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A 3D Efficient Procedure for Shepard Interpolants on Tetrahedra

  • Roberto Cavoretto
  • Alessandra De RossiEmail author
  • Francesco Dell’Accio
  • Filomena Di Tommaso
Conference paper
  • 52 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11973)

Abstract

The need of scattered data interpolation methods in the multivariate framework and, in particular, in the trivariate case, motivates the generalization of the fast algorithm for triangular Shepard method. A block-based partitioning structure procedure was already applied to make the method very fast in the bivariate setting. Here the searching algorithm is extended, it allows to partition the domain and nodes in cubic blocks and to find the nearest neighbor points that need to be used in the tetrahedral Shepard interpolation.

Keywords

Scattered data interpolation Tetrahedral Shepard operator Fast algorithms Approximation algorithms 

Notes

Acknowledgments

The authors acknowledge support from the Department of Mathematics “Giuseppe Peano” of the University of Torino via Project 2019 “Mathematics for applications”. Moreover, this work was partially supported by INdAM – GNCS Project 2019 “Kernel-based approximation, multiresolution and subdivision methods and related applications”. This research has been accomplished within RITA (Research ITalian network on Approximation).

References

  1. 1.
    Allasia, G., Cavoretto, R., De Rossi, A.: Hermite-Birkhoff interpolation on scattered data on the sphere and other manifolds. Appl. Math. Comput. 318, 35–50 (2018)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cavoretto, R., De Rossi, A., Perracchione, E.: Efficient computation of partition of unity interpolants through a block-based searching technique. Comput. Math. Appl. 71, 2568–2584 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cavoretto, R., De Rossi, A., Dell’Accio, F., Di Tommaso, F.: Fast computation of triangular Shepard interpolants. J. Comput. Appl. Math. 354, 457–470 (2019)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cavoretto R., De Rossi A., Dell’Accio F., Di Tommaso F.: An efficient trivariate algorithm for tetrahedral Shepard interpolation (2019, submitted)Google Scholar
  5. 5.
    Dell’Accio, F., Di Tommaso, F., Hormann, K.: On the approximation order of triangular Shepard interpolation. IMA J. Numer. Anal. 36, 359–379 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Fasshauer, G.E.: Meshfree Approximation Methods with Matlab. World Scientific Publishing Co., Singapore (2007)CrossRefGoogle Scholar
  7. 7.
    Fasshauer, G.E., McCourt, M.J.: Kernel-Based Approximation Methods using Matlab. Interdisciplinary Mathematical Sciences, vol. 19, World Scientific Publishing Co., Singapore (2015)Google Scholar
  8. 8.
    Little, F.F.: Convex combination surfaces. In: Barnhill R.E., Boehm, W. (eds.) Surfaces in Computer Aided Geometric Design, Amsterdam, North-Holland, pp. 99–108 (1983)Google Scholar
  9. 9.
    Renka, R.J.: Multivariate interpolation of large sets of scattered data. ACM Trans. Math. Softw. 14, 139–148 (1988)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Wendland, H.: Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics, vol. 17. Cambridge University Press, Cambridge (2005)Google Scholar
  11. 11.
    Zhang, M., Liang, X.-Z.: On a Hermite interpolation on the sphere. Appl. Numer. Math. 61, 666–674 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mathematics “Giuseppe Peano”University of TurinTurinItaly
  2. 2.Department of Mathematics and Computer ScienceUniversity of CalabriaRende (CS)Italy

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