Finding Numerical Derivatives for Unstructured and Noisy Data by Multiscale Kernels

Leevan Ling

doi:10.1137/050630246

Finding Numerical Derivatives for Unstructured and Noisy Data by Multiscale Kernels

Leevan Ling^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

12 Citations (Scopus)

30 Downloads (Pure)

Abstract

The recently developed multiscale kernel of R. Opfer [Adv. Comput. Math., 25 (2006), pp. 357–380] is applied to approximate numerical derivatives. The proposed method is truly mesh‐free and can handle unstructured data with noise in any dimension. The method of Tikhonov and the method of L‐curve are employed for regularization; no information about the noise level is required. An error analysis is provided in a general setting for all dimensions. Numerical comparisons are given in two dimensions which show competitive results with recently published thin plate spline methods.

Original language	English
Pages (from-to)	1780-1800
Number of pages	21
Journal	SIAM Journal on Numerical Analysis
Volume	44
Issue number	4
DOIs	https://doi.org/10.1137/050630246
Publication status	Published - 26 Sept 2006

Scopus Subject Areas

Numerical Analysis
Computational Mathematics
Applied Mathematics

User-Defined Keywords

Inverse problems
L-curve
Multiscale kernel
Multivariate interpolation
Numerical differentiation
Tikhonov regularization
Unstructured data

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10.1137/050630246

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title = "Finding Numerical Derivatives for Unstructured and Noisy Data by Multiscale Kernels",

abstract = "The recently developed multiscale kernel of R. Opfer [Adv. Comput. Math., 25 (2006), pp. 357–380] is applied to approximate numerical derivatives. The proposed method is truly mesh‐free and can handle unstructured data with noise in any dimension. The method of Tikhonov and the method of L‐curve are employed for regularization; no information about the noise level is required. An error analysis is provided in a general setting for all dimensions. Numerical comparisons are given in two dimensions which show competitive results with recently published thin plate spline methods.",

keywords = "Inverse problems, L-curve, Multiscale kernel, Multivariate interpolation, Numerical differentiation, Tikhonov regularization, Unstructured data",

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Finding Numerical Derivatives for Unstructured and Noisy Data by Multiscale Kernels

Abstract

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