Node generation for rbf-fd methods by qr factorization

Research output: Contribution to journalArticlepeer-review

Abstract

Polyharmonic spline (PHS) radial basis functions (RBFs) have been used in conjunction with polynomials to create RBF finite-difference (RBF-FD) methods. In 2D, these methods are usually implemented with Cartesian nodes, hexagonal nodes, or most commonly, quasi-uniformly distributed nodes generated through fast algorithms. We explore novel strategies for computing the placement of sampling points for RBF-FD methods in both 1D and 2D while investigating the benefits of using these points. The optimality of sampling points is determined by a novel piecewise-defined Lebesgue constant. Points are then sampled by modifying a simple, robust, column-pivoting QR algorithm previously implemented to find sets of near-optimal sampling points for polynomial approximation. Using the newly computed sampling points for these methods preserves accuracy while reducing computational costs by mitigating stencil size restrictions for RBF-FD methods. The novel algorithm can also be used to select boundary points to be used in conjunction with fast algorithms that provide quasi-uniformly distributed nodes.

Original languageEnglish (US)
Article number1845
JournalMathematics
Volume9
Issue number16
DOIs
StatePublished - Aug 2 2021

Keywords

  • Complex regions
  • Finite-difference methods
  • Lebesgue constant
  • Node sampling
  • RBF-FD
  • Radial basis functions

ASJC Scopus subject areas

  • Mathematics(all)

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