Approximation of a thin plate spline smoother using continuous piecewise polynomial functions

SIAM Journal on Numerical Analysis

Volumn 41, Issue 1, 2003, Pages 208-234

  Roberts, Stephen ^a   Hegland, Markus ^a   Altas, Irfan ^b  

^a AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

^b CHARLES STURT UNIVERSITY   (Australia)

Author keywords

Finite element methods; Surface smoothing; Thin plate splines

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; LEAST SQUARES APPROXIMATIONS; PIECEWISE LINEAR TECHNIQUES; PLATES (STRUCTURAL COMPONENTS); POLYNOMIALS; SPLINES; SURFACE STRUCTURE;

BIHARMONIC EQUATION; CONTINOUS PIECEWISE POLYNOMIAL FUNCTIONS; FIRST ORDER SYSTEMS LEAST SQUARES; SURFACE SMOOTHING; THIN PLATE SPLINE SMOOTHER;

FINITE ELEMENT METHOD;

EID: 1842563900     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142901383296     Document Type: Article

References (29)

1. R. A. Adams, Sobolev Spaces, Academic Press, New York, London, 1975.
2. k à partir d'un grand nombre de donnèes de Lagrange, Math. Model. Numer. Anal., 21 (1987), pp. 529-555.
3. n, Publication URA CNRS 1204 no 86/2, University of Pau, Pau, France, 1986.
4. m-splines, in Mathematical Methods in Computer Aided Geometric Design, T. Lyche, and L. L. Schumaker, eds., Academic Press, New York, 1989, pp. 35-44.
5. R. Arcangéli, R. Manzanilla, and J. J. Torrens, Approximation spline de surfaces de type explicite comportant des failles, Math. Model. Numer. Anal., 31 (1997), pp. 643-676.
6. Australian Society of Exploration Geophysicists, Magnet Data for the Ebagoola Region of Queensland, Australia, http:://www.aseg.org.au/asegeduc/fr26.htm.
7. R. K. Beatson, G. Goodsell, and M. J. D. Powell, On multigrid techniques for thin plate spline interpolation in two dimensions, in The Mathematics of Numerical Analysis, Lectures in Appl. Math. 32, AMS, Providence, RI, 1996, pp. 77-97.
8. R. K. Beatson and W. A. Light, Fast evaluation of radial basis functions: Methods for two-dimensional polyharmonic splines, IMA J. Numer. Anal., 17 (1997), pp. 343-372.
9. R. K. Beatson and G. N. Newsam, Fast evaluation of radial basis functions, Comput. Math. Appl., 24 (1992), pp. 7-19.
10. R. K. Beatson and M. J. D. Powell, An iterative method for thin plate spline interpolation that employs approximations to Lagrange functions. in Numerical Analysis 1993 (Dundee, 1993), Pitman Res. Notes Math. Ser. 303, Longman Scientific and Technical, Harlow, UK, 1994, pp. 17-39.
11. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
12. Z. Cai, T. A. Manteuffel, and S. F. McCormick, First-order system least squares for velocity-vorticity-pressure form of the Stokes equations, with application to linear elasticity, Electron. Trans. Numer. Anal., 3 (1995), pp. 150-159 (electronic).
13. Z. Cai, T. A. Manteuffel, and S. F. McCormick, First-order system least squares for second-order partial differential equations: Part II, SIAM J. Numer. Anal., 34 (1997), pp. 425-454.
14. Z. Cai, T. A. Manteuffel, and S. F. McCormick, First-order system least squares for the Stokes equations, with application to linear elasticity, SIAM J. Numer. Anal., 34 (1994), pp. 1727-1741.
15. Z. Cai, T. A. Manteuffel, S. F. McCormick, and S. V. Parter, First-order system least squares (FOSLS) for planar linear elasticity: Pure traction problem, SIAM J. Numer. Anal., 35 (1998), pp. 320-335.
16. P. Christen, I. Altas, M. Hegland, S. Roberts, K. Burrage, and R. Sidje, Parallelization of a finite element surface fitting algorithm for data mining, ANZIAM J., 42 (2000), pp. C385-C399.
17. P. Christen, M. Hegland, O. Nielsen, S. Roberts, P. Strazdins, and I. Altas, Scalable parallel algorithms for surface fitting and data mining, Parallel Computing, 27 (2001), pp. 941-961.
18. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
19. P. Craven and G. Wahba, Smoothing noisy data with spline functions. Estimating the correct degree of smoothing by the method of generalized cross-validation, Numer. Math., 31 (1978/79), pp. 377-403.
20. J. Duchon, Splines minimizing rotation-invariant semi-norms in Sobolev spaces, in Constructive Theory of Functions of Several Variables, Lecture Notes in Math. 571, 1977, pp. 85-100.
21. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer-Verlag, Berlin, 1986.
22. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.
23. M. Hegland, S. Roberts, and I. Altas, Finite element thin plate splines for surface fitting, in Computational Techniques and Applications: CTAC97 (Singapore, 1997), B. Noye, M. Teubner, and A. Gill, eds., World Scientific, River Edge, NJ, pp. 289-296.
24. M. Hegland, S. Roberts, and I. Altas, Finite element thin plate splines for data mining applications, in Mathematical Methods for Curves and Surfaces, II (Lillehammer, 1997) Vanderbilt University Press, Nashville, TN, 1998, pp. 245-252.
25. M. F. Hutchinson, A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines, Comm. Statist. Simulation Comput., 19 (1990), pp. 433-450.
26. R. Sibson and G. Stone, Computation of thin-plate splines, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 1304-1313.
27. m-splines: Applications to surface fitting, in Mathematical Methods for Curves and Surfaces, II (Lillehammer, 1997), Vanderbilt University Press, Nashville, TN, 1998, pp. 477-484.
28. F. I. Utreras, Convergence rates for multivariate smoothing spline functions, J. Approx. Theory, 52 (1988), pp. 1-27.
29. G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conf. Ser. in Appl. Math. 59, SIAM, Philadelphia, 1990.