Document Type

Journal Article

Department/Unit

Department of Mathematics

Language

English

Abstract

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.

Keywords

adaptive trial space selection, Kansa method, overdetermined collocation, radial basis function

Publication Date

4-27-2018

Source Publication Title

International Journal for Numerical Methods in Engineering

Volume

114

Issue

4

Start Page

454

End Page

467

Publisher

Wiley

Peer Reviewed

1

Copyright

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.

Funder

Hong Kong Research Grant Council GRF; Hong Kong Baptist University FRG Grants; National Natural Science Foundation of China. Grant Number: 11528205

DOI

10.1002/nme.5750

Link to Publisher's Edition

http://dx.doi.org/10.1002/nme.5750

ISSN (print)

00295981

ISSN (electronic)

10970207

Available for download on Wednesday, May 01, 2019

Included in

Mathematics Commons

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