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Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods

Jan S. Hesthaven, Benjamin Stamm, Shun Zhang (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We propose two new algorithms to improve greedy sampling of high-dimensional functions. While the techniques have a substantial degree of generality, we frame the discussion in the context of methods for empirical interpolation and the development of reduced basis techniques for high-dimensional parametrized functions. The first algorithm, based on a saturation assumption of the error in the greedy algorithm, is shown to result in a significant reduction of the workload over the standard greedy...

Efficient numerical solution of mixed finite element discretizations by adaptive multilevel methods

Ronald H.W. Hoppe, Barbara Wohlmuth (1995)

Applications of Mathematics

We consider mixed finite element discretizations of second order elliptic boundary value problems. Emphasis is on the efficient iterative solution by multilevel techniques with respect to an adaptively generated hierarchy of nonuniform triangulations. In particular, we present two multilevel solvers, the first one relying on ideas from domain decomposition and the second one resulting from mixed hybridization. Local refinement of the underlying triangulations is done by efficient and reliable a...

Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations

Martin A. Grepl, Yvon Maday, Ngoc C. Nguyen, Anthony T. Patera (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems involving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function...

Electrowetting of a 3D drop: numerical modelling with electrostatic vector fields

Patrick Ciarlet Jr., Claire Scheid (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The electrowetting process is commonly used to handle very small amounts of liquid on a solid surface. This process can be modelled mathematically with the help of the shape optimization theory. However, solving numerically the resulting shape optimization problem is a very complex issue, even for reduced models that occur in simplified geometries. Recently, the second author obtained convincing results in the 2D axisymmetric case. In this paper, we propose and analyze a method that is suitable...

Embedding and a priori wavelet-adaptivity for Dirichlet problems

Andreas Rieder (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The accuracy of the domain embedding method from [A. Rieder, Modél. Math. Anal. Numér.32 (1998) 405-431] for the solution of Dirichlet problems suffers under a coarse boundary approximation. To overcome this drawback the method is furnished with an a priori (static) strategy for an adaptive approximation space refinement near the boundary. This is done by selecting suitable wavelet subspaces. Error estimates and numerical experiments validate the proposed adaptive scheme. In contrast to similar,...

Energetics and switching of quasi-uniform states in small ferromagnetic particles

François Alouges, Sergio Conti, Antonio DeSimone, Yvo Pokern (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present a numerical algorithm to solve the micromagnetic equations based on tangential-plane minimization for the magnetization update and a homothethic-layer decomposition of outer space for the computation of the demagnetization field. As a first application, detailed results on the flower-vortex transition in the cube of Micromagnetic Standard Problem number 3 are obtained, which confirm, with a different method, those already present in the literature, and validate our method and code. We...

Energetics and switching of quasi-uniform states in small ferromagnetic particles

François Alouges, Sergio Conti, Antonio DeSimone, Yvo Pokern (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a numerical algorithm to solve the micromagnetic equations based on tangential-plane minimization for the magnetization update and a homothethic-layer decomposition of outer space for the computation of the demagnetization field. As a first application, detailed results on the flower-vortex transition in the cube of Micromagnetic Standard Problem number 3 are obtained, which confirm, with a different method, those already present in the literature, and validate our method and...

Enrichissement des interpolations d’éléments finis en utilisant des méthodes sans maillage

Antonio Huerta, Sonia Fernández-Méndez, Pedro Díez (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Les méthodes sans maillage emploient une interpolation associée à un ensemble de particules : aucune information concernant la connectivité ne doit être fournie. Un des atouts de ces méthodes est que la discrétisation peut être enrichie d’une façon très simple, soit en augmentant le nombre de particules (analogue à la stratégie de raffinement h ), soit en augmentant l’ordre de consistance (analogue à la stratégie de raffinement p ). Néanmoins, le coût du calcul des fonctions d’interpolation est très...

Enrichissement des interpolations d'éléments finis en utilisant des méthodes sans maillage

Antonio Huerta, Sonia Fernández-Méndez, Pedro Díez (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Les méthodes sans maillage emploient une interpolation associée à un ensemble de particules : aucune information concernant la connectivité ne doit être fournie. Un des atouts de ces méthodes est que la discrétisation peut être enrichie d'une façon très simple, soit en augmentant le nombre de particules (analogue à la stratégie de raffinement h), soit en augmentant l'ordre de consistance (analogue à la stratégie de raffinement p). Néanmoins, le coût du calcul des fonctions d'interpolation est...

Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes

Martin Vohralík (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the lowest-order Raviart–Thomas mixed finite element method for second-order elliptic problems on simplicial meshes in two and three space dimensions. This method produces saddle-point problems for scalar and flux unknowns. We show how to easily and locally eliminate the flux unknowns, which implies the equivalence between this method and a particular multi-point finite volume scheme, without any approximate numerical integration. The matrix of the final linear system is sparse, positive...

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