Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
Radial basis function (RBF) interpolation can be very effective for scattered data in any number of dimensions. As one of their many applications, RBFs can provide highly accurate collocation-type numerical solutions to several classes of PDEs. To better understand the accuracy that can be obtained, we survey here derivative approximations based on RBFs using a similar Fourier analysis approach that has become the standard way for assessing the accuracy of finite difference schemes. We find that the accuracy is directly linked to the decay rate, at large arguments, of the (generalized) Fourier transform of the radial function. Three different types of convergence rates can be distinguished as the node density increases – polynomial, spectral, and superspectral, as exemplified, for example, by thin plate splines, multiquadrics, and Gaussians respectively.
Supplementary Material (0)
- M.D. Buhmann, Radial Basis Functions (Cambridge Univ. Press, Cambridge, 2003).
- M.D. Buhmann and N. Dyn, Spectral convergence of multiquadric interpolation, in: Multivariate Approximation: From CAGD to Wavelets, eds. K. Jetter and F.I. Utreras (World Scientific, Singapore, 1993) pp. 35–75.
- M.D. Buhmann and M.J.D. Powell, Radial basis function interpolation on an infinite regular grid, in: Algorithms for Approximation, Vol. II, eds. M.G. Cox and J.C. Mason (Chapman & Hall, London, 1990) pp. 146–169.
- R.E. Carlson and T.A. Foley, The parameter R 2 in multiquadric interpolation, Comput. Math. Appl. 21 (1991) 29–42.
- T.A. Driscoll and B. Fornberg, Interpolation in the limit of increasingly flat radial basis functions, Comput. Math. Appl. 43 (2002) 413–422.
- T.A. Foley, Near optimal parameter selection for multiquadric interpolation, J. Appl. Sci. Comput. 1 (1994) 54–69.
- B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge Univ. Press, Cambridge, 1996).
- B. Fornberg and G. Wright, Stable computation of multiquadric interpolants for all values of the shape parameter, Comput. Math. Appl. (2004) to appear.
- B. Fornberg, G. Wright and E. Larsson, Some observations regarding interpolants in the limit of flat radial basis functions, Comput. Math. Appl. 47 (2004) 37–55.
- K. Jetter, Multivariate approximation from the cardinal point of view, in: Approximation Theory, Vol. VII, eds. E.W. Cheney, C.K. Chui and L.L. Schumaker (Academic Press, New York, 1992) pp. 131–161.
- D.S. Jones, Generalized Functions (McGraw-Hill, New York, 1966).
- E.J. Kansa, A scattered data approximation scheme with applications to computational fluid dynamics. I. Surface approximations and partial derivative estimates, Comput. Math. Appl. 19(8/9) (1990) 127–145.
- E.J. Kansa, Multiquadrics – a scattered data approximation scheme with applications to computational fluid dynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl. 19(8/9) (1990) 147–161.
- E. Larsson and B. Fornberg, A numerical study of radial basis function based solution methods for elliptic PDEs, Comput. Math. Appl. 46 (2003) 891–902.
- M.J. Lighthill, Fourier Analysis and Generalised Functions (Cambridge Univ. Press, Cambridge, 1958).
- W.R. Madych, Miscellaneous error bounds for multiquadric and related interpolators, Comput. Math. Appl. 24 (1992) 121–138.
- W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive functions I, Approx. Theory Appl. 4 (1988) 77–89.
- W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive functions II, Approx. Theory Appl. 4, and Math. Comp. 54 (1990) 211–230.
- S. Rippa, An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv. Comput. Math. 11 (1999) 193–210.
- J. Yoon, Spectral approximation orders of radial basis function interpolation on the Sobolev space, SIAM J. Math. Anal. 33(4) (2001) 946–958.
About this Article
- Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids
Advances in Computational Mathematics
Volume 23, Issue 1-2 , pp 5-20
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links