A least-squares preconditioner for radial basis functions collocation methods

Advances in Computational Mathematics

Volumn 23, Issue 1-2, 2005, Pages 31-54

  Ling, Leevan ^a   Kansa, Edward J ^b  

^a SIMON FRASER UNIVERSITY   (Canada)

^b EMBRY RIDDLE AERONAUTICAL UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 16244376155     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10444-004-1809-5     Document Type: Article

References (34)

1. 1/2 as c → ∞, Comput. Math. Appl. 24(12) (1992) 1-6.
2. R.K. Beatson, J.B. Cherrie and C.T. Mouat, Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration, Adv. Comput. Math. 11 (1999) 253-270.
3. n, SIAM J. Sci. Comput. 23(5) (2002) 1549-1571.
4. R.K. Beatson and W.A. Light, Fast evaluation of radial basis function: Methods for two-dimensional polyharmonic splines, IMA J. Numer. Anal. 17 (1997) 343-372.
5. R.K. Beatson, W.A. Light and S. Billings, Fast solution of the radial basis function interpolation methods: Domain decomposition methods, SIAM J. Sci. Comput. 22(5) (2000) 1717-1740.
6. R.K. Beatson and G.N. Newsam, Fast evaluation of radial basis functions I, Comput. Math. Appl. 24(12) (1992) 7-19.
7. M. Bozzini, L. Lenarduzzi and R. Schaback, Kernel B-splines and adaptive interpolation, Manuscript (2002)
8. P.N. Brown and H.F. Walker, GMRES on (nearly) singular systems, SIAM J. Matrix. Anal. Appl. 18 (1997) 37-51.
9. M.D. Buhmann, Multivariate interpolation in odd-dimensional Euclidean spaces using multiquadrics, Constr. Approx. 6(1) (1990) 21-34.
10. M.D. Buhmann, Multivariate cardinal interpolation with radial-basis functions, Constr. Approx. 6(3) (1990) 225-255.
11. M.D. Buhmann and C.A. Micchelli, Multiquadric interpolation improved, Comput. Math. Appl. 24(12) (1992) 21-25.
12. A.H.D. Cheng, M.A. Golberg, E.J. Kansa and T. Zammito, Exponential convergence and h-c mul-tiquadric collocation method for partial differential equations, Numer. Methods Partial Differential Equations (to appear)
13. G.E. Fasshauer, On the numerical solution of differential equations with radial basis functions, in: Boundary Element Technology, Vol. XIII, eds. C.S. Chen, C.A. Brebbia and D.W. Pepper (WIT Press, 1999) pp. 291-300.
14. A.C. Faul, Iterative techniques for radial basis function interpolation, Ph.D. thesis, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England (2001).
15. A.C. Faul and M.J.D. Powell, Krylov subspace methods for radial basis function interpolation, in: Numerical Analysis 1999, Dundee, Research Notes in Mathematics, Vol. 420 (Chapman & Hall/CRC, Boca Raton, FL, 2000) pp. 115-141.
16. A.I. Fedoseyev, M.J. Friedman and E.J. Kansa, Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary, Comput. Math. Appl. 43(3-5) (2002) 439-455.
17. B. Fornberg, T.A. Driscoll, G. Wright and R. Charles, Observations on the behavior of radial basis functions near boundaries, Comput. Math. Appl. 43 (2002) 473-490.
18. B. Fornberg and G. Wright, Stable computation of multiquadric interpolants for all values of the shape parameter, BIT (to appear)
19. G.H. Golub and C.F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins Univ. Press, Baltimore/London, 1996).
20. R.L. Hardy, Theory and applications of the multiquadric-biharmonic method. 20 years of discovery 1968-1988, Comput. Math. Appl. 10(8/9) (1990) 163-208.
21. Y.C. Hon and Z. Wu, Additive Schwarz domain decomposition with radial basis approximation, Internat. J. Appl. Math. 4 (2002) 81-98.
22. E.J. Kansa, 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.
23. 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.
24. E.J. Kansa and Y.C. Hon, 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.
25. E.J. Kansa, H. Power, G.E. Fasshauer and L. Ling, A volumetric integral radial basis function method for time dependent partial differential equations I. Formulation, Engrg. Anal. Bound. Elem. (in review).
26. E. Larsson and B. Fornberg, A numerical study of radial basis functions based solution methods for elliptic PDEs, Comput. Math. Appl. (in press).
27. C.L. Lawson and R.J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1974).
28. J. Li and Y.C. Hon, Domain decomposition for radial basis meshless methods, Numer. Methods Partial Differential Equations (in review)
29. W.R. Madych, Miscellaneous error bounds for multiquadric and related interpolators, Comput. Math. Appl. 24 (1992) 121-138.
30. C.T. Mouat, Fast algorithms and preconditioning techniques for fitting radial basis functions. Ph.D. thesis, Mathematics Department, University of Canterbury, Christchurch, New Zealand (2001)
31. R. Sibson and G. Stone, Computation of thin-plate splines, SIAM J. Sci. Statist. Comput. 12 (1991) 1304-1313.
32. B.F. Smith, P.E. Bjørstad and W.D. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations (Cambridge Univ. Press, Cambridge, UK, 1996).
33. L.N. Trefethen and D. Bau III, Numerical Linear Algebra (SIAM, Philadelphia, PA, 1997).
34. X. Zhou, Y.C. Hon and J. Li, Overlapping domain decomposition method by radial basis functions, Appl. Numer. Math, (in press)