Discrete least-squares radial basis functions approximations

Siqing Li*, Leevan LING, Ka Chun Cheung

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

We consider discrete least-squares methods using radial basis functions. A general ℓ 2 -Tikhonov regularization with W 2 m -penalty is considered. We provide error estimates that are comparable to kernel-based interpolation in cases which the function it is approximating is within and is outside of the native space of the kernel. Our proven theories concern the denseness condition of collocation points and selection of regularization parameters. In particular, the unregularized least-squares method is shown to have W 2 μ (Ω) convergence for μ > d/2 on smooth domain Ω⊂R d . For any properly regularized least-squares method, the same convergence estimates hold for a large range of μ ≥ 0. These results are extended to the case of noisy data. Numerical demonstrations are provided to verify the theoretical results. In terms of applications, we also apply the proposed method to solve a heat equation whose initial condition has huge oscillation in the domain.

Original languageEnglish
Pages (from-to)542-552
Number of pages11
JournalApplied Mathematics and Computation
Volume355
DOIs
Publication statusPublished - 15 Aug 2019

Scopus Subject Areas

  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Error estimate
  • Kernel methods
  • Meshfree approximation
  • Noisy data
  • Tikhonov regularization

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