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Construction de bases d'ondelettes α-höldériennes.

Albert Cohen — 1990

Revista Matemática Iberoamericana

Dans cet article, nous reprenons une méthode due à Ingrid Daubechies pour générer des bases orthonormales de fonctions dans L(R) de la forme {2 ψ (2x - k)} à partir de filtres miroir en quadrature (QMF) tels que l'ondelette ψ ait de bonnes propriétés de régularité. Une estimation de l'exposant de Hölder global optimal est obtenue en caractérisant précisément la decroissance de la fonction ψ'. Nous précisons finalement les liens exacts entre la régularité de l'ondelette et son ordre de cancellation...

Greedy Algorithms for Adaptive Approximation

Albert Cohen — 2009

Bollettino dell'Unione Matematica Italiana

We discuss the performances of greedy algorithms for two problems of numerical approximation. The first one is the best approximation of an arbitrary function by an N-terms linear combination of simple functions adaptively picked within a large dictionary. The second one is the approximation of an arbitrary function by a piecewise polynomial function on an optimally adapted triangulation of cardinality N. Performance is measured in terms of convergence rate with respect to the number of element...

Ondelettes, espaces d’interpolation et applications

Albert Cohen

Séminaire Équations aux dérivées partielles

Nous établissons des résultats d’interpolation non-standards entre les espaces de Besov et les espaces L 1 et B V , avec des applications aux lemmes de régularité en moyenne et aux inégalités de type Gagliardo-Nirenberg. La preuve de ces résultats utilise les décompositions dans des bases d’ondelettes.

Non-separable bidimensional wavelet bases.

Albert CohenIngrid Daubechies — 1993

Revista Matemática Iberoamericana

We build orthonormal and biorthogonal wavelet bases of L(R) with dilation matrices of determinant 2. As for the one dimensional case, our construction uses a scaling function which solves a two-scale difference equation associated to a FIR filter. Our wavelets are generated from a single compactly supported mother function. However, the regularity of these functions cannot be derived by the same approach as in the one dimensional case. We review existing techniques to evaluate the regularity of...

A new technique to estimate the regularity of refinable functions.

Albert CohenIngrid Daubechies — 1996

Revista Matemática Iberoamericana

We study the regularity of refinable functions by analyzing the spectral properties of special operators associated to the refinement equation; in particular, we use the Fredholm determinant theory to derive numerical estimates for the spectral radius of these operators in certain spaces. This new technique is particularly useful for estimating the regularity in the cases where the refinement equation has an infinite number of nonzero coefficients and in the multidimensional cases.

Adaptivity and variational stabilization for convection-diffusion equations

Albert CohenWolfgang DahmenGerrit Welper — 2012

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

In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies, based on...

Régularité des bases d'ondelettes et mesures ergodiques.

Albert CohenJean-Pierre Conze — 1992

Revista Matemática Iberoamericana

Nous reprenons la construction des bases orthonormées d'ondelettes à partir des filtres miroirs en quadrature tel qu'elle apparaît dans [4]. Nous montrons que leur régularité est liée à une mesure invariante pour la transformation ω → 2ω mod-2π. Cette méthode permet d'obtenir le facteur exact qui relie asymptotiquement la régularité des ondelettes constriutes dans [4] à la taille de leur support.

Adaptivity and variational stabilization for convection-diffusion equations

Albert CohenWolfgang DahmenGerrit Welper — 2012

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies, based on...

Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs

Abdellah ChkifaAlbert CohenRonald DeVoreChristoph Schwab — 2013

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

The numerical approximation of parametric partial differential equations is a computational challenge, in particular when the number of involved parameter is large. This paper considers a model class of second order, linear, parametric, elliptic PDEs on a bounded domain with diffusion coefficients depending on the parameters in an affine manner. For such models, it was shown in [9, 10] that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations...

Harmonic analysis of the space BV.

Albert CohenWolfgang DahmenIngrid DaubechiesRonald DeVore — 2003

Revista Matemática Iberoamericana

We establish new results on the space BV of functions with bounded variation. While it is well known that this space admits no unconditional basis, we show that it is almost characterized by wavelet expansions in the following sense: if a function f is in BV, its coefficient sequence in a BV normalized wavelet basis satisfies a class of weak-l1 type estimates. These weak estimates can be employed to prove many interesting results. We use them to identify the interpolation spaces between BV and Sobolev...

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