Department of Mathematics
On meshfree numerical differentiation
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 deriva- tives 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 pro- pose 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 behaviour and effectiveness of the proposed numerical differentiation scheme.
Source Publication Title
Analysis and Applications
World Scientific Publishing
This work was partially supported by a Hong Kong Re- search Grant Council GRF Grant, a Hong Kong Baptist University FRG Grant, the “Thousand Talents Program” of China, the National Natural Science Foundation of China (11601162), and the South China Normal University Grant.
Link to Publisher's Edition
Ling, Leevan, and Qi Ye. "On meshfree numerical differentiation." Analysis and Applications (2018).