Title :

Differentiation by integration with Jacobi polynomials

Author(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]

Journal title :

Journal of Computational and Applied Mathematics

Pages :

Pages 3015-3032

Publisher :

Elsevier

Publication date :

2011

ISSN :

0377-0427

English keyword(s) :

Numerical differentiation Ill-posed problems Jacobi orthogonal polynomials Orthogonal series

HAL domain(s) :

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

English abstract : [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. ...
Show more >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.Show less >

Language :

Anglais

Popular science :

Non

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL