Node Generation for RBF-FD Methods by QR Factorization

Tony Liu and Rodrigo B. Platte
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Tony Liu: Department of Mathematics and Statistics, Air Force Institute of Technology, Dayton, OH 45433, USA
Rodrigo B. Platte: School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85281, USA

Mathematics, 2021, vol. 9, issue 16, 1-25

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.

Keywords: radial basis functions; RBF-FD; node sampling; lebesgue constant; complex regions; finite-difference methods (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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