, we prove that the splitting sequences so generated are convergent to the equilibrium and the Lagrange multipliers, respectively. We also develop variational inequality formulations for migration networks and spatial market systems with goaling constraints. Based on the given economic equilibrium conditions, we establish the corresponding variational inequality formulations. In the second case, we provide direct equivalence proof that is motivated by the governing economic conditions. Essentially, we establish that the economic conditions are the dual forms of the corresponding variational inequalities. By applying the theory of variational inequalities, we then study the qualitative properties of these spatial equilibrium systems. In particular, we show the existence and uniqueness of the equilibrium in each case, assuming some monotonicity conditions that can be interpreted economically. We then apply the above numerical scheme to the variational inequality formulations of spatial equilibrium systems. As a result, we obtain a class of methods of multipliers for the computation of the studied economic equilibria. The methods so derived have an important feature that they require only monotonicity instead of strong monotonicity of supply price functions and demand price functions. They still require strong monotonicity of transaction cost functions. Finally, since they are splitting algorithms, they are suitable for decomposing large-scale problems. With a sequence of penalty parameters being set properly, each split part can then be computed sequentially or parallelly.
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Porwal, Kamana. "A Posteriori Error Analysis of Discontinuous Galerkin Methods for Elliptic Variational Inequalities." Thesis, 2014. http://hdl.handle.net/2005/3107.
Full textAbstract:
The main emphasis of this thesis is to study a posteriori error analysis of discontinuous Galerkin (DG) methods for the elliptic variational inequalities. The DG methods have become very pop-ular in the last two decades due to its nature of handling complex geometries, allowing irregular meshes with hanging nodes and different degrees of polynomial approximation on different ele-ments. Moreover they are high order accurate and stable methods. Adaptive algorithms refine the mesh locally in the region where the solution exhibits irregular behaviour and a posteriori error estimates are the main ingredients to steer the adaptive mesh refinement.
The solution of linear elliptic problem exhibits singularities due to change in boundary con-ditions, irregularity of coefficients and reentrant corners in the domain. Apart from this, the solu-tion of variational inequality exhibits additional irregular behaviour due to occurrence of the free boundary (the part of the domain which is a priori unknown and must be found as a component of the solution). In the lack of full elliptic regularity of the solution, uniform refinement is inefficient and it does not yield optimal convergence rate. But adaptive refinement, which is based on the residuals ( or a posteriori error estimator) of the problem, enhance the efficiency by refining the mesh locally and provides the optimal convergence. In this thesis, we derive a posteriori error estimates of the DG methods for the elliptic variational inequalities of the first kind and the second kind.
This thesis contains seven chapters including an introductory chapter and a concluding chap-ter. In the introductory chapter, we review some fundamental preliminary results which will be used in the subsequent analysis. In Chapter 2, a posteriori error estimates for a class of DG meth-ods have been derived for the second order elliptic obstacle problem, which is a prototype for elliptic variational inequalities of the first kind. The analysis of Chapter 2 is carried out for the general obstacle function therefore the error estimator obtained therein involves the min/max func-tion and hence the computation of the error estimator becomes a bit complicated. With a mild assumption on the trace of the obstacle, we have derived a significantly simple and easily com-putable error estimator in Chapter 3. Numerical experiments illustrates that this error estimator indeed behaves better than the error estimator derived in Chapter 2. In Chapter 4, we have carried out a posteriori analysis of DG methods for the Signorini problem which arises from the study of the frictionless contact problems. A nonlinear smoothing map from the DG finite element space to conforming finite element space has been constructed and used extensively, in the analysis of Chapter 2, Chapter 3 and Chapter 4. Also, a common property shared by all DG methods allows us to carry out the analysis in unified setting. In Chapter 5, we study the C0 interior penalty method for the plate frictional contact problem, which is a fourth order variational inequality of the second kind. In this chapter, we have also established the medius analysis along with a posteriori analy-sis. Numerical results have been presented at the end of every chapter to illustrate the theoretical results derived in respective chapters. We discuss the possible extension and future proposal of the work presented in the Chapter 6. In the last chapter, we have documented the FEM codes used in the numerical experiments.
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Sedebo, Getachew Temesgen. "The dynamics of a forced and damped two degrees of freedom spring pendulum." 2013. http://encore.tut.ac.za/iii/cpro/DigitalItemViewPage.external?sp=1001091.
Full textAbstract:
M. Tech. Mathematical Technology.Discusses the main problems in terms of how to derive mathematical models for a free, a forced and a damped spring pendulum and determining numerical solutions using a computer algebra system (CAS), because exact analytical solutions are not obvious. Hence this mini-dissertation mainly deals with how to derive mathematical models for the spring pendulum using the Euler-Lagrange equations both in the Cartesian and polar coordinate systems and finding solutions numerically. Derivation of the equations of motion are done for the free, forced and damped cases of the spring pendulum. The main objectives of this mini-dissertation are: firstly, to derive the equations of motion governing the oscillatory and rotational components of the spring pendulum for the free, the forced and damped cases of the spring pendulum ; secondly, to solve these equations numerically by writing the equations as initial value problems (IVP); and finally, to introduce a novel way of incorporating nonlinear damping into the Euler-Lagrange equations of motion as introduced by Joubert, Shatalov and Manzhirov (2013, [20]) for the spring pendulum and interpreting the numerical solutions using CAS-generated graphics.
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"Non-linear functional analysis and vector optimization." 1999. http://library.cuhk.edu.hk/record=b5889846.
Full textAbstract:
by Yan Shing.Thesis (M.Phil.)--Chinese University of Hong Kong, 1999.
Includes bibliographical references (leaves 78-80).
Abstract also in Chinese.
Chapter 1 --- Admissible Points of Convex Sets --- p.7
Chapter 1.1 --- Introduction and Notations --- p.7
Chapter 1.2 --- The Main Result --- p.7
Chapter 1.2.1 --- The Proof of Theoreml.2.1 --- p.8
Chapter 1.3 --- An Application --- p.10
Chapter 2 --- A Generalization on The Theorems of Admissible Points --- p.12
Chapter 2.1 --- Introduction and Notations --- p.12
Chapter 2.2 --- Fundamental Lemmas --- p.14
Chapter 2.3 --- The Main Result --- p.16
Chapter 3 --- Introduction to Variational Inequalities --- p.21
Chapter 3.1 --- Variational Inequalities in Finite Dimensional Space --- p.21
Chapter 3.2 --- Problems Which Relate to Variational Inequalities --- p.25
Chapter 3.3 --- Some Variations on Variational Inequality --- p.28
Chapter 3.4 --- The Vector Variational Inequality Problem and Its Relation with The Vector Optimization Problem --- p.29
Chapter 3.5 --- Variational Inequalities in Hilbert Space --- p.31
Chapter 4 --- Vector Variational Inequalities --- p.36
Chapter 4.1 --- Preliminaries --- p.36
Chapter 4.2 --- Notations --- p.37
Chapter 4.3 --- Existence Results of Vector Variational Inequality --- p.38
Chapter 5 --- The Generalized Quasi-Variational Inequalities --- p.44
Chapter 5.1 --- Introduction --- p.44
Chapter 5.2 --- Properties of The Class F0 --- p.46
Chapter 5.3 --- Main Theorem --- p.53
Chapter 5.4 --- Remarks --- p.58
Chapter 6 --- A set-valued open mapping theorem and related re- sults --- p.61
Chapter 6.1 --- Introduction and Notations --- p.61
Chapter 6.2 --- An Open Mapping Theorem --- p.62
Chapter 6.3 --- Main Result --- p.63
Chapter 6.4 --- An Application on Ordered Normed Spaces --- p.66
Chapter 6.5 --- An Application on Open Decomposition --- p.70
Chapter 6.6 --- An Application on Continuous Mappings from Order- infrabarreled Spaces --- p.72
Bibliography
42
Ptashnyk, Mariya [Verfasser]. "Nonlinear pseudoparabolic equations and variational inequalities / vorgelegt von Mariya Ptashnyk." 2004. http://d-nb.info/972054200/34.
Full text
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Krebs, Andreas [Verfasser]. "On solving nonlinear variational inequalities by p-version finite elements / von Andreas Krebs." 2004. http://d-nb.info/974375454/34.
Full text
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"Functions of bounded variation and the isoperimetric inequality." 2013. http://library.cuhk.edu.hk/record=b5884417.
Full textAbstract:
Lin, Jessey.Thesis (M.Phil.)--Chinese University of Hong Kong, 2013.
Includes bibliographical references (leaves 79-80).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
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Lkhamsuren, Altangerel [Verfasser]. "A duality approach to gap functions for variational inequalities and equilibrium problems / vorgelegt von Lkhamsuren Altangerel." 2006. http://d-nb.info/980955939/34.
Full text