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Kunert, Gerd, Zoubida Mghazli, and Serge Nicaise. "A posteriori error estimation for a finite volume discretization on anisotropic meshes." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601352.
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A singularly perturbed reaction diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertex-centered finite volume method. The anisotropy of the solution is reflected by using \emph{anisotropic meshes} which can improve the accuracy of the discretization considerably. The main focus is on \emph{a posteriori} error estimation. A residual type error estimator is proposed and rigorously analysed. It is shown to be robust with respect to the small perturbation parameter. The estimator is also robust with respect to the mesh anisotropy as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution (which is almost always the case for sensible discretizations). Altogether, reliable and efficient \emph{a posteriori} error estimation is achieved for the finite volume method on anisotropic meshes.
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Rankin, Richard Andrew Robert. "Fully computable a posteriori error bounds for noncomforming and discontinuous galekin finite elemant approximation." Thesis, University of Strathclyde, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.501776.
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We obtain fully computable constant free a posteriori error bounds on the broken energy seminorm of the error in nonconforming and discontinuous Galerkin finite element approximations of a linear second ore elliptic problem on meshes omprised of triangular elements. We do this for nonconforming finite element approximations of uniform arbitrary order as well as for non-uniform order symmetric interior penalty Galerkin, non-symmetric interior penalty Galerkin and ncomplete interior penalty Galerkin finite element approximations.
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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.<br>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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Camacho, Fernando F. "A Posteriori Error Estimates for Surface Finite Element Methods." UKnowledge, 2014. http://uknowledge.uky.edu/math_etds/21.
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Problems involving the solution of partial differential equations over surfaces appear in many engineering and scientific applications. Some of those applications include crystal growth, fluid mechanics and computer graphics. Many times analytic solutions to such problems are not available. Numerical algorithms, such as Finite Element Methods, are used in practice to find approximate solutions in those cases.
In this work we present L2 and pointwise a posteriori error estimates for Adaptive Surface Finite Elements solving the Laplace-Beltrami equation −△Γ u = f . The two sources of errors for Surface Finite Elements are a Galerkin error, and a geometric error that comes from replacing the original surface by a computational mesh. A posteriori error estimates on flat domains only have a Galerkin component. We use residual type error estimators to measure the Galerkin error. The geometric component of our error estimate becomes zero if we consider flat domains, but otherwise has the same order as the residual one. This is different from the available energy norm based error estimates on surfaces, where the importance of the geometric components diminishes asymptotically as the mesh is refined. We use our results to implement an Adaptive Surface Finite Element Method.
An important tool for proving a posteriori error bounds for non smooth functions is the Scott-Zhang interpolant. A refined version of a standard Scott-Zhang interpolation bound is also proved during our analysis. This local version only requires the interpolated function to be in a Sobolev space defined over an element T instead of an element patch containing T.
In the last section we extend our elliptic results to get estimates for the surface heat equation ut − △Γ u = f using the elliptic reconstruction technique.
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Ainsworth, Mark. "A posteriori error estimation in the finite element method." Thesis, Durham University, 1989. http://etheses.dur.ac.uk/6326/.
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The work broadly consists of two parts. In the first part we construct a framework for analyzing and developing a posteriori error estimators for use in the finite element solution of elliptic partial differential equations which have smooth solutions. The analysis makes use of complementary variational principles and the superconvergence phenomenon associated with the finite element method. The second part generalizes these results to the important case when the solution of the boundary value problem contains singularities. It is shown how the classical techniques may be easily modified to perform satisfactorily for the singular case.
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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.<br>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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Chow, Chak-On 1968. "On a posteriori finite element bound procedures for nonsymmetric Eigenvalue problems." Thesis, Massachusetts Institute of Technology, 1999. http://hdl.handle.net/1721.1/85266.
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Pled, Florent. "Vers une stratégie robuste et efficace pour le contrôle des calculs par éléments finis en ingénierie mécanique." Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2012. http://tel.archives-ouvertes.fr/tel-00776633.
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Ce travail de recherche vise à contribuer au développement de nouveaux outils d'estimation d'erreur globale et locale en ingénierie mécanique. Les estimateurs d'erreur globale étudiés reposent sur le concept d'erreur en relation de comportement à travers des techniques spécifiques de construction de champs admissibles, assurant l'aspect conservatif ou garanti de l'estimation. Une nouvelle méthode de construction de champs admissibles est mise en place et comparée à deux autres méthodes concurrentes, en matière de précision, coût de calcul et facilité d'implémentation dans les codes éléments finis. Une amélioration de cette nouvelle méthode hybride fondée sur une minimisation locale de l'énergie complémentaire est également proposée. Celle-ci conduit à l'introduction et à l'élaboration de critères géométriques et énergétiques judicieux, permettant un choix approprié des régions à sélectionner pour améliorer localement la qualité des champs admissibles. Dans le cadre des estimateurs d'erreur locale basés sur l'utilisation conjointe des outils d'extraction et des estimateurs d'erreur globale, deux nouvelles techniques d'encadrement de l'erreur en quantité d'intérêt sont proposées. Celles-ci sont basées sur le principe de Saint-Venant à travers l'emploi de propriétés spécifiques d'homothétie, afin d'améliorer la précision des bornes d'erreur locale obtenues à partir de la technique d'encadrement classique fondée sur l'inégalité de Cauchy-Schwarz. Les diverses études comparatives sont menées dans le cadre des problèmes d'élasticité linéaire en quasi-statique. Le comportement des différents estimateurs d'erreur est illustré et discuté sur des exemples numériques tirés d'applications industrielles. Les travaux réalisés constituent des éléments de réponse à la problématique de la vérification dans un contexte industriel.
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Apel, Thomas, and Cornelia Pester. "Clément-type interpolation on spherical domains - interpolation error estimates and application to a posteriori error estimation." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601335.
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In this paper, a mixed boundary value problem for
the Laplace-Beltrami operator is considered for
spherical domains in $R^3$, i.e. for domains on
the unit sphere. These domains are parametrized
by spherical coordinates (\varphi, \theta),
such that functions on the unit sphere are
considered as functions in these coordinates.
Careful investigation leads to the introduction
of a proper finite element space corresponding to
an isotropic triangulation of the underlying
domain on the unit sphere. Error estimates are
proven for a Clément-type interpolation operator,
where appropriate, weighted norms are used.
The estimates are applied to the deduction of
a reliable and efficient residual error estimator
for the Laplace-Beltrami operator.
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Kunert, Gerd. "A posteriori error estimation for convection dominated problems on anisotropic meshes." Universitätsbibliothek Chemnitz, 2002. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200200255.
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A singularly perturbed convection-diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously.
The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated.
The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers.
Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis.
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Russant, Stuart. "A-posteriori error estimation using higher moments in computational fluid dynamics." Thesis, University of Manchester, 2015. https://www.research.manchester.ac.uk/portal/en/theses/aposteriori-error-estimation-using-higher-moments-in-computational-fluid-dynamics(77bdb9c6-e99a-490d-9624-fdc61525d039).html.
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In industrial situations time is expensive and simulation accuracy is not always investigated because it requires grid refinement studies or other time consuming methods. With this in mind the goal of this research is to develop a method to assess the errors and uncertainties on computational fluid dynamics (CFD) simulations that can be adopted by industry to meet their requirements and time constraints. In a CFD calculation there are a number of sources of errors and uncertainties. An uncertainty is a potential deficiency that is due to a lack of knowledge of an activity of the modelling process, for example turbulence modelling. An error is defined as a recognisable deficiency that is not due to a lack of knowledge, for example numerical discretisation error. The process of determining the level of errors and uncertainties is termed verification and validation. The work aims to define an error estimation method for verification of numerical errors that can be produced during one simulation on a single grid. The second moment solution error estimate for scalar and vector quantities was proposed to meet these requirements. Where the governing equations of CFD, termed the first moments, represent the transport of primary variables such as the velocity, the second moments represents the transport of the primary variables squared such as the total kinetic energy. The second moments are formed by a rearrangement of the first moments. Based on a mathematical justification, an error estimate for vector or scalar quantities was defined from combinations of the solutions to the first and second moments. The error estimate was highly successful when applied to six test cases using laminar flow and scalar transport. These test cases used either central differencing with Gaussian elimination, or the finite volume method with the CFD solver Code_Saturne to conduct the simulations, demonstrating the applicability of the error estimate across solution methods. Comparisons were made to the numerical simulation errors, which were found using either the analytical or refined solutions. The comparisons were aided by the normalised cross correlation coefficient, which compared the similarity of the shape prediction, and the averaged summation coefficients, which compared the scale prediction. When using the first order upwind scheme the method consistently produced good predictions of the locations of error. When using the second order centred or second order linear upwind schemes there was similar success, but limited by influences from solution unboundedness, non-resolution of the boundary layer, the near-wall gradient approximation, and numerical pressure error. At high Reynolds numbers these caused the prediction of the location of error to degrade. This effect was made worse when using the second order schemes in conjunction with the constant value boundary condition. This was the case for the scalar or velocity simulations, and is caused by the unavoidable drop to first order accuracy during the near-wall gradient approximation that is required for the second moment source term approximation. The prediction of the scale demonstrated a dependence on the cell Peclet number. Below cell Peclet number 4 the increase of the estimate scale was linearly related to the increase of the error scale. The estimate scale consistently over-predicts by up to a factor of 3. This allows confidence that the true error level is below that which is predicted by the error estimate. At cell Peclet numbers greater than 4 the relationship between the scales remained linear, however, the estimate begins to under-predict the estimate. The exact relation becomes case dependent, and the highest under-prediction was by a factor of 10. In such circumstances a computationally inexpensive calibration can be done.
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Giacomini, Matteo. "Quantitative a posteriori error estimators in Finite Element-based shape optimization." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLX070/document.
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Les méthodes d’optimisation de forme basées sur le gradient reposent sur le calcul de la dérivée de forme. Dans beaucoup d’applications, la fonctionnelle coût dépend de la solution d’une EDP. Il s’en suit qu’elle ne peut être résolue exactement et que seule une approximation de celle-ci peut être calculée, par exemple par la méthode des éléments finis. Il en est de même pour la dérivée de forme. Ainsi, les méthodes de gradient en optimisation de forme - basées sur des approximations du gradient - ne garantissent pas a priori que la direction calculée à chaque itération soit effectivement une direction de descente pour la fonctionnelle coût. Cette thèse est consacrée à la construction d’une procédure de certification de la direction de descente dans des algorithmes de gradient en optimisation de forme grâce à des estimations a posteriori de l’erreur introduite par l’approximation de la dérivée de forme par la méthode des éléments finis. On présente une procédure pour estimer l’erreur dans une Quantité d’Intérêt et on obtient une borne supérieure certifiée et explicitement calculable. L’Algorithme de Descente Certifiée (CDA) pour l’optimisation de forme identifie une véritable direction de descente à chaque itération et permet d’établir un critère d’arrêt fiable basé sur la norme de la dérivée de forme. Deux applications principales sont abordées dans la thèse. Premièrement, on considère le problème scalaire d’identification de forme en tomographie d’impédance électrique et on étudie différentes estimations d’erreur. Une première approche est basée sur le principe de l’énergie complémentaire et nécessite la résolution de problèmes globaux additionnels. Afin de réduire le coût de calcul de la procédure de certification, une estimation qui dépend seulement de quantités locales est dérivée par la reconstruction des flux équilibrés. Après avoir validé les estimations de l’erreur pour un cas bidimensionnel, des résultats numériques sont présentés pour tester les méthodes discutées. Une deuxième application est centrée sur le problème vectoriel de la conception optimale des structures élastiques. Dans ce cadre figure, on calcule l’expression volumique de la dérivée de forme de la compliance à partir de la formulation primale en déplacements et de la formulation duale mixte pour l’équation de l’élasticité linéaire. Quelques résultats numériques préliminaires pour la minimisation de la compliance sous une contrainte de volume en 2D sont obtenus à l’aide de l’Algorithme de Variation de Frontière et une estimation a posteriori de l’erreur de la dérivée de forme basée sur le principe de l’énergie complémentaire est calculée<br>Gradient-based shape optimization strategies rely on the computation of the so-called shape gradient. In many applications, the objective functional depends both on the shape of the domain and on the solution of a PDE which can only be solved approximately (e.g. via the Finite Element Method). Hence, the direction computed using the discretized shape gradient may not be a genuine descent direction for the objective functional. This Ph.D. thesis is devoted to the construction of a certification procedure to validate the descent direction in gradient-based shape optimization methods using a posteriori estimators of the error due to the Finite Element approximation of the shape gradient.By means of a goal-oriented procedure, we derive a fully computable certified upper bound of the aforementioned error. The resulting Certified Descent Algorithm (CDA) for shape optimization is able to identify a genuine descent direction at each iteration and features a reliable stopping criterion basedon the norm of the shape gradient.Two main applications are tackled in the thesis. First, we consider the scalar inverse identification problem of Electrical Impedance Tomography and we investigate several a posteriori estimators. A first procedure is inspired by the complementary energy principle and involves the solution of additionalglobal problems. In order to reduce the computational cost of the certification step, an estimator which depends solely on local quantities is derived via an equilibrated fluxes approach. The estimators are validated for a two-dimensional case and some numerical simulations are presented to test the discussed methods. A second application focuses on the vectorial problem of optimal design of elastic structures. Within this framework, we derive the volumetric expression of the shape gradient of the compliance using both H 1 -based and dual mixed variational formulations of the linear elasticity equation. Some preliminary numerical tests are performed to minimize the compliance under a volume constraint in 2D using the Boundary Variation Algorithm and an a posteriori estimator of the error in the shape gradient is obtained via the complementary energy principle
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Castellazzi, Giovanni <1975>. "Verification in computational structural mechanics: recovery-based a posteriori error estimation." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2007. http://amsdottorato.unibo.it/441/1/Giovanni_Castellazzi.pdf.
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Castellazzi, Giovanni <1975>. "Verification in computational structural mechanics: recovery-based a posteriori error estimation." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2007. http://amsdottorato.unibo.it/441/.
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Bacharach, Lucien. "Caractérisation des limites fondamentales de l'erreur quadratique moyenne pour l'estimation de signaux comportant des points de rupture." Thesis, Université Paris-Saclay (ComUE), 2018. http://www.theses.fr/2018SACLS322/document.
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Cette thèse porte sur l'étude des performances d'estimateurs en traitement du signal, et s'attache en particulier à étudier les bornes inférieures de l'erreur quadratique moyenne (EQM) pour l'estimation de points de rupture, afin de caractériser le comportement d'estimateurs, tels que celui du maximum de vraisemblance (dans le contexte fréquentiste), mais surtout du maximum a posteriori ou de la moyenne conditionnelle (dans le contexte bayésien). La difficulté majeure provient du fait que, pour un signal échantillonné, les paramètres d'intérêt (à savoir les points de rupture) appartiennent à un espace discret. En conséquence, les résultats asymptotiques classiques (comme la normalité asymptotique du maximum de vraisemblance) ou la borne de Cramér-Rao ne s'appliquent plus. Quelques résultats sur la distribution asymptotique du maximum de vraisemblance provenant de la communauté mathématique sont actuellement disponibles, mais leur applicabilité à des problèmes pratiques de traitement du signal n'est pas immédiate. Si l'on décide de concentrer nos efforts sur l'EQM des estimateurs comme indicateur de performance, un travail important autour des bornes inférieures de l'EQM a été réalisé ces dernières années. Plusieurs études ont ainsi permis de proposer des inégalités plus précises que la borne de Cramér-Rao. Ces dernières jouissent en outre de conditions de régularité plus faibles, et ce, même en régime non asymptotique, permettant ainsi de délimiter la plage de fonctionnement optimal des estimateurs. Le but de cette thèse est, d'une part, de compléter la caractérisation de la zone asymptotique (en particulier lorsque le rapport signal sur bruit est élevé et/ou pour un nombre d'observations infini) dans un contexte d'estimation de points de rupture. D'autre part, le but est de donner les limites fondamentales de l'EQM d'un estimateur dans la plage non asymptotique. Les outils utilisés ici sont les bornes inférieures de l’EQM de la famille Weiss-Weinstein qui est déjà connue pour être plus précise que la borne de Cramér-Rao dans les contextes, entre autres, de l’analyse spectrale et du traitement d’antenne. Nous fournissons une forme compacte de cette famille dans le cas d’un seul et de plusieurs points de ruptures puis, nous étendons notre analyse aux cas où les paramètres des distributions sont inconnus. Nous fournissons également une analyse de la robustesse de cette famille vis-à-vis des lois a priori utilisées dans nos modèles. Enfin, nous appliquons ces bornes à plusieurs problèmes pratiques : données gaussiennes, poissonniennes et processus exponentiels<br>This thesis deals with the study of estimators' performance in signal processing. The focus is the analysis of the lower bounds on the Mean Square Error (MSE) for abrupt change-point estimation. Such tools will help to characterize performance of maximum likelihood estimator in the frequentist context but also maximum a posteriori and conditional mean estimators in the Bayesian context. The main difficulty comes from the fact that, when dealing with sampled signals, the parameters of interest (i.e., the change points) lie on a discrete space. Consequently, the classical large sample theory results (e.g., asymptotic normality of the maximum likelihood estimator) or the Cramér-Rao bound do not apply. Some results concerning the asymptotic distribution of the maximum likelihood only are available in the mathematics literature but are currently of limited interest for practical signal processing problems. When the MSE of estimators is chosen as performance criterion, an important amount of work has been provided concerning lower bounds on the MSE in the last years. Then, several studies have proposed new inequalities leading to tighter lower bounds in comparison with the Cramér-Rao bound. These new lower bounds have less regularity conditions and are able to handle estimators’ MSE behavior in both asymptotic and non-asymptotic areas. The goal of this thesis is to complete previous results on lower bounds in the asymptotic area (i.e. when the number of samples and/or the signal-to-noise ratio is high) for change-point estimation but, also, to provide an analysis in the non-asymptotic region. The tools used here will be the lower bounds of the Weiss-Weinstein family which are already known in signal processing to outperform the Cramér-Rao bound for applications such as spectral analysis or array processing. A closed-form expression of this family is provided for a single and multiple change points and some extensions are given when the parameters of the distributions on each segment are unknown. An analysis in terms of robustness with respect to the prior influence on our models is also provided. Finally, we apply our results to specific problems such as: Gaussian data, Poisson data and exponentially distributed data
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Tempone, Olariaga Raul. "Numerical Complexity Analysis of Weak Approximation of Stochastic Differential Equations." Doctoral thesis, KTH, Numerisk analys och datalogi, NADA, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3413.
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The thesis consists of four papers on numerical complexityanalysis of weak approximation of ordinary and partialstochastic differential equations, including illustrativenumerical examples. Here by numerical complexity we mean thecomputational work needed by a numerical method to solve aproblem with a given accuracy. This notion offers a way tounderstand the efficiency of different numerical methods. The first paper develops new expansions of the weakcomputational error for Ito stochastic differentialequations using Malliavin calculus. These expansions have acomputable leading order term in a posteriori form, and arebased on stochastic flows and discrete dual backward problems.Beside this, these expansions lead to efficient and accuratecomputation of error estimates and give the basis for adaptivealgorithms with either deterministic or stochastic time steps.The second paper proves convergence rates of adaptivealgorithms for Ito stochastic differential equations. Twoalgorithms based either on stochastic or deterministic timesteps are studied. The analysis of their numerical complexitycombines the error expansions from the first paper and anextension of the convergence results for adaptive algorithmsapproximating deterministic ordinary differential equations.Both adaptive algorithms are proven to stop with an optimalnumber of time steps up to a problem independent factor definedin the algorithm. The third paper extends the techniques to theframework of Ito stochastic differential equations ininfinite dimensional spaces, arising in the Heath Jarrow Mortonterm structure model for financial applications in bondmarkets. Error expansions are derived to identify differenterror contributions arising from time and maturitydiscretization, as well as the classical statistical error dueto finite sampling. The last paper studies the approximation of linear ellipticstochastic partial differential equations, describing andanalyzing two numerical methods. The first method generates iidMonte Carlo approximations of the solution by sampling thecoefficients of the equation and using a standard Galerkinfinite elements variational formulation. The second method isbased on a finite dimensional Karhunen- Lo`eve approximation ofthe stochastic coefficients, turning the original stochasticproblem into a high dimensional deterministic parametricelliptic problem. Then, adeterministic Galerkin finite elementmethod, of either h or p version, approximates the stochasticpartial differential equation. The paper concludes by comparingthe numerical complexity of the Monte Carlo method with theparametric finite element method, suggesting intuitiveconditions for an optimal selection of these methods. 2000Mathematics Subject Classification. Primary 65C05, 60H10,60H35, 65C30, 65C20; Secondary 91B28, 91B70.<br>QC 20100825
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Wilkins, Catherine. "Adaptive finite element methods for the damped wave equation." Thesis, University of Oxford, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.302398.
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Ohlberger, Mario. "A posteriori error estimates and adaptive methods for convection dominated transport processes." [S.l. : s.n.], 2001. http://deposit.ddb.de/cgi-bin/dokserv?idn=961616245.
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Creusé, Emmanuel, Gerd Kunert, and Serge Nicaise. "A posteriori error estimation for the Stokes problem: Anisotropic and isotropic discretizations." Universitätsbibliothek Chemnitz, 2003. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200300057.
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The paper presents a posteriori error estimators for the stationary Stokes problem. We consider anisotropic finite element discretizations (i.e. elements with very large aspect ratio) where conventional, isotropic error estimators fail.
Our analysis covers two- and three-dimensional domains, conforming and nonconforming discretizations as well as different elements.
This large variety of settings requires different approaches and results in different estimators. Furthermore many examples of finite element pairs that are covered by the analysis are presented.
Lower and upper error bounds form the main result with minimal assumptions on the elements. The lower error bound is uniform with respect to the mesh anisotropy with the exception of nonconforming 3D discretizations made of pentahedra or hexahedra. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation.
In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. Some of the corresponding results seem to be novel (in particular for 3D domains), and cover element pairs of practical importance.
The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimators.
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Grepl, Martin A. (Martin Alexander) 1974. "Reduced-basis approximation a posteriori error estimation for parabolic partial differential equations." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/32387.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2005.<br>Includes bibliographical references (p. 243-251).<br>Modern engineering problems often require accurate, reliable, and efficient evaluation of quantities of interest, evaluation of which demands the solution of a partial differential equation. We present in this thesis a technique for the prediction of outputs of interest of parabolic partial differential equations. The essential ingredients are: (i) rapidly convergent reduced-basis approximations - Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter-time space; (ii) a posteriori error estimation - relaxations of the error-residual equation that provide rigorous and sharp bounds for the error in specific outputs of interest: the error estimates serve a priori to construct our samples and a posteriori to confirm fidelity; and (iii) offline-online computional procedures - in the offline stage the reduced- basis approximation is generated; in the online stage, given a new parameter value, we calculate the reduced-basis output and associated error bound. The operation count for the online stage depends only on N (typically small) and the parametric complexity of the problem; the method is thus ideally suited for repeated, rapid, reliable evaluation of input-output relationships in the many-query or real-time contexts. We first consider parabolic problems with affine parameter dependence and subsequently extend these results to nonaffine and certain classes of nonlinear parabolic problems.<br>(cont.) To this end, we introduce a collateral reduced-basis expansion for the nonaffine and nonlinear terms and employ an inexpensive interpolation procedure to calculate the coefficients for the function approximation - the approach permits an efficient offline-online computational decomposition even in the presence of nonaffine and highly nonlinear terms. Under certain restrictions on the function approximation, we also introduce rigorous a posteriori error estimators for nonaffine and nonlinear problems. Finally, we apply our methods to the solution of inverse and optimal control problems. While the efficient evaluation of the input-output relationship is essential for the real-time solution of these problems, the a posteriori error bounds let us pursue a robust parameter estimation procedure which takes into account the uncertainty due to measurement and reduced-basis modeling errors explicitly (and rigorously). We consider several examples: the nondestructive evaluation of delamination in fiber-reinforced concrete, the dispersion of pollutants in a rectangular domain, the self-ignition of a coal stockpile, and the control of welding quality. Numerical results illustrate the applicability of our methods in the many-query contexts of optimization, characterization, and control.<br>by Martin A. Grepl.<br>Ph.D.
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Mavriplis, Cathy. "Nonconforming discretizations and a posteriori error estimators for adaptive spectral element techniques." Thesis, Massachusetts Institute of Technology, 1989. http://hdl.handle.net/1721.1/14526.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1989.<br>Includes bibliographical references (leaves 151-157).<br>by Catherine Andria Mavriplis.<br>Ph.D.
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Simões, Eduardo Tenório. "Linear and nonlinear hirarchical plate models and a posteriori kinematical error estimator." Universidade de São Paulo, 2015. http://www.teses.usp.br/teses/disponiveis/3/3144/tde-26072016-151855/.
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This study explores the use of hierarchical models to represent three-dimensional solids in a computationally inexpensive way. First, it is investigated the choice of the finite element spaces and how it affects the convergence in relation to the thickness parameter. It was studied three different models. It was shown that the best lowest order suitable combination of spaces grows in all fields as the model order is enriched. After, it is presented a theory to evaluate the error in the discretization and the kinematical hypothesis. It is shown that the implemented error in discretization technique is capable of capturing the boundary layer in automated way for any model. It is also given a posteriori error procedure for kinematical hypothesis. The method is based on the equilibrium error of higher order models. Good results are shown. In the end, it is presented a geometrical nonlinear hierarchical shell model and its discretization. It is shown that the model succeeds in representing the three-dimensional solution when compared with solid elements in a commercial code.<br>Este estudo explora o uso de modelos hierárquicos para representar sólidos tridimensionais de forma computacionalmente barata. Em primeiro lugar, é explorada a escolha dos espaços de elementos finitos e como isso afeta a convergência em relação ao parâmetro da espessura. Foram estudados três modelos diferentes. Mostrou-se que a menor ordem adequada do espaço de discretização cresce para todos os campos conforme a ordem do modelo é enriquecida . Isso impõe um problema, já que um maior polinômio exige maior custo computacional e modelos de alta ordem só são necessários perto do contorno. Depois, são usados estimadores de erro na discretização e na hipótese cinemática. Mostra-se que o erro implementado na discretização é capaz de capiturar a camada limite de forma automatizada para qualquer modelo. Também é apresentada uma técnica de erro a posteriri na hipótese cinemática com base no erro no equilíbrio de modelos de ordem superior. No final, é apresentado um modelo hierárquico de casca geométricamente não linear e sua discretização. Mostra-se que o modelo consegue representar a solução tridimensional quando comparado com o um software comercial.
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Kunert, Gerd. "A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes." Doctoral thesis, [S.l. : s.n.], 1999. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB10324701.
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Ren, Chengfang. "Caractérisation des performances minimales d'estimation pour des modèles d'observations non-standards." Thesis, Paris 11, 2015. http://www.theses.fr/2015PA112167/document.
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Dans le contexte de l'estimation paramétrique, les performances d'un estimateur peuvent être caractérisées, entre autre, par son erreur quadratique moyenne (EQM) et sa résolution limite. La première quantifie la précision des valeurs estimées et la seconde définit la capacité de l'estimateur à séparer plusieurs paramètres. Cette thèse s'intéresse d'abord à la prédiction de l'EQM "optimale" à l'aide des bornes inférieures pour des problèmes d'estimation simultanée de paramètres aléatoires et non-aléatoires (estimation hybride), puis à l'extension des bornes de Cramér-Rao pour des modèles d'observation moins standards. Enfin, la caractérisation des estimateurs en termes de résolution limite est également étudiée. Ce manuscrit est donc divisé en trois parties :Premièrement, nous complétons les résultats de littérature sur les bornes hybrides en utilisant deux bornes bayésiennes : la borne de Weiss-Weinstein et une forme particulière de la famille de bornes de Ziv-Zakaï. Nous montrons que ces bornes "étendues" sont plus précises pour la prédiction de l'EQM optimale par rapport à celles existantes dans la littérature.Deuxièmement, nous proposons des bornes de type Cramér-Rao pour des contextes d'estimation moins usuels, c'est-à-dire : (i) Lorsque les paramètres non-aléatoires sont soumis à des contraintes d'égalité linéaires ou non-linéaires (estimation sous contraintes). (ii) Pour des problèmes de filtrage à temps discret où l'évolution des états (paramètres) est régit par une chaîne de Markov. (iii) Lorsque la loi des observations est différente de la distribution réelle des données.Enfin, nous étudions la résolution et la précision des estimateurs en proposant un critère basé directement sur la distribution des estimées. Cette approche est une extension des travaux de Oh et Kashyap et de Clark pour des problèmes d'estimation de paramètres multidimensionnels<br>In the parametric estimation context, estimators performances can be characterized, inter alia, by the mean square error and the resolution limit. The first quantities the accuracy of estimated values and the second defines the ability of the estimator to allow a correct resolvability. This thesis deals first with the prediction the "optimal" MSE by using lower bounds in the hybrid estimation context (i.e. when the parameter vector contains both random and non-random parameters), second with the extension of Cramér-Rao bounds for non-standard estimation problems and finally to the characterization of estimators resolution. This manuscript is then divided into three parts :First, we fill some lacks of hybrid lower bound on the MSE by using two existing Bayesian lower bounds: the Weiss-Weinstein bound and a particular form of Ziv-Zakai family lower bounds. We show that these extended lower bounds are tighter than the existing hybrid lower bounds in order to predict the optimal MSE.Second, we extend Cramer-Rao lower bounds for uncommon estimation contexts. Precisely: (i) Where the non-random parameters are subject to equality constraints (linear or nonlinear). (ii) For discrete-time filtering problems when the evolution of states are defined by a Markov chain. (iii) When the observation model differs to the real data distribution.Finally, we study the resolution of the estimators when their probability distributions are known. This approach is an extension of the work of Oh and Kashyap and the work of Clark to multi-dimensional parameters estimation problems
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Ludwig, Marcus John. "Finite element error estimation and adaptivity for problems of elasticity." Thesis, Brunel University, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.246151.
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Zhu, Liang. "Robust a posteriori error estimation for discontinuous Galerkin methods for convection diffusion problems." Thesis, University of British Columbia, 2010. http://hdl.handle.net/2429/23337.
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The present thesis is concerned with the development and practical implementation of robust a-posteriori error estimators for discontinuous Galerkin (DG) methods for convection-diffusion problems.
It is well-known that solutions to convection-diffusion problems may have boundary and internal layers of small width where their gradients change rapidly. A powerful approach to numerically resolve these layers is based on using hp-adaptive finite element methods, which control and minimize the discretization errors by locally adapting the mesh sizes (h-refinement) and the approximation orders (p-refinement) to the features of the problems. In this work, we choose DG methods to realize adaptive algorithms. These methods yield stable and robust discretization schemes for convection-dominated problems, and are naturally suited to handle local variations in the mesh sizes and approximation degrees as required for hp-adaptivity.
At the heart of adaptive finite element methods are a-posteriori error estimators. They provide information on the errors on each element and indicate where local refinement/derefinement should be applied. An efficient error estimator should always yield an upper and lower bound of the discretization error in a suitable norm. For convection-diffusion problems, it is desirable that the estimator is also robust, meaning that the upper and lower bounds differ by a factor that is independent of the mesh Peclet number of the problem.
We develop a new approach to obtain robust a-posteriori error estimates for convection-diffusion problems for h-version and hp-version DG methods. The main technical tools in our analysis are new hp-version approximation results of an averaging operator, which are derived for irregular hexahedral meshes in three dimensions, as well as for irregular anisotropic rectangular meshes in two dimensions.
We present a series of numerical examples based on C++ implementations of our methods. The numerical results indicate that the error estimator is effective in locating and resolving interior and boundary layers. For the hp-adaptive algorithms, once the local mesh size is of the same order as the width of boundary or interior layers, both the energy error and the error estimator are observed to converge exponentially fast in the number of degrees of freedom.
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Bridgeman, Leila. "Stability and a posteriori error analysis of discontinious Galerkin methods for linearized elasticity." Thesis, McGill University, 2010. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=95054.
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We consider discontinuous Galerkin finite element methods for the discretization of linearized elasticity problems in two space dimensions. Inf-sup stability results on the continuous and discrete level are provided. Furthermore, we derive lower and upper a posteriori error bounds that are robust with respect to nearly incompressible materials, and can easily be implemented within an automatic mesh refinement procedure. The theoretical results are illustrated with a series of numerical experiments.<br>Nous considérons les méthodes de Galerkin pour la discrétisation des relations déformations-déplacements linéaires en deux dimensions d'espace. Des résultats du stabilité inf-sup sur les niveaux continus et discrets sont fournis. En plus, nous dérivons des limites inférieurs et supérieures pour l'erreur a posteriori qui peuvent être utilisées dans des procédures de maillage automatisées sans difficulté et qui demeurent robustes dans le cas des matériaux qui ne sont presque pas compressibles. Les résultats théoriques sont illustrés par des expériences numériques.
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Andrews, J. G. "An a posteriori error indicator and its application to adaptive methods in CFD." Thesis, University of Oxford, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.319051.
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Effland, Alexander [Verfasser]. "Discrete Riemannian Calculus and A Posteriori Error Control on Shape Spaces / Alexander Effland." Bonn : Universitäts- und Landesbibliothek Bonn, 2018. http://d-nb.info/1150777796/34.
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Yu, Peng. "Isogeometric analysis with local adaptivity based on a posteriori error estimation for elastodynamics." Thesis, Cardiff University, 2019. http://orca.cf.ac.uk/119867/.
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IsoGeometric Analysis (IGA) was invented to integrate the Computer-Aided Design (CAD) and Computer-Aided Engineering (CAE) into a unified process. According to the recent research, IGA performs a super convergence in case of vibration, and especially, it perfectly addresses the Gibbs phenomenon (fluctuation) occurring in discrete spectra when using standard Finite Element Method (FEM). However, due to the tensor-product structure of Non-Uniform Rational B-Splines (NURBS), it fails to achieve the local refinement, which restricts its application to engineering fields performing local characteristics that require local refinement, such as sharp geometrical feature and/or varying material properties. In this context, the first goal of thesis is to extend the recently proposed paradigm, called Geometry Independent Field approximaTion (GIFT), to be applied in the scheme of dynamics. The GIFT methodology allows geometry of structure to be described within the NURBS provided directly by the existing CAD software, and solution field to be approximated by the Polynomial splines over Hierarchical Tmeshes (PHT) with the feature of local refinement meanwhile. Subsequently, in the framework of GIFT, an adaptivity technique based on hierarchical a posteriori error estimation on the modal vector is established for the free vibration of thick plate. The proposed adaptive mesh achieves a faster convergence than uniform refinement. Especially, the employment of Modal Assurance Criterion (MAC)-style strategy is able to better determine the modal correspondence between coarse and fine discretizations than Frequency Error Criterion (FEC) method. Furthermore, based on hierarchical a posteriori error estimation strategy, three types of adaptivity algorithms are constructed to deal with the space-time refinement. Specially, unidirectional multi-level space-time adaptive GIFT/Newmark (UM-STAGN) well catches stress wave propagation but fails in error information transfer. Energybased space-time adaptive GIFT/Newmark (E-STAGN) can reassess the error but cannot uncover the source of error. Dual weighted residual adaptive GIFT/Newmark (DWR-STAGN) methods are error-sensitive so that it leads to the best convergence among these three approaches.
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Lins, Rafael Marques. "A posteriori error estimations for the generalized finite element method and modified versions." Universidade de São Paulo, 2015. http://www.teses.usp.br/teses/disponiveis/18/18134/tde-03092015-083839/.
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This thesis investigates two a posteriori error estimators, based on gradient recovery, aiming to fill the gap of the error estimations for the Generalized FEM (GFEM) and, mainly, its modified versions called Corrected XFEM (C-XFEM) and Stable GFEM (SGFEM). In order to reach this purpose, firstly, brief reviews regarding the GFEM and its modified versions are presented, where the main advantages attributed to each numerical method are highlighted. Then, some important concepts related to the error study are presented. Furthermore, some contributions involving a posteriori error estimations for the GFEM are shortly described. Afterwards, the two error estimators hereby proposed are addressed focusing on linear elastic fracture mechanics problems. The first estimator was originally proposed for the C-XFEM and is hereby extended to the SGFEM framework. The second one is based on a splitting of the recovered stress field into two distinct parts: singular and smooth. The singular part is computed with the help of the J integral, whereas the smooth one is calculated from a combination between the Superconvergent Patch Recovery (SPR) and Singular Value Decomposition (SVD) techniques. Finally, various numerical examples are selected to assess the robustness of the error estimators considering different enrichment types, versions of the GFEM, solicitant modes and element types. Relevant aspects such as effectivity indexes, error distribution and convergence rates are used for describing the error estimators. The main contributions of this thesis are: the development of two efficient a posteriori error estimators for the GFEM and its modified versions; a comparison between the GFEM and its modified versions; the identification of the positive features of each error estimator and a detailed study concerning the blending element issues.<br>Esta tese investiga dois estimadores de erro a posteriori, baseados na recuperação do gradiente, visando preencher o hiato das estimativas de erro para o Generalized FEM (GFEM) e, sobretudo, suas versões modificadas denominadas Corrected XFEM (C-XFEM) e Stable GFEM (SGFEM). De modo a alcançar este objetivo, primeiramente, breves revisões a respeito do GFEM e suas versões modificadas são apresentadas, onde as principais vantagens atribuídas a cada método são destacadas. Em seguida, alguns importantes conceitos relacionados ao estudo do erro são apresentados. Além disso, algumas contribuições envolvendo estimativas de erro a posteriori para o GFEM são brevemente descritas. Posteriormente, os dois estimadores de erro propostos neste trabalho são abordados focando em problemas da mecânica da fratura elástico linear. O primeiro estimador foi originalmente proposto para o C-XFEM e por este meio é estendido para o âmbito do SGFEM. O segundo é baseado em uma divisão do campo de tensões recuperadas em duas partes distintas: singular e suave. A parte singular é calculada com o auxílio da integral J, enquanto que a suave é calculada a partir da combinação entre as técnicas Superconvergent Patch Recovery (SPR) e Singular Value Decomposition (SVD). Finalmente, vários exemplos numéricos são selecionados para avaliar a robustez dos estimadores de erro considerando diferentes tipos de enriquecimento, versões do GFEM, modos solicitantes e tipos de elemento. Aspectos relevantes tais como índices de efetividade, distribuição do erro e taxas de convergência são usados para descrever os estimadores de erro. As principais contribuições desta tese são: o desenvolvimento de dois eficientes estimadores de erro a posteriori para o GFEM e suas versões modificadas; uma comparação entre o GFEM e suas versões modificadas; a identificação das características positivas de cada estimador de erro e um estudo detalhado sobre a questão dos elementos de mistura.
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Buß, Hinderk M. "A posteriori error estimators based on duality techniques from the calculus of variations." [S.l. : s.n.], 2003. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB10790752.
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ElSheikh, Ahmed H. Chidiac S. E. Smith Spencer B. "Multiscale a posteriori error estimation and mesh adaptivity for reliable finite element analysis." *McMaster only, 2007.
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Houston, Paul D. "Lagrange-Galerkin methods for unsteady convection-diffusion problems : a posteriori error analysis and adaptivity." Thesis, University of Oxford, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.337607.
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Kunert, Gerd [Verfasser]. "A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes / Gerd Kunert." Chemnitz : Universitätsbibliothek Chemnitz, 1999. http://d-nb.info/1210931834/34.
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Moldenhauer, Marcel [Verfasser], and Gerhard [Akademischer Betreuer] Starke. "Stress reconstruction and a-posteriori error estimation for elasticity / Marcel Moldenhauer ; Betreuer: Gerhard Starke." Duisburg, 2020. http://d-nb.info/1221061712/34.
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Lin, Shan. "Analysing Generalisation Error Bounds For Convolutional Neural Networks." Thesis, The University of Sydney, 2018. http://hdl.handle.net/2123/20315.
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Analysing Generalisation Error Bounds for Convolutional Neural Networks Abstract: Convolutional neural networks (CNNs) have achieved breakthrough performance in a wide range of applications including image classification, semantic segmentation, and object detection. Previous research on characterising the generalisability of neural networks has mostly focused on fully connected neural networks (FNNs), with CNNs regarded as a special case of FNNs without taking into account the special structure of convolutional layers; therefore, the CNN bounds may not be as tight as in FNNs. Here we propose a generalisation bound for CNNs by exploiting the sparse and shared structure of weight matrices for convolutional layers. As the new generalisation bound relies on the spectral norm of weight matrices, we further discuss the spectral norms for three convolution operations including standard convolution, depthwise convolution, and pointwise convolution. We show that our new bound for CNNs is indeed tighter than previously proposed under certain conditions.
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Kieweg, Michael. "An a posteriori error analysis for distributed elliptic optimal control problems with pointwise state constraints." kostenfrei kostenfrei, 2007. http://nbn-resolving.de/urn:nbn:de:bvb:384-opus-7184.
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Pester, Cornelia. "A residual a posteriori error estimator for the eigenvalue problem for the Laplace-Beltrami operator." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601556.
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The Laplace-Beltrami operator corresponds to the Laplace operator on curved surfaces. In this paper, we consider an eigenvalue problem for the Laplace-Beltrami operator on subdomains of the unit sphere in $\R^3$. We develop a residual a posteriori error estimator for the eigenpairs and derive a reliable estimate for the eigenvalues. A global parametrization of the spherical domains and a carefully chosen finite element discretization allows us to use an approach similar to the one for the two-dimensional case. In order to assure results in the quality of those for plane domains, weighted norms and an adapted Clément-type interpolation operator have to be introduced.
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Wu, Heng. "An a-posteriori finite element error estimator for adaptive grid computation of viscous incompressible flows." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape2/PQDD_0021/NQ53797.pdf.
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Trenz, Stefan [Verfasser]. "POD-Based A-posteriori Error Estimation for Control Problems Governed by Nonlinear PDEs / Stefan Trenz." Konstanz : Bibliothek der Universität Konstanz, 2017. http://d-nb.info/1142113868/34.
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Brenner, Andreas [Verfasser], Eberhard [Gutachter] Bänsch, and Charalambos [Gutachter] Makridakis. "A-posteriori error estimates for pressure-correction schemes / Andreas Brenner ; Gutachter: Eberhard Bänsch, Charalambos Makridakis." Erlangen : Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), 2016. http://d-nb.info/1114499692/34.
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Kirby, Robert Charles. "Local time stepping and a posteriori error estimates for flow and transport in porous media /." Digital version accessible at:, 2000. http://wwwlib.umi.com/cr/utexas/main.
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Kunert, Gerd. "Robust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshes." Universitätsbibliothek Chemnitz, 2000. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200000867.
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We consider a singularly perturbed reaction-diffusion problem and
derive and rigorously analyse an a posteriori residual error
estimator that can be applied to anisotropic finite element meshes.
The quotient of the upper and lower error bounds is the so-called
matching function which depends on the anisotropy (of the
mesh and the solution) but not on the small perturbation parameter.
This matching function measures how well the anisotropic finite
element mesh corresponds to the anisotropic problem.
Provided this correspondence is sufficiently good, the matching
function is O(1).
Hence one obtains tight error bounds, i.e. the error estimator
is reliable and efficient as well as robust with respect to the
small perturbation parameter.
A numerical example supports the anisotropic error analysis.
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Grosman, Serguei. "The robustness of the hierarchical a posteriori error estimator for reaction-diffusion equation on anisotropic meshes." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601418.
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Singularly perturbed reaction-diffusion problems
exhibit in general solutions with anisotropic
features, e.g. strong boundary and/or interior
layers. This anisotropy is reflected in the
discretization by using meshes with anisotropic
elements. The quality of the numerical solution
rests on the robustness of the a posteriori error
estimator with respect to both the perturbation
parameters of the problem and the anisotropy of the
mesh. The simplest local error estimator from the
implementation point of view is the so-called
hierarchical error estimator. The reliability
proof is usually based on two prerequisites:
the saturation assumption and the strengthened
Cauchy-Schwarz inequality. The proofs of these
facts are extended in the present work for the
case of the singularly perturbed reaction-diffusion
equation and of the meshes with anisotropic elements.
It is shown that the constants in the corresponding
estimates do neither depend on the aspect ratio
of the elements, nor on the perturbation parameters.
Utilizing the above arguments the concluding
reliability proof is provided as well as the
efficiency proof of the estimator, both
independent of the aspect ratio and perturbation
parameters.
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Kunert, Gerd. "A posteriori H^1 error estimation for a singularly perturbed reaction diffusion problem on anisotropic meshes." Universitätsbibliothek Chemnitz, 2001. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200100730.
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The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H^1 seminorm that can be applied to anisotropic finite element meshes.
A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably.
Furthermore three modifications of these estimators are introduced and discussed.
Numerical experiments for all estimators complement and confirm the theoretical results.
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Grosz, Lutz [Verfasser]. "A-posteriori error estimates for the finite element solution on non-linear variational problems / Lutz Grosz." Karlsruhe : KIT-Bibliothek, 1997. http://d-nb.info/1013872436/34.
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Lu, James 1977. "An a posteriori error control framework for adaptive precision optimization using discontinuous Galerkin finite element method." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/34134.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2005.<br>Includes bibliographical references (leaves 169-178).<br>Introduction: Aerodynamic design optimization has seen significant development over the past decade. Adjoint-based shape design for elliptic systems was first proposed by Pironneau and applied to transonic flow by Jameson . A review of the aerodynamic shape optimization literature and a large list of references is given in. Over the years much technology has been developed, allowing engineers to contemplate applying optimization methods to a wide variety of problems. In the context of structured grids, adjoint-based applications include multipoint, multi-objective airfoil design using compressible Navier-Stokes equations and 3D multipoint design of aircraft configurations using inviscid Euler equations. There have also been significant effort in applying adjoint methods to the unstructured grid setting. In this context, Newman et al., Elliot and Peraire were among the first to develop discrete adjoint approaches for the inviscid Euler equations.<br>by James Ching-Chi<br>Ph.D.
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Sen, Sugata 1977. "Reduced basis approximation and a posteriori error estimation for non-coercive elliptic problems : applications to acoustics." Thesis, Massachusetts Institute of Technology, 2007. http://hdl.handle.net/1721.1/39355.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2007.<br>Includes bibliographical references (p. 251-261).<br>Modern engineering problems often require accurate, reliable, and efficient evaluation of quantities of interest, evaluation of which demands the solution of a partial differential equation. We present in this thesis a general methodology for the predicition of outputs of interest of non-coercive elliptic partial differential equations. The essential ingredients are: (i) rapidly convergent reduced basis approximations - Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter-time space; (ii) a posteriori error estimation - relaxations of the error-residual equation that provide rigorous and sharp bounds for the error in specific outputs of interest; and (iii) offline-online computational procedures - in the offline stage the reduced basis approximation is generated; in the online stage, given a new parameter value, we calculate the reduced basis output and associated error bound. The operation count for the online stage depends only on N (typically small) and the parametric complexity of the problem; the method is thus ideally suited for repeated, rapid, reliable evaluation of input-output relationships in the many-query or real-time contexts. We consider the crucial ingredients for the treatment of acoustics problems<br>(cont.) - simultaneous treatment of non-coercive (and near-resonant), non-Hermitian elliptic operators, complex-valued fields, often unbounded domains, and quadratic outputs of interest. We introduce the successive constraint approach to approximate lower bounds to the inf-sup stability constant, a key ingredient of our rigorous a posteriori output error estimator. We develop a novel expanded formulation that enables treatment of quadratic outputs as linear compliant outputs. We also build on existing ideas in domain truncation to develop a radiation boundary condition to truncate unbounded domains. We integrate the different theoretical contributions and apply our methods as proof of concept to some representative applications in acoustic filter design and characterization. In the online stage, we achieve O(10) computational economies of cost while demonstrating both the rapid convergence of the reduced basis approximation, and the sharpness of our error estimators ([approx.] O(20)). The obtained computational economies are expected to be significantly greater for problems of larger size. We thus emphasize the feasibility of our methods in the many-query contexts of optimization, characterization, and control.<br>by Sugata Sen.<br>Ph.D.
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Frankenbach, Matthias [Verfasser]. "An Adjoint Based A Posteriori Error Estimator for Moving Meshes in Large Eddy Simulations / Matthias Frankenbach." München : Verlag Dr. Hut, 2014. http://d-nb.info/1055863877/34.
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