Fully adaptive kernel-based methods

Leevan Ling*, Sung Nok Chiu

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

3 Citations (Scopus)
32 Downloads (Pure)


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.

Original languageEnglish
Pages (from-to)454-467
Number of pages14
JournalInternational Journal for Numerical Methods in Engineering
Issue number4
Early online date15 Dec 2017
Publication statusPublished - 27 Apr 2018

Scopus Subject Areas

  • Numerical Analysis
  • Engineering(all)
  • Applied Mathematics

User-Defined Keywords

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


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