Interpolation in the limit of increasingly flat radial basis functions

By Tobin A. Driscoll, Bengt Fornberg. 2002

361 citations recorded by the source corpus.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/ccaaca3e5d753d6b541cbde64077b86ab2133a22

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In Computational Mechanics

In Electrical and Electronic Engineering

In Mechanics of Materials

Citations in the corpus

Multivariate Interpolation by Polynomials and Radial Basis Functions (2005, neutral): “To see a bound on the degree, consider |ρ| = k and conclude that dρ(X,Y ) := det ( ‖xi − yj‖2ρi2 ) 1≤i,j≤M = ∑ π∈SM (−1)π M∏ j=1 ‖xj − yπ(j)‖ 2ρj 2 has total degree at most 2|ρ| = 2k. Altogether we have Lemma 5 The polynomials pk(X,Y ) have maximal degree 2k as polynomials in X and Y .”
Integrated radial basis functions‐based differential quadrature method and its performance (2007, methodology): “After that, the RBFs attract more and more attentions on solving PDEs as a global method, and a series of papers were published in the literature [6–12].”
DSC‐Ritz method for the free vibration analysis of Mindlin plates (2005, background): “Certainly, a detailed study of the impact of the parameter space as that given by Driscoll and Fornberg for their radial basis functions [51] would aid our understanding.”
Gaussian RBF-FD weights and its corresponding local truncation errors (2012, neutral): “well known that RBF-FD formulas approach standard finite difference formulas [6].”
DSC‐Ritz method for high‐mode frequency analysis of thick shallow shells (2005, methodology): “It is to point out that albeit the proposed DSC-Ritz method has a distinct mathematical foundation, it shares some common features with a class of meshless methods, such as the smooth particle hydrodynamics [40, 41], reproducing kernel particle method [42], radial basis functions [43], and element-f”
A numerical study of some radial basis function based solution methods for elliptic PDEs (2003, background): “1 (for some analysis of approximations in the limit ε → 0, see [15]).”
Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and RBF-PS discretizations with optimal shape parameter (2008, background): “[43,44]) and Schaback [45] showed that certain limiting cases of radial basis functions correspond to polynomial interpolants.”
RBF-based meshless method for large deflection of thin plates (2007, background): “Furthermore, previous studies [13] have shown that even values of n produced inaccurate solutions.”
The Runge phenomenon and spatially variable shape parameters in RBF interpolation (2007, methodology): “It was demonstrated in [9] that, in the limit of ε → 0, RBF interpolants in 1-D in general converge to the Lagrange interpolating polynomial.”
On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs (2010, methodology): “Driscoll and Fornberg [16] in fact reported this result: using MQ interpolation on a 1⁄20,1 1⁄20,1 square using a uniform grid with h1⁄41/5, they found that the error diverges at the rate of”
Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations (2003, methodology): “Other great contributions in using RBFs to solve PDEs include the work of Fornberg [13], Hon and Wu [14], Chen and Tanaka [15], Chen et al.”
Stable Computations with Gaussian Radial Basis Functions (2011, methodology): “1) form two successive ill-conditioned numerical steps in obtaining a quantity s(x, ε) that we know in general depends in a wellconditioned way on the data {xk, fk} [4], [6], [14], [19], [26].”
On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy (2016, methodology): “When calculating in the small ε regime with a stable algorithm, it is not surprising that added polynomials offer no benefit, since the GA themselves virtually span the same function space and exactly the same space in the ε→ 0 limit [8, 21, 28].”
Solving PDEs with radial basis functions * (2015, methodology): “RBF-Direct uses ill-conditioned expansion coefficients λk as intermediate quantities for arriving at what should be a well-conditioned result (Driscoll and Fornberg 2002, Fornberg, Wright and Larsson 2004).”