A matrix decomposition RBF algorithm: Approximation of functions and their derivatives

Applied Numerical Mathematics

Volumn 57, Issue 3, 2007, Pages 304-319

  Karageorghis, Andreas ^a   Chen, C S ^b   Smyrlis, Yiorgos Sokratis ^a  

^a UNIVERSITY OF CYPRUS   (Cyprus)

^b The University of Southern Mississippi   (United States)

Author keywords

Boundary meshless methods; Circulant matrices; Elliptic boundary value problems; Radial basis functions

Indexed keywords

APPROXIMATION THEORY; FUNCTIONS; INTERPOLATION; PROBLEM SOLVING; RADIAL BASIS FUNCTION NETWORKS;

BLOCK CIRCULANT STRUCTURES; BOUNDARY MESHLESS METHODS; CIRCULANT MATRICES; ELLIPTIC BOUNDARY VALUE PROBLEMS; RADIAL BASIS FUNCTIONS (RBF);

ALGORITHMS;

EID: 33846783143     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2006.03.028     Document Type: Article

References (36)

1. Baxter B.J.C. Preconditioned conjugate gradients, radial basis functions, and Toeplitz matrices. Comput. Math. Appl. 43 (2002) 305-318
2. Beatson R.K., Cherrie J.B., and Mouat C.T. Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration. Adv. Comput. Math. 11 (1999) 253-270
3. Beatson R.K., Cherrie J.B., and Ragozin D.L. Fast evaluation of radial basis functions: Methods for four-dimensional polyharmonic splines. SIAM J. Math. Anal. 32 (2001) 1272-1310
4. Beatson R.K., and Greengard L. A short course on fast multipole methods. In: Ainsworth M., Levesley J., Light W.A., and Marletta M. (Eds). Wavelets Multilevel Methods and Elliptic PDEs (1997), Oxford University Press, Oxford 1-37
5. Beatson R.K., and Light W.A. Fast solution of the radial basis function interpolation equations: Domain decomposition methods. SIAM J. Sci. Comput. 22 (2000) 1717-1740
6. Buhmann M.D. A new class of radial basis functions with compact support. Math. Comp. 70 (2001) 307-318
7. Buhmann M.D. New developments in the theory of radial Basis Functions interpolation. In: Jetter K., and Utreras F.I. (Eds). Multivariate Approximation: From CAGD to Wavelets (1993), World Scientific, Singapore 35-75
8. Buhmann M.D. Radial Basis Functions (2003), Cambridge University Press, Cambridge
9. Davis P.J. Circulant Matrices (1979), John Wiley & Sons, New York
10. Dyn N. Interpolation of Scattered data by radial basis functions. In: Chui C.K., Schumaker L.L., and Utreras F.I. (Eds). Topics in Multivariate Approximation (1987), Academic Press, New York 47-61
11. Dyn N. Interpolation and approximation by radial and related functions. In: Chui C.K., Schumaker L.L., and Ward J.D. (Eds). Approximation Theory VI: vol. 1 (1989), Academic Press, New York 211-234
12. Fairweather G., and Karageorghis A. The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9 (1998) 69-95
13. Fairweather G., Karageorghis A., and Martin P.A. The method of fundamental solutions for scattering and radiation problems. Engrg. Anal. Boundary Elements 27 (2003) 759-769
14. Franke R. Scattered data interpolation: Tests of some methods. Math. Comp. 38 (1982) 181-200
15. Golberg M.A., and Chen C.S. The method of fundamental solutions for potential, Helmholtz and diffusion problems. In: Golberg M.A. (Ed). Boundary Integral Methods and Mathematical Aspects (1999), WIT Press/Computational Mechanics Publications, Boston 103-176
16. Hickernell F.J., and Hon Y.C. Radial basis function approximation as smoothing splines. Appl. Math. Comput. 102 (1999) 1-24
17. IMSL Fortran Library (2002), Visual Numerics Inc, Houston, TX
18. Ingber M.S., Chen C.S., and Tanski J.A. A mesh free approach using radial basis functions and parallel domain decomposition for solving three-dimensional diffusion equations. Int. J. Numer. Methods Engrg. 60 (2004) 2183-2201
19. Jetter K., and Stöckler J. Algorithms for cardinal interpolation using box splines and radial basis functions. Numer. Math. 60 (1991) 97-114
20. Kansa E.J. Multiquadrics - A scattered data approximation scheme with applications to computational fluid-dynamics (I). Comput. Math. Appl. 19 (1990) 127-145
21. Ling L., and Kansa E.J. Least-squares preconditioner for radial basis functions collocation methods. Adv. Comput. Math. 23 (2005) 35-54
22. LINPACK library. Available from:. http://www.netlib.org/linkpack
23. MathWorks: Developers of MATLAB and Simulink. http://www.mathworks.com
24. Meyer C.D. Matrix Analysis and Applied Linear Algebra (2000), SIAM, Philadelphia, PA
25. Micchelli C.A. Interpolation of scattered data: Distance matrices and conditionally positive definite functions. Constructive Approximation 2 (1986) 11-12
26. Numerical Algorithms Group Library Mark 20 (2001), NAG Ltd, Wilkinson House, Jordan Hill Road, Oxford, UK
27. NETLIB library. Available from:. http://netlib.bell-labs.com/netlib/bihar
28. Powell M.J.D. The theory of radial basis function approximation in 1990. In: Light W.A. (Ed). Advances in Numerical Analysis II: Wavelets, Subdivision, Algorithms, and Radial Basis Functions (1992), Oxford University Press, Oxford 105-210
29. Schaback R. Creating surfaces from scattered data using radial basis functions. In: Dæhlen M., Lyche T., and Schumaker L. (Eds). Mathematical Methods for Curves and Surfaces (1995), Vanderbilt University Press, Nashville 477-496
30. Schaback R. Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 3 (1995) 251-264
31. Smyrlis Y.-S., and Karageorghis A. Some aspects of the method of fundamental solutions for certain harmonic problems. J. Sci. Comput. 16 (2001) 341-371
32. Smyrlis Y.-S., and Karageorghis A. Numerical analysis of the MFS for certain harmonic problems. M2AN Math. Model. Numer. Anal. 38 (2004) 495-517
33. Wendland H. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4 (1995) 389-396
34. Wu Z. Multivariate compactly supported positive definite radial functions. Adv. Comput. Math. 4 (1995) 283-292
35. Li X., Ho C.H., and Chen C.S. Computational test of approximation of functions and their derivatives by radial basis functions. Neural Parallel Sci. Comput. 10 (2002) 25-46
36. Zhou X., Hon Y.C., and Li J. Overlapping domain decomposition method by radial basis functions. Appl. Numer. Math. 44 (2003) 241-255