Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T01:54:33.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Bivariate Polynomial Interpolation over Nonrectangular Meshes

Part of: Numerical approximation and computational geometry

Published online by Cambridge University Press:  17 November 2016

Jiang Qian ,
Sujuan Zheng ,
Fan Wang  and
Zhuojia Fu
Jiang Qian*
Affiliation:
College of Sciences, Hohai University, Nanjing 211000, China
Sujuan Zheng*
Affiliation:
College of Sciences, Hohai University, Nanjing 211000, China
Fan Wang*
Affiliation:
College of Engineering, Nanjing Agricultural University, Nanjing 210031, China
Zhuojia Fu*
Affiliation:
College of Mechanics and Materials, Hohai University, Nanjing 211000, China
*
*Corresponding author. Email addresses:[email protected] (J. Qian), [email protected] (S.-J. Zheng), [email protected] (F. Wang), [email protected] (Z.-J. Fu)
*Corresponding author. Email addresses:[email protected] (J. Qian), [email protected] (S.-J. Zheng), [email protected] (F. Wang), [email protected] (Z.-J. Fu)
*Corresponding author. Email addresses:[email protected] (J. Qian), [email protected] (S.-J. Zheng), [email protected] (F. Wang), [email protected] (Z.-J. Fu)
*Corresponding author. Email addresses:[email protected] (J. Qian), [email protected] (S.-J. Zheng), [email protected] (F. Wang), [email protected] (Z.-J. Fu)
Get access

Abstract

In this paper, bymeans of a new recursive algorithm of non-tensor-product-typed divided differences, bivariate polynomial interpolation schemes are constructed over nonrectangular meshes firstly, which is converted into the study of scattered data interpolation. And the schemes are different as the number of scattered data is odd and even, respectively. Secondly, the corresponding error estimation is worked out, and an equivalence is obtained between high-order non-tensor-product-typed divided differences and high-order partial derivatives in the case of odd and even interpolating nodes, respectively. Thirdly, several numerical examples illustrate the recursive algorithms valid for the non-tensor-product-typed interpolating polynomials, and disclose that these polynomials change as the order of the interpolating nodes, although the node collection is invariant. Finally, from the aspect of computational complexity, the operation count with the bivariate polynomials presented is smaller than that with radial basis functions.

Keywords

Bivariate polynomial interpolationNonrectangular meshesScattered data interpolationThe conformality of smoothing cofactor methodRadial basis function

MSC classification

Secondary: 65D05: Interpolation
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 4 , November 2016 , pp. 549 - 578
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Allasia, G., and Bracco, C., Multivariate Hermite-Birkhoff interpolation by a class of cardinal basis function, Appl. Math. Comput., vol. 218 (2012), pp. 9248–9260.Google Scholar
[2] Bailey, B. A., Multivariate polynomial interpolation and sampling in Paley-Wiener spaces, J. Approx. Theory, vol. 164 (2012), pp. 460–487.Google Scholar
[3] Chai, J., Lei, N., Li, Y., and Xia, P., The proper interpolation space for multivariate Birkhoff interpolation, J. Comput. Appl. Math., vol. 235 (2011), pp. 3207–3214.Google Scholar
[4] Cui, K., and Lei, N., Stable monomial basis for multivariate Birkhoff interpolation problems, J. Comput. Appl. Math., vol. 277 (2015), pp. 162–170.Google Scholar
[5] Dyn, N., and Floater, M. S., Multivariate polynomial interpolation on lower sets, J. Approx. Theory, vol. 177 (2014), pp. 34–42.Google Scholar
[6] Gasca, M., Sauer, T., On the history of multivariate polynomial interpolation, J. Comput. and Appl. Math., vol. 122 (2000), pp. 23–35.CrossRefGoogle Scholar
[7] Lai, M. J., Convex preserving scattered data interpolation using bivariate C1 cubic splines, J. Comput. Appl. Math., vol. 119 (2000), pp. 249–258.CrossRefGoogle Scholar
[8] Li, C. J., and Wang, R. H.,, Bivariate cubic spline space and bivariate cubic NURBS surfaces, Proceedings of Geometric Modeling and Processing 2004, IEEE Computer Society Pressvol, pp. 115–123.Google Scholar
[9] Madych, W. R., An estimate for multivariate interpolation II, J. Approx. Theory, vol. 142 (2006), pp. 116–128.Google Scholar
[10] Mazroui, A., Sbibih, D., and Tijini, A., Recursive computation of bivariate Hermite spline interpolants, Appl. Numer. Math., vol. 57 (2007), pp. 962–973.Google Scholar
[11] Qian, J., Wang, F., On the approximation of the derivatives of spline quasi-interpolation in cubic spline space , Numer. Math. Theor. Meth. Appl. vol., 7(1) (2014), pp. 1–22.Google Scholar
[12] Qian, J., Wang, R. H., Li, C. J., The bases of the Non-uniform cubic spline space , Numer. Math. Theor. Meth. Appl. vol. 5(4) (2012), pp. 635–652.Google Scholar
[13] Qian, J., Wang, R. H., Zhu, C. G., Wang, F., On spline quasi-interpolation in cubic spline space , Sci Sin Math, vol. 44(7) (2014), pp. 769–778, (in Chinese).Google Scholar
[14] Salzer, H.E., Some new divided difference algorithm for two variables, in: Langer, R.E. (Ed.) On Numerical Approximation, 1959.Google Scholar
[15] Sauer, T., Numerical Analysis, China Machine Press, Beijing, 2012.Google Scholar
[16] Wang, R. H., Numerical Approximation, Beijing, High Education Press, 1999.Google Scholar
[17] Wang, R. H., and Li, C. J., Bivariate quartic spline spaces and quasi-interpolation operators, J. Comput. and Appl. Math., vol. 190, (2006), pp. 325–338.Google Scholar
[18] Wang, R. H., Qian, J., On branched continued fractions rational interpolation over pyramid-typed grids, Numer. Algor., vol. 54 (2010), pp. 47–72.Google Scholar
[19] Wang, R. H., Qian, J., Bivariate polynomial and continued fraction interpolation over orthotriples, Applied Mathematics and Computation, vol. 217 (2011), pp. 7620–7635.Google Scholar
[20] Wang, R. H., Shi, X. Q., Luo, Z. X., and Su, Z. X., Multivariate Spline Functions and Their Applications, Science Press/Kluwer Academic Publishers, Beijing, New York, Dordrecht, Boston, London, 2001.Google Scholar
[21] Wendland, H., Scattered Data Approximation, Cambridge University Press, 2005.Google Scholar
[22] Zhou, T. H., and Lai, M. J., Scattered data interpolation by bivariate splines with higher approximation order, J. Comput. Appl. Math., vol. 242 (2013), pp. 125–140.Google Scholar
[23] Zhu, C. G., and Wang, R. H., Lagrange interpolation by bivariate splines on cross-cut partitions, J. Comp. Appl. Math., vol. 195 (2006), pp. 326–340.CrossRefGoogle Scholar