3 research outputs found
Canonical sets of best L1-approximation
In mathematics, the term approximation usually means either interpolation on a point set or approximation with respect to a given distance. There is a concept, which joins the two approaches together, and this is the concept of characterization of the best approximants via interpolation. It turns out that for some large classes of functions the best approximants with respect to a certain distance can be constructed by interpolation on a point set that does not depend on the choice of the function to be approximated. Such point sets are called canonical sets of best approximation. The present paper summarizes results on canonical sets of best L1-approximation with emphasis on multivariate interpolation and best L1-approximation by blending functions. The best L1-approximants are characterized as transfinite interpolants on canonical sets. The notion of a Haar-Chebyshev system in the multivariate case is discussed also. In this context, it is shown that some multivariate interpolation spaces share properties of univariate Haar-Chebyshev systems. We study also the problem of best one-sided multivariate L 1-approximation by sums of univariate functions. Explicit constructions of best one-sided L1-approximants give rise to well-known and new inequalities
Eliminating Gibbs phenomena: A non-linear PetrovâGalerkin method for the convectionâdiffusionâreaction equation
In this article we consider the numerical approximation of the convection-diffusion-reaction equation. One of the main challenges of designing a numerical method for this problem is that boundary layers occurring in the convection-dominated case can lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The idea of this article is to consider the approximation problem as a residual minimization in dual norms in Lq-type Sobolev spaces, with 1 < q < . We then apply a non-standard, non-linear PetrovGalerkin discretization, that is applicable to reflexive Banach spaces such that the space itself and its dual are strictly convex. Similar to discontinuous Petrov-Galerkin methods, this method is based on minimizing the residual in a dual norm. Replacing the intractable dual norm by a suitable discrete dual norm gives rise to a non-linear inexact mixed method. This generalizes the Petrov-Galerkin framework developed in the context of discontinuous Petrov-Galerkin methods to more general Banach spaces. For the convection-diffusion-reaction equation, this yields a generalization of a similar approach from the L2-setting to the Lq-setting. A key advantage of considering a more general Banach space setting is that, in certain cases, the oscillations in the numerical approximation vanish as q tends to 1, as we will demonstrate using a few simple numerical examples
Approximation de fonctions et de données discrÚtes au sens de la norme L1 par splines polynomiales
Data and function approximation is fundamental in application domains like path planning or signal processing (sensor data). In such domains, it is important to obtain curves that preserve the shape of the data. Considering the results obtained for the problem of data interpolation, L1 splines appear to be a good solution. Contrary to classical L2 splines, these splines enable to preserve linearities in the data and to not introduce extraneous oscillations when applied on data sets with abrupt changes. We propose in this dissertation a study of the problem of best L1 approximation. This study includes developments on best L1 approximation of functions with a jump discontinuity in general spaces called Chebyshev and weak-Chebyshev spaces. Polynomial splines fit in this framework. Approximation algorithms by smoothing splines and spline fits based on a sliding window process are introduced. The methods previously proposed in the littĂ©rature can be relatively time consuming when applied on large datasets. Sliding window algorithm enables to obtain algorithms with linear complexity. Moreover, these algorithms can be parallelized. Finally, a new approximation approach with prescribed error is introduced. A pure algebraic algorithm with linear complexity is introduced. This algorithm is then applicable to real-time application.L'approximation de fonctions et de donnĂ©es discrĂštes est fondamentale dans des domaines tels que la planification de trajectoire ou le traitement du signal (donnĂ©es issues de capteurs). Dans ces domaines, il est important d'obtenir des courbes conservant la forme initiale des donnĂ©es. L'utilisation des splines L1 semble ĂȘtre une bonne solution au regard des rĂ©sultats obtenus pour le problĂšme d'interpolation de donnĂ©es discrĂštes par de telles splines. Ces splines permettent notamment de conserver les alignements dans les donnĂ©es et de ne pas introduire d'oscillations rĂ©siduelles comme c'est le cas pour les splines d'interpolation L2. Nous proposons dans cette thĂšse une Ă©tude du problĂšme de meilleure approximation au sens de la norme L1. Cette Ă©tude comprend des dĂ©veloppements thĂ©oriques sur la meilleure approximation L1 de fonctions prĂ©sentant une discontinuitĂ© de type saut dans des espaces fonctionnels gĂ©nĂ©raux appelĂ©s espace de Chebyshev et faiblement Chebyshev. Les splines polynomiales entrent dans ce cadre. Des algorithmes d'approximation de donnĂ©es discrĂštes au sens de la norme L1 par procĂ©dĂ© de fenĂȘtre glissante sont dĂ©veloppĂ©s en se basant sur les travaux existants sur les splines de lissage et d'ajustement. Les mĂ©thodes prĂ©sentĂ©es dans la littĂ©rature pour ces types de splines peuvent ĂȘtre relativement couteuse en temps de calcul. Les algorithmes par fenĂȘtre glissante permettent d'obtenir une complexitĂ© linĂ©aire en le nombre de donnĂ©es. De plus, une parallĂ©lisation est possible. Enfin, une approche originale d'approximation, appelĂ©e interpolation Ă delta prĂšs, est dĂ©veloppĂ©e. Nous proposons un algorithme algĂ©brique avec une complexitĂ© linĂ©aire et qui peut ĂȘtre utilisĂ© pour des applications temps rĂ©el