Numerical differentiation by radial basis functions approximation

Advances in Computational Mathematics

Volumn 27, Issue 3, 2007, Pages 247-272

  Wei, T ^a,b   Hon, Y C ^b  

^a LANZHOU UNIVERSITY   (China)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

numerical differentiation; radial basis functions; Tikhonov regularization

Indexed keywords

EID: 34548265741     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10444-005-9001-0     Document Type: Article

References (24)

1. R.A. Adams, Sobolev Spaces (Academie Press [A subsidiary of Harcourt Brace Jovanovich], New York-London, 1975). Pure and Applied Mathematics, Vol. 65.
2. R.S. Anderssen and M. Hegland, For numerical differentiation, dimensionality can be a blessing! Math. Comput. 68(227) (1999) 1121-1141.
3. J. Cheng, Y.C. Hon and Y.B. Wang, A numerical method for the discontinuous solutions of Abel integral equations, in: Inverse Problems and Spectral Theory, volume 348 of Contemp. Math., pp. 233-243. Am. Math. Soc, Providence, Rhode Island, 2004.
4. J. Cullum, Numerical differentiation and regularization, SIAM J. Numer. Anal. 8 (1971) 254-265.
5. S.R. Deans, The Radon Transform and Some of its Applications, A Wiley-Interscience Publication (Wiley, New York, 1983).
6. J. Duchon, Splines minimizing rotation-invariant semi-norms in Sobolev spaces, in: Constructive Theory of Functions of Several Variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976), Lecture Notes in Math., Vol. 571. (Springer, Berlin Heidelberg New York, 1977) pp. 85-100.
7. m-splines. RAIRO Anal. Numér. 12(4) (1978) 325-334, vi.
8. R. Gorenflo and S. Vessella, Abel Integral Equations, volume 1461 of Lecture Notes in Mathematics (Springer, Berlin Heidelberg New York, 1991), Analysis and applications.
9. C.W. Groetsch, Differentiation of approximately specified functions, Am. Math. Mon. 98(9) (1991) 847-850
10. M. Hanke and O. Scherzer, Error analysis of an equation error method for the identification of the diffusion coefficient in a quasi-linear parabolic differential equation, SIAM J. Appl. Math. 59(3) (1999) 1012-1027 (electronic).
11. M. Hanke and O. Scherzer, Inverse problems light: numerical differentiation, Am. Math. Mon. 108(6) (2001) 512-521.
12. F.J. Hickernell and Y.C. Hon, Radial basis function approximations as smoothing splines, Appl. Math. Comput. 102(1) (1999) 1-24.
13. W. Light and H. Wayne, On power functions and error estimates for radial basis function interpolation, J. Approx. Theory 92(2) (1998) 245-266.
14. L.T. Luh, The embedding theory of native spaces, Approx. Theory its Appl. (N.S.) 17(4) (2001) 90-104.
15. W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive definite functions. II, Math. Comput. 54(189) (1990) 211-230.
16. M.J.D. Powell, The uniform convergence of thin plate spline interpolation in two dimensions, Numer. Math. 68(1) (1994) 107-128.
17. A.G. Ramm and A.B. Smirnova, On stable numerical differentiation, Math. Comput. 70(235) (2001) 1131-1153 (electronic).
18. R. Schaback, Improved error bounds for scattered data interpolation by radial basis functions, Math. Comput. 68(225) (1999) 201-216.
19. A.N. Tikhonov and V.Y. Arsenin, Solutions of Ill-posed Problems, (V. H. Winston & Sons, Washington, District of Columbia: Wiley, New York, 1977). Translated from the Russian, Preface by translation editor Fritz John, Scripta Series in Mathematics.
20. G. Wahba and J. Wendelberger, Some new mathematical methods for variational objective analysis using splines and cross validation, Mon. Weather Rev. 108 (1980) 1122-1143.
21. Y.B. Wang, X.Z. Jia and J. Cheng, A numerical differentiation method and its application to reconstruction of discontinuity, Inverse Problems 18(6) (2002) 1461-1476.
22. Z.M. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13(1) (1993) 13-27.
23. p-error estimates for "shifted" surface spline interpolation on Sobolev space, Math. Comput. 72(243) (2003) 1349-1367 (electronic).
24. p-approximation power to derivatives by radial basis function interpolation, Appl. Math. Comput. 150(3) (2004) 875-887.