On meshfree numerical differentiation

Leevan Ling, Qi Ye*

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

1 Citation (Scopus)


We combine techniques in meshfree methods and Gaussian process regressions to construct kernel-based estimators for numerical derivatives from noisy data. Specially, we construct meshfree estimators from normal random variables, which are defined by kernel-based probability measures induced from symmetric positive definite kernels, to reconstruct the unknown partial derivatives from scattered noisy data. Our developed theories give rise to Tikhonov regularization methods with a priori parameter, but the shape parameters of the kernels remain tunable. For that, we propose an error measure that is computable without the exact values of the derivative. This allows users to obtain a quasi-optimal kernel-based estimator by comparing the approximation quality of kernel-based estimators. Numerical examples in two dimensions and three dimensions are included to demonstrate the convergence behavior and effectiveness of the proposed numerical differentiation scheme.

Original languageEnglish
Pages (from-to)717-739
Number of pages23
JournalAnalysis and Applications
Issue number5
Publication statusPublished - 1 Sept 2018

Scopus Subject Areas

  • Analysis
  • Applied Mathematics

User-Defined Keywords

  • kernel-based approximation method
  • Meshfree method
  • numerical derivative
  • scattered noisy data


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