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Xuelei Chen - One of the best experts on this subject based on the ideXlab platform.
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cosmological models with Lagrange multiplier field
Physics Letters B, 2011Co-Authors: Changjun Gao, Yan Gong, Xin Wang, Xuelei ChenAbstract:We first consider the Einstein-aether theory with a gravitational coupling and a Lagrange multiplier field, and then consider the non-minimally coupled quintessence field theory with Lagrange multiplier field. We study the influence of the Lagrange multiplier field on these models. We show that the energy density evolution of the Einstein-aether field and the quintessence field are significantly modified. The energy density of the Einstein-aether is nearly a constant during the entire history of the Universe. The energy density of the quintessence field can also be kept nearly constant in the matter dominated Universe, or even exhibit a phantom-like behavior for some models. This suggests a possible dynamical origin of the cosmological constant or dark energy. Further more, for the canonical quintessence in the absence of gravitational coupling, we find that the quintessence scalar field can play the role of cold dark matter with the introduction of a Lagrange multiplier field. We conclude that the Lagrange multiplier field could play a very interesting and important role in the construction of cosmological models. (C) 2011 Elsevier B.V. All rights reserved.
Lloyd N Trefethen - One of the best experts on this subject based on the ideXlab platform.
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barycentric Lagrange interpolation
Siam Review, 2004Co-Authors: Jeanpaul Berrut, Lloyd N TrefethenAbstract:Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable. It deserves to be known as the standard method of polynomial interpolation.
Bishnu P Lamichhane - One of the best experts on this subject based on the ideXlab platform.
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higher order mortar finite elements with dual Lagrange multiplier spaces and applications
2006Co-Authors: Bishnu P LamichhaneAbstract:The numerical approximation of partial differential equations coming from physical and engineering modeling is often a challenging task. Most often these partial differential equations are discretized with finite elements and can be solved by modern super-computers. Working with different discretization techniques in different subdomains or independent triangulations, the challenging task is to couple these different discretization schemes or non-matching triangulation without losing the optimality of the approach. Mortar methods yield optimal and flexible coupling techniques for different discretization schemes. Especially when combined with dual Lagrange multiplier spaces, the efficient realization of the weak matching condition is possible, and efficient multigrid methods can be adapted to the non-conforming situation. In this thesis, we concentrate on higher order dual Lagrange multiplier spaces for mortar finite elements. These non-standard Lagrange multipliers show the same qualitative a priori estimates and quantitative numerical results as the standard ones and yield locally supported basis functions for the constrained space leading to an efficient numerical realization. Working with abstract assumptions on Lagrange multiplier spaces, we prove optimal a priori estimates for mortar finite elements allowing that the dimension of the Lagrange multiplier space can be smaller than the dimension of the trace space of the finite element space from the slave side (with zero boundary condition on the interface). Geometrically non-conforming decompositions and locally refined meshes are also covered. In two dimensions, we show that a dual Lagrange multiplier space can be constructed for a finite element space of any order satisfying these abstract assumptions. In contrast to earlier approaches, these Lagrange multiplier basis functions have the same support as the nodal finite element basis functions. Using an interesting relation between biorthogonality and quadrature formulas, we prove that an optimal dual Lagrange multiplier space for a finite element space can be constructed if and only if the finite element space is based on Gau-Lobatto nodes. The two-dimensional construction can easily be extended to the three-dimensional case for a finite element space with tensor product structure. If a finite element space does not have the tensor product structure, e.g., serendipity elements on hexahedra or conforming finite elements on simplices, the situation is more difficult. To deal with this problem, we generalize the idea of a dual Lagrange multiplier space by introducing a quasi-dual Lagrange multiplier space for quadratic serendipity elements. Furthermore, working with a more general assumption that the Lagrange multiplier space can have smaller dimension than the trace space of the finite element space at the slave side (with zero boundary condition on the interface), we introduce dual Lagrange multiplier spaces for quadratic tetrahedral and serendipity elements. Numerical results are presented to illustrate the performance of our approach. We also study interface problems arising from heat conduction with a discontinuous flux and solution within the framework of mortar finite element methods. Taking into account non-homogeneous jumps of the solution and the flux across the interface, we give a saddle point formulation of the interface problems. Optimal a priori estimates are proved and numerical results are provided. We have applied mortar finite elements to couple different physical models, material laws and discretization schemes. Furthermore, time-dependent heat transfer problems with sliding meshes are also considered. Another particular interest for us is the locking phenomenon in linear and nonlinear elasticity. We analyze low order finite element methods based on the Hu-Washizu formulation in linear elasticity and prove the robust and optimal convergence of the finite element approximation of the displacement for the nearly incompressible case. A three-field mixed formulation for finite elasticity is also introduced and numerical results are presented. Mortar finite elements for coupling two different materials in elasticity, where one is a nearly incompressible material and the other one is a compressible material are analyzed for the linear elastic case, and numerical results are provided to verify the theoretical results. In dieser Arbeit stehen duale Lagrange-Multiplikatorraume hoherer Ordnung fur Mortar-Finite-Elemente im Mittelpunkt. Diese speziellen Lagrange-Multiplikatoren weisen die gleichen qualitativen und quantitativen numerischen Eigenschaften auf wie Standard-Lagrange-Multiplikatoren und liefern Basisfunktionen mit lokalem Trager fur den die Kopplungsbedingung respektierenden Raum, was zu einer effizienten numerischen Unsetzung fuhrt. Indem mit abstrakten Annahmen fur die Lagrange-Multiplikatorraume gearbeitet wird, konnen a priori Abschatzungen fur Mortar-Finite-Elemente gezeigt werden, die es erlauben, dass die Dimension des Lagrange-Multiplikatorraumes kleiner sein darf als die Dimension der Spur des finite Element-Raumes, der die Null-Randbedingung am Interface der Slave-Seite erfullt. Geometrisch nicht-konforme Zerlegungen und lokal verfeinerte Gitter sind auch verwendbar. Fur zweidimensionale Mortar-Finite-Elemente wird gezeigt, dass ein dualer Lagrange-Multiplikatorraum fur einen Finite-Element-Raum beliebiger Ordnung konstruiert werden kann, der diesen abstrakten Annahmen genugt. Im Gegensatz zu fruheren Ansatzen haben diese Lagrange-Multiplikator-Basisfunktionen den gleichen Trager wie die nodalen Finiten-Element-Basisfunktionen. Indem eine interessante Beziehung zwischen der Biorthogonalitat und Quadraturformeln verwendet wird, wird bewiesen, dass ein optimaler dualer Lagrange-Multiplikatorraum fur einen Finite-Element-Raum nur dann konstruiert werden kann, wenn der Finite-Element-Raum auf Gau-Lobatto-Knoten basiert. Das zweidimensionale Konstruktionsschema kann leicht auf den dreidimensionalen Fall erweitert werden, sofern ein Finite-Element-Raum mit Tensorprodukt-Struktur vorliegt. Wenn ein Finite-Element-Raum keine Tensorprodukt-Struktur aufweist, wie z.B. bei Serendipity-Elementen auf Hexaedernetz oder bei konforme simplizialen Elementen, ist die Lage schwieriger. Um dieses Problem zu behandeln, wird die Idee des dualen Lagrange-Multiplikatorraums verallgemeinert, indem ein quasi-dualer Lagrange-Multiplikatorraum fur quadratische Serendipity-Elemente eingefuhrt wird. Ferner werden anhand der allgemeineren Annahme, dass der Lagrange-Multiplikatorraum eine kleinere Dimension als der Spur-Raum des Approximationsraums auf der Slave-Seite (mit Null-Randbedingung am Interface) haben kann, duale Lagrange-Multiplikatorraume fur quadratische simpliziale- und Serendipity-Elemente eingefuhrt. Numerische Ergebnisse demonstrieren die Effizienz des Ansatzes. Des Weiteren werden Interface-Probleme im Rahmen der Mortar-Finite-Element-Methoden behandelt, die aus der Warmeleitung mit unstetigem Fluss herruhren. Unter Berucksichtigung von inhomogenen Sprungen in der Losung und im Fluss am Interface wird eine Sattelpunktformulierung des Interface-Problemes hergeleitet. Deren Optimalitat wird bewiesen und durch numerische Ergebnisse untermauert. Zusatzlich werden Mortar-Finite-Elemente zur Kopplung verschiedener physikalischer Modelle, Materialgesetze und Diskretisierungs-Schemata angewandt. Der letzte wichtige Punkt, der in dieser Arbeit beleuchtet wird, ist das sogenannte 'Locking'-Phanomen in der linearen und nichtlinearen Elastizitat. Analysiert werden Finite-Element-Methoden von niedrigster Ordnung, die auf der Hu-Washizu-Formulierung basieren. Es wird gezeigt, dass die numerische Approximation robust und optimal konvergiert. Eine Drei-Feld-gemischte Formulierung fur nichtlineare (finite) Elastizitat wird ebenfalls eingefuhrt und numerische Ergebnisse dazu prasentiert. Der Fall finiter Elemente zur Kopplung zweier verschiedener Materialien, wobei eines davon nahezu inkompressibel und das andere kompressibel ist, wird fur den linear-elastischen Fall analysiert; numerische Ergebnisse bestatigen die theoretischen Aussagen.
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higher order mortar finite element methods in 3d with dual Lagrange multiplier bases
Numerische Mathematik, 2005Co-Authors: Bishnu P Lamichhane, Rob Stevenson, Barbara WohlmuthAbstract:Mortar methods with dual Lagrange multiplier bases provide a flexible, efficient and optimal way to couple different discretization schemes or nonmatching triangulations. Here, we generalize the concept of dual Lagrange multiplier bases by relaxing the condition that the trace space of the approximation space at the slave side with zero boundary condition on the interface and the Lagrange multiplier space have the same dimension. We provide a new theoretical framework within this relaxed setting, which opens a new and simpler way to construct dual Lagrange multiplier bases for higher order finite element spaces. As examples, we consider quadratic and cubic tetrahedral elements and quadratic serendipity hexahedral elements. Numerical results illustrate the performance of our approach.
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higher order dual Lagrange multiplier spaces for mortar finite element discretizations
Calcolo, 2002Co-Authors: Bishnu P Lamichhane, Barbara WohlmuthAbstract:Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements. In particular, we focus on dual Lagrange multiplier spaces. These non-standard Lagrange multiplier spaces yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces. As a result, standard efficient iterative solvers as multigrid methods can be easily adapted to the nonconforming situation. We construct locally supported and continuous dual basis functions for quadratic finite elements starting from the discontinuous quadratic dual basis functions for the Lagrange multiplier space. In particular, we compare different dual Lagrange multiplier spaces and piecewise linear and quadratic finite elements. The optimality of the associated mortar method is shown. Numerical results illustrate the performance of our approach.
Vesa Valimaki - One of the best experts on this subject based on the ideXlab platform.
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a computationally efficient coefficient update technique for Lagrange fractional delay filters
International Conference on Acoustics Speech and Signal Processing, 2008Co-Authors: A Haghparast, Vesa ValimakiAbstract:A new algorithm for coefficient update of the Lagrange fractional delay FIR filter is proposed, which reduces the computational complexity dramatically. It is based on rearranging the polynomial terms of the Lagrange interpolation formula and computing the common product terms only once. Reordering the Lagrange interpolation formula yields two other methods for updating the coefficients, the direct and the division- based methods. The division-based method uses only one division per coefficient. The two latter methods reduce the computational load, although they are not as efficient as the new algorithm. Finally, the superiority of the direct form FIR implementation of the Lagrange fractional delay filter and the new coefficient update method over other existing methods is demonstrated in an audio signal processing application.
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Fractional Delay Filter Design Based on Truncated Lagrange Interpolation
IEEE Signal Processing Letters, 2007Co-Authors: Vesa Valimaki, Azadeh HaghparastAbstract:A new design method for fractional delay filters based on truncating the impulse response of the Lagrange interpolation filter is presented. The truncated Lagrange fractional delay filter introduces a wider approximation bandwidth than the Lagrange filter. However, because of truncation, a ripple caused by the Gibbs phenomenon appears in the filter's frequency response. Proper choices of filter order and prototype filter order allow adjusting the overshoot to a desired level and simultaneously reducing the overall frequency-response error. The design of the proposed filter is computationally efficient, because it is based on polynomial formulas, which have common terms for all coefficients.
Tiejun Huang - One of the best experts on this subject based on the ideXlab platform.
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a background proportion adaptive Lagrange multiplier selection method for surveillance video on hevc
International Conference on Multimedia and Expo, 2013Co-Authors: Long Zhao, Xianguo Zhang, Yonghong Tian, Ronggang Wang, Tiejun HuangAbstract:In the recent video coding standards, the selection of Lagrange multiplier is crucial to achieve trade-off between the choices of low-distortion and low-bitrate prediction modes. For surveillance video coding, the rate-distortion analysis shows that, a larger Lagrange multiplier should be used if the background in a coding unit took a larger proportion. Therefore, a modified Lagrange multiplier might be better for rate-distortion optimization. To address this problem, we perform an in-depth analysis on the relationship between the optimal Lagrange multiplier and the background proportion, and then propose a Lagrange multiplier selection model to obtain the optimal coding performance for surveillance videos. Following this, we further develop a Lagrange multiplier optimized video coding method. Experimental results show that our coding method can averagely achieve 18.07% bitrate saving on CIF sequences and 11.88% on SD sequences against the background-irrelevant Lagrange multiplier selection method.
