Journal of Applied Mathematics

Piecewise Bivariate Hermite Interpolations for Large Sets of Scattered Data

Renzhong Feng and Yanan Zhang

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The requirements for interpolation of scattered data are high accuracy and high efficiency. In this paper, a piecewise bivariate Hermite interpolant satisfying these requirements is proposed. We firstly construct a triangulation mesh using the given scattered point set. Based on this mesh, the computational point (x,y) is divided into two types: interior point and exterior point. The value of Hermite interpolation polynomial on a triangle will be used as the approximate value if point (x,y) is an interior point, while the value of a Hermite interpolation function with the form of weighted combination will be used if it is an exterior point. Hermite interpolation needs the first-order derivatives of the interpolated function which is not directly given in scatted data, so this paper also gives the approximate derivative at every scatted point using local radial basis function interpolation. And numerical tests indicate that the proposed piecewise bivariate Hermite interpolations are economic and have good approximation capacity.

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J. Appl. Math., Volume 2013 (2013), Article ID 239703, 10 pages.

First available in Project Euclid: 14 March 2014

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Feng, Renzhong; Zhang, Yanan. Piecewise Bivariate Hermite Interpolations for Large Sets of Scattered Data. J. Appl. Math. 2013 (2013), Article ID 239703, 10 pages. doi:10.1155/2013/239703.

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  • W. Xiong, W. Fan, and R. Ding, “Least-squares parameter estimation algorithm for a class of input nonlinear systems,” Journal of Applied Mathematics, vol. 2012, Article ID 684074, 14 pages, 2012.
  • F. Ding, H. Chen, and M. Li, “Multi-innovation least squares identification methods based on the auxiliary model for MISO systems,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 658–668, 2007.
  • F. Ding, P. X. Liu, and G. Liu, “Multi-innovation least-squares identification for system modeling,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 40, no. 3, pp. 767–778, 2010.
  • R. Z. Feng and R. H. Wang, “Closed smooth surface defined from cubic triangular splines,” Journal of Computational Mathematics, vol. 23, no. 1, pp. 67–74, 2005.
  • R. Z. Feng and R. H. Wang, “Smooth spline surfaces over arbitrary topological triangular meshes,” Journal of Software, vol. 14, no. 4, pp. 830–837, 2003.
  • S. Lee, Ge. Wolberg, and S. Y. Shin, “Scattered data interpolation with multilevel B-splines,” IEEE Transactions on Visualization and Computer Graphics, vol. 3, no. 3, pp. 228–244, 1997.
  • W. Z. Xu, L. T. Guan, and Y. X. Xu, “Smoothing of space scattered data by polynomial natural splines,” Acta Scientiarum Naturalium Universitatis Sunyatseni, vol. 49, no. 6, pp. 20–30, 2010.
  • Z. M. Wu, “Radial basis functions in scattered data interpolation and the meshless method of numerical solution of PDEs,” Chinese Journal of Engineering Mathematics, vol. 19, no. 2, pp. 1–12, 2002.
  • D. Lazzaro and L. B. Montefusco, “Radial basis functions for the multivariate interpolation of large scattered data sets,” Journal of Computational and Applied Mathematics, vol. 140, no. 1-2, pp. 521–536, 2002.
  • R. Feng and L. Xu, “Large scattered data fitting based on radial basis functions,” Computer Aided Drafting, Design and Manufacturing, vol. 17, no. 1, pp. 66–72, 2007.
  • R. Franke and H. Hagen, “Least squares surface approximation using multiquadrics and parametric domain distortion,” Computer Aided Geometric Design, vol. 16, no. 3, pp. 177–196, 1999.
  • T. Sauer and Y. Xu, “On multivariate Hermite interpolation,” Advances in Computational Mathematics, vol. 4, no. 3, pp. 207–259, 1995.
  • Z. M. Wu, “Hermite-Birkhoff interpolation of scattered data by radial basis functions,” Approximation Theory and Its Applications, vol. 8, no. 2, pp. 1–10, 1992.
  • L. Zha and R. Feng, “A scattered hermite interpolation using radial basis Functions,” Journal of Information Computational Science, vol. 4, pp. 361–369, 2007.
  • B. Delaunay, “Sur la sphère vide,” Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk, vol. 7, pp. 793–800, 1934.
  • R. Franke and G. Nielson, “Smooth interpolation of large sets of scattered data,” International Journal for Numerical Methods in Engineering, vol. 15, no. 11, pp. 1691–1704, 1980.
  • D. Shepard, “A two-dimensional interpolation function for irregularly spaced data,” in Proceedings of the 23rd ACM National Conference, pp. 517–524, ACM, 1968.