Skip to main content
SpringerLink
Log in

Comparison of Shepard’s Like Methods with Different Basis Functions

  • Conference paper
  • First Online:
Book cover Numerical Computations: Theory and Algorithms (NUMTA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11973))

  • 1022 Accesses

Abstract

The problem of reconstruction of an unknown function from a finite number of given scattered data is well known and well studied in approximation theory. The methods developed with this goal are several and are successfully applied in different contexts. Due to the need of fast and accurate approximation methods, in this paper we numerically compare some variation of the Shepard method obtained by considering different basis functions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Caira, R., Dell’Accio, F., Di Tommaso, F.: On the bivariate Shepard-Lidstone operators. J. Comput. Appl. Math. 236(7), 1691–1707 (2012)

    Article  MathSciNet  Google Scholar 

  2. Cătinaş, T.: The combined Shepard-Lidstone bivariate operator. In: Mache, D.H., Szabados, J., de Bruin, M.G. (eds.) Trends and Applications in Constructive Approximation, pp. 77–89. Birkhäuser Basel, Basel (2005)

    Chapter  Google Scholar 

  3. Cătinaş, T.: The bivariate Shepard operator of Bernoulli type. Calcolo 44(4), 189–202 (2007)

    Article  MathSciNet  Google Scholar 

  4. 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)

    Article  MathSciNet  Google Scholar 

  5. Cavoretto, R., De Rossi, A., Perracchione, E.: Efficient computation of partition of unity interpolants through a block-based searching technique. Comput. Math. Appl. 71(12), 2568–2584 (2016)

    Article  MathSciNet  Google Scholar 

  6. Costabile, F.A., Dell’Accio, F., Di Tommaso, F.: Enhancing the approximation order of local Shepard operators by Hermite polynomials. Comput. Math. Appl. 64(11), 3641–3655 (2012)

    Article  MathSciNet  Google Scholar 

  7. Costabile, F.A., Dell’Accio, F., Di Tommaso, F.: Complementary Lidstone interpolation on scattered data sets. Numer. Algorithms 64(1), 157–180 (2013)

    Article  MathSciNet  Google Scholar 

  8. Dell’Accio, F., Di Tommaso, F., Hormann, K.: On the approximation order of triangular Shepard interpolation. IMA J. Numer. Anal. 36, 359–379 (2016)

    MathSciNet  MATH  Google Scholar 

  9. Dell’Accio, F., Di Tommaso, F.: Complete Hermite-Birkhoff interpolation on scattered data by combined Shepard operators. J. Comput. Appl. Math. 300, 192–206 (2016)

    Article  MathSciNet  Google Scholar 

  10. Dell’Accio, F., Di Tommaso, F.: Scattered data interpolation by Shepard’s like methods: classical results and recent advances. Dolomites Res. Notes Approx. 9, 32–44 (2016)

    MathSciNet  MATH  Google Scholar 

  11. Fasshauer, G.E.: Meshfree Approximation Methods with Matlab. World Scientific Publishing Co. Inc., Singapore (2007)

    Book  Google Scholar 

  12. Franke, R.: Scattered data interpolation: tests of some methods. Math. Comput. 38(157), 181–200 (1982)

    MathSciNet  MATH  Google Scholar 

  13. Little, F.: Convex combination surfaces. In: Barnhill, R.E., Boehm, W. (eds.) Surfaces in Computer Aided Geometric Design, North-Holland, vol. 1479, pp. 99–108 (1983)

    Google Scholar 

  14. Renka, R.J.: Algorithm 790: CSHEP2D: cubic Shepard method for bivariate interpolation of scattered data. ACM Trans. Math. Softw. 25(1), 70–73 (1999)

    Article  Google Scholar 

  15. Senger, S., Sun, X., Wu, Z.: Optimal order Jackson type inequality for scaled Shepard approximation. J. Approx. Theor. 227, 37–50 (2018)

    Article  MathSciNet  Google Scholar 

  16. Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 1968 23rd ACM National Conference, ACM 1968, pp. 517–524. ACM, New York (1968)

    Google Scholar 

  17. Wendland, H.: Scattered Data Approximation, vol. 17. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  18. Wong, T.T., Luk, W.S., Heng, P.A.: Sampling with Hammersley and Halton points. J. Graph. Tools 2(2), 9–24 (1997)

    Article  Google Scholar 

Download references

Acknowledgments

This work was partially supported by the INdAM-GNCS 2019 research project “Kernel-based approximation, multiresolution and subdivision methods and related applications”. This research has been accomplished within RITA (Research ITalian network on Approximation).

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Calabria, Via P. Bucci, Cubo 30A, Rende, Italy

    Francesco Dell’Accio & Filomena Di Tommaso

  2. NTT DATA Italia, Contrada Cutura, Rende, Italy

    Domenico Gonnelli

Authors
  1. Francesco Dell’Accio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Filomena Di Tommaso
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Domenico Gonnelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Filomena Di Tommaso .

Editor information

Editors and Affiliations

  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Dell’Accio, F., Di Tommaso, F., Gonnelli, D. (2020). Comparison of Shepard’s Like Methods with Different Basis Functions. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11973. Springer, Cham. https://doi.org/10.1007/978-3-030-39081-5_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-39081-5_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-39080-8

  • Online ISBN: 978-3-030-39081-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation