Titre :

Differentiation by integration with Jacobi polynomials

Auteur(s) :

Liu, Da-Yan [Auteur]
Laboratoire d'Automatique, Génie Informatique et Signal [LAGIS]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Non-Asymptotic estimation for online systems [NON-A]
Gibaru, Olivier [Auteur]
Non-Asymptotic estimation for online systems [NON-A]
Laboratoire de Métrologie et de Mathématiques Appliquées [L2MA]
Perruquetti, Wilfrid [Auteur]
Non-Asymptotic estimation for online systems [NON-A]
Systèmes Non Linéaires et à Retards [SyNeR]

Titre de la revue :

Journal of Computational and Applied Mathematics

Pagination :

Pages 3015-3032

Éditeur :

Elsevier

Date de publication :

2011

ISSN :

0377-0427

Mot(s)-clé(s) en anglais :

Numerical differentiation Ill-posed problems Jacobi orthogonal polynomials Orthogonal series

Discipline(s) HAL :

Mathématiques [math]/Analyse numérique [math.NA]

Résumé en anglais : [en]

In this paper, the numerical differentiation by integration method based on Jacobi polynomials originally introduced by Mboup, Fliess and Join is revisited in the central case where the used integration window is centered. ...
Lire la suite >In this paper, the numerical differentiation by integration method based on Jacobi polynomials originally introduced by Mboup, Fliess and Join is revisited in the central case where the used integration window is centered. Such method based on Jacobi polynomials was introduced through an algebraic approach and extends the numerical differentiation by integration method introduced by Lanczos. The here proposed method is used to estimate the $n^{th}$ ($n \in \mathbb{N}$) order derivative from noisy data of a smooth function belonging to at least $C^{n+1+q}$ $(q \in \mathbb{N})$. In the recent paper of Mboup, Fliess and Join , where the causal and anti-causal case were investigated, the mismodelling due to the truncation of the Taylor expansion was investigated and improved allowing a small time-delay in the derivative estimation. Here, for the central case, we show that the bias error is $O(h^{q+2})$ where $h$ is the integration window length for $f\in C^{n+q+2}$ in the noise free case and the corresponding convergence rate is $O(\delta^{\frac{q+1}{n+1+q}})$ where $\delta$ is the noise level for a well chosen integration window length. Numerical examples show that this proposed method is stable and effective.Lire moins >

Langue :

Anglais

Vulgarisation :

Non

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL