Department of Mathematics
By exploiting the meshless property of kernel‐based collocation methods, we propose a fully automatic numerical recipe for solving interpolation/regression and boundary value problems adaptively. The proposed algorithm is built upon a least squares collocation formulation on some quasi‐random point sets with low discrepancy. A novel strategy is proposed to ensure that the fill distances of data points in the domain and on the boundary are in the same order of magnitude. To circumvent the potential problem of ill‐conditioning due to extremely small separation distance in the point sets, we add an extra dimension to the data points for generating shape parameters such that nearby kernels are of distinctive shape. This effectively eliminates the needs of shape parameter identification. Resulting linear systems were then solved by a greedy trial space algorithm to improve the robustness of the algorithm. Numerical examples are provided to demonstrate the efficiency and accuracy of the proposed methods.
adaptive trial space selection, Kansa method, overdetermined collocation, radial basis function
Source Publication Title
International Journal for Numerical Methods in Engineering
Copyright © 2017 John Wiley & Sons, Ltd.
This is the peer reviewed version of the following article: Ling L, Chiu SN. Fully adaptive kernel‐based methods. Int J Numer Methods Eng. 2018;114:454–467. https://doi.org/10.1002/nme.5750, which has been published in final form at http://dx.doi.org/10.1002/nme.5750. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.
Hong Kong Research Grant Council GRF; Hong Kong Baptist University FRG Grants; National Natural Science Foundation of China. Grant Number: 11528205
Link to Publisher's Edition
Ling, L., & Chiu, S. (2018). Fully adaptive kernel‐based methods. International Journal for Numerical Methods in Engineering, 114 (4), 454-467. https://doi.org/10.1002/nme.5750