The Runge phenomenon and spatially variable shape parameters in RBF interpolation

Computers and Mathematics with Applications

Volumn 54, Issue 3, 2007, Pages 379-398

  Fornberg, Bengt ^a   Zuev, Julia ^a  

^a UNIVERSITY OF COLORADO   (United States)

Author keywords

Conditioning; Radial basis functions; RBF; Runge phenomenon; Shape parameter

Indexed keywords

FINITE ELEMENT METHOD; INTERPOLATION; MESH GENERATION; POLYNOMIALS;

NUMERICAL CONDITIONING; SHAPE PARAMETERS;

RADIAL BASIS FUNCTION NETWORKS;

EID: 34250838673     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2007.01.028     Document Type: Article

References (34)

1. J. Zuev, Topics in Numerical Methods for solving PDEs, Ph.D. Thesis, University of Colorado at Boulder, Dept. of Applied Mathematics, 2007
2. Powell M.J.D. The theory of radial basis function approximation in 1990. Advances in Numerical Analysis. In: Light W. (Ed). Wavelets, Subdivision Algorithms and Radial Functions Vol. II (1990), Oxford University Press, Oxford 105-210
3. Fornberg B. A Practical Guide to Pseudospectral Methods (1996), Cambridge University Press, Cambridge
4. Trefethen L.N. Spectral Methods in Matlab (2000), SIAM, Philadelphia
5. Powell M.J.D. Approximation Theory and Methods (1981), Cambridge University Press, Cambridge
6. Duchon J. Splines minimizing rotation-invariant seminorms in Sobolev spaces. Constructive Theory of Functions of Several Variables. In: Schempp W., and Zeller K. (Eds). Lecture Notes in Mathematics vol. 571 (1977), Springer Verlag, Berlin 85-100
7. Madych W.R., and Nelson S.A. Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation. J. Approx. Theory 70 (1992) 94-114
8. Yoon J. Spectral approximation orders of radial basis function interpolation on the Sobolev space. SIAM J. Math. Anal. 23 (2001) 946-958
9. Driscoll T.A., and Fornberg B. Interpolation in the limit of increasingly flat radial basis functions. Comput. Math. Appl. 43 (2002) 413-422
10. Fornberg B., Wright G., and Larsson E. Some observations regarding interpolants in the limit of flat radial basis functions. Comput. Math. Appl. 47 (2004) 37-55
11. Fornberg B., and Wright G. Stable computation of multiquadric interpolants for all values of the shape parameter. Comput. Math. Appl. 48 (2004) 853-867
12. B. Fornberg, C. Piret, A stable algorithm for flat radial basis functions on a sphere, SIAM J. Sci. Comput. (submitted for publication)
13. E. Larsson, B. Fornberg, A stable algorithm for flat radial basis functions, to be submitted
14. Larsson E., and Fornberg B. A numerical study of radial basis function based solution methods for elliptic PDEs. Comput. Math. Appl. 46 (2003) 891-902
15. Kansa E.J., and Hon Y.C. Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations. Comput. Math. Appl. 39 7-8 (2000) 123-137
16. Ling L., and Hon Y.C. Improved numerical solver for Kansa's method based on affine space decomposition. Eng. Anal. with Bound. Elem. 29 (2005) 1077-1085
17. Carlson R.E., and Foley T.A. The parameter R2 in multiquadric interpolation. Comput. Math. Appl. 21 (1991) 29-42
18. Foley T.A. Interpolation and approximation of 3-D and 4-D scattered data. Comput. Math. Appl. 13 (1987) 711-740
19. Goldberg M.A., Chen C.S., and Karur S.R. Improved multiquadric approximation for partial differential equations. Eng. Anal. Bound. Elem. 18 (1996) 9-17
20. Hardy R.L. Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res. 76 (1971) 1905-1915
21. Kansa E.J., and Carlson R.E. Improved accuracy of multiquadric interpolation using variable shape parameters. Comput. Math. Appl. 24 12 (1992) 99-120
22. Rippa S. An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv. Comput. Math. 11 (1999) 193-210
23. Larsson E., and Fornberg B. Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions. Comput. Math. Appl. 49 (2005) 103-130
24. Fornberg B., Driscoll T.A., Wright G., and Charles R. Observations on the behavior of radial basis functions near boundaries. Comput. Math. Appl. 43 (2002) 473-490
25. Kansa E.J. Multiquadrics - 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
26. Bozzini M., Lenarduzzi L., and Schaback R. Adaptive interpolation by scaled multiquadrics. Adv. Comput. Math. 16 (2002) 375-387
27. T.A. Driscoll, A. Heryudono, Adaptive residual subsampling methods for radial basis function interpolation and collocation problems, Comput. Math. Appl. (in press)
28. Wertz J., Kansa E.J., and Ling L. The role of the multiquadric shape parameters in solving elliptic partial differential equations. Comput. Math. Appl. 51 (2006) 1335-1348
29. Schaback R. Multivariate interpolation by polynomials and radial basis functions. Constr. Approx. 21 (2005) 293-317
30. Vértesi P. Optimal Lebesgue constant for Lagrange interpolation. SIAM J. Numer. Anal. 27 (1990) 1322-1331
31. Platte R.B., and Driscoll T.A. Polynomials and potential theory for Gaussian radial basis function interpolation. SIAM J. Numer. Anal. 43 (2005) 750-766
32. Kirkpatrick S., Gelatt C.D., and Vecci M.P. Optimization by simulated annealing. Science 220 (1983) 671-680
33. Press W.H., Flannery B.P., Teukolsky S.A., and Vetterling W.T. Numerical Recipes (1986), Cambridge University Press
34. Dongarra J., and Sullivan F. Top ten algorithms of the century. Computing in Science and Engineering (2000) 22-23