An insight into RBF-FD approximations augmented with polynomials

Abstract

Radial basis function-generated finite differences (RBF-FD) based on the combination of polyharmonic splines (PHS) with high degree polynomials have recently emerged as a powerful and robust numerical approach for the local interpolation and derivative approximation of functions over scattered node layouts. Among the key features, (i) high orders of accuracy can be achieved without the need of selecting a shape parameter or the issues related to numerical ill-conditioning, and (ii) the harmful edge effects associated to the use of high order polynomials (better known as Runge's phenomenon) can be overcome by simply increasing the stencil size for a fixed polynomial degree. The present study complements our previous results, providing an analytical insight into RBF-FD approximations augmented with polynomials. It is based on a closed-form expression for the interpolant, which reveals the mechanisms underlying these features, including the role of polynomials and RBFs in the interpolant, the approximation error, and the behavior of the cardinal functions near boundaries. Numerical examples are included for illustration.

Two-Dimensional RBF-ENO Method on Unstructured Grids

“…Based on [11,12] it is known that the combination of polyharmonics and Gaussians with polynomials overcomes the stagnation error. Bayona [3] shows that under certain assumptions the order of convergence is ensured by the polynomial part.…”

“…Considering ansatz (8) for the interpolation problem (7), (9) Bayona shows in [3], under the assumption of full rank of A and P, that the order of convergence is at least O(h l+1 ) based on the polynomial part. With similar techniques we can relax the assumptions of full rank of A by assuming ϕ to be a conditionally positive definite RBF of order l + 1.…”

Comparison of Moving Least Squares and RBF+poly for Interpolation and Derivative Approximation

“…If the node distribution is such that A and P are both full rank matrices, it was proven in [4] that the exact expressions for the λ and β coefficients can be obtained via block decomposition as…”

“…Essentially, it is a generalization of the classical finite difference (FD) method to scattered node layouts, yielding highly sparse differentiation matrices (better-conditioned than in the global approach) and high-order algebraic convergence. More recently, it has been found the advantages of PHS+poly generated RBF-FD weights over the standard approach based on infinitely smooth RBFs [3,4,5,6,18,19]. Among the key features, (i) high orders of accuracy can be achieved without the need of selecting a shape parameter or the issues related to numerical ill-conditioning, and (ii) the harmful edge effects associated to the use of high order polynomials (better known as Runge's phenomenon) can be overcome by simply increasing the stencil size for a fixed polynomial degree.…”

“…Following the numerical tests from [4,5,6,19], we then compare their performances for scattered data interpolation and derivative approximation in Section 4. Heuristic perspectives and numerical demonstrations are provided in 1-D and 2-D.…”