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Spherical interpolation of scattered data using least squares thin-plate spline and inverse multiquadric functions

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Abstract

We construct some smooth functions defined over a sphere that interpolate large sets of scattered data, using some modified Shepard methods, the least squares thin-plate spline and the inverse multiquadric functions. We illustrate the benefits of our methods in numerical examples for some test functions and some real data applications.

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References

  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)

    MathSciNet  Google Scholar 

  2. Baxter, B.J.C., Hubbert, S.: Radial basis functions for the sphere. In: Recent Progress in Multivariate Approximation, Witten-Bommerholz, 2000. Internat. Ser. Numer. Math., vol. 137, pp. 33-47. Birkhäuser, Basel (2001)

  3. Cătinaş, T.: The combined Shepard-Abel-Goncharov univariate operator. Rev. Anal. Numér. Théor. Approx. 32, 11–20 (2003)

    Article  Google Scholar 

  4. Cătinaş, T.: The combined Shepard-Lidstone bivariate operator. In: Trends and Applications in Constructive Approximation. International Series of Numerical Mathematics (de Bruin, M.G. et al. (eds.)), 151, pp. 77–89, Springer Group-Birkhäuser Verlag (2005)

  5. Cătinaş, T.: Bivariate interpolation by combined Shepard operators. In: Proceedings of \(17^{th}\) IMACS World Congress, Scientific Computation, Applied Mathematics and Simulation (Borne, P., Benrejeb, M., Dangoumau, N., Lorimier, L. (eds.)), 7 pp., Paris, July 11–15 (2005)

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

    Article  MathSciNet  Google Scholar 

  7. De Rossi, A.: Spherical interpolation of large scattered data sets using zonal basis functions. In: Mathematical Methods for Curves and Surfaces (Daehlen, M., Morken, K., Schumaker, L. (eds.)), pp. 125–134, TrØmso 2004. Nashboro Press (2005)

  8. De Rossi, A.: Hybrid spherical approximation. arXiv:1404.1475 (2014)

  9. Cavoretto, R., De Rossi, A.: Fast and accurate interpolation of large scattered data sets on the sphere. J. Comput. Appl. Math. 234, 1505–1521 (2010)

    Article  MathSciNet  Google Scholar 

  10. Cavoretto, R., De Rossi, A.: Numerical comparison of different weights in Shepard’s interpolants on the sphere. Appl. Math. Sci. 4, 3425–3435 (2010)

    MathSciNet  Google Scholar 

  11. Cavoretto, R., De Rossi, A.: Spherical interpolation using the partition of unity method: an efficient and flexible algorithm. Appl. Math. Lett. 25, 1251–1256 (2012)

    Article  MathSciNet  Google Scholar 

  12. Cavoretto, R., De Rossi, A.: Achieving accuracy and efficiency in spherical modelling of real data. Math. Methods Appl. Sci. 37, 1449–1459 (2014)

    Article  MathSciNet  Google Scholar 

  13. Farwig, R.: Rate of convergence of Shepard’s global interpolation formula. Math. Comp. 46, 577–590 (1986)

    MathSciNet  Google Scholar 

  14. Fasshauer, G.E., Zhang, J.G.: On choosing “optimal’’ shape parameters for RBF approximation. Numer. Algor. 45, 345–368 (2007)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  16. Franke, R., Nielson, G.: Smooth interpolation of large sets of scattered data. Int. J. Numer. Meths. Engrg. 15, 1691–1704 (1980)

    Article  MathSciNet  Google Scholar 

  17. Le Gia, Q.T., Sloan, I.H., Wendland, H.: Multiscale analysis in Sobolev spaces on the sphere. SIAM J. Numer. Anal. 48(6), 2065–2090 (2010)

    Article  MathSciNet  Google Scholar 

  18. Hardy, R.L.: Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res. 76, 1905–1915 (1971)

    Article  Google Scholar 

  19. Hubbert, S., Le Gia, Q.T., Morton, T.: Spherical radial basis functions. Springer, Theory and Applications (2015)

    Book  Google Scholar 

  20. Lazzaro, D., Montefusco, L.B.: Radial basis functions for multivariate interpolation of large scattered data sets. J. Comput. Appl. Math. 140, 521–536 (2002)

    Article  MathSciNet  Google Scholar 

  21. Renka, R.J., Cline, A.K.: A triangle-based \(C^{1}\) interpolation method. Rocky Mountain J. Math. 14, 223–237 (1984)

    Article  MathSciNet  Google Scholar 

  22. Renka, R.J.: Multivariate interpolation of large sets of scattered data. ACM Trans. Math. Software 14, 139–148 (1988)

    Article  MathSciNet  Google Scholar 

  23. Saff, E.B., Kuijlaars, A.B.J.: Distributing many points on a sphere. Math. Intelligencer. 19, 5–11 (1997)

    Article  MathSciNet  Google Scholar 

  24. Shepard, D.: A two dimensional interpolation function for irregularly spaced data. In: Proceedings of the 1968 23rd ACM National Conference, pp. 517–523. ACM (1968)

  25. Sloan, I.H., Sommariva, A.: Approximation on the sphere using radial basis functions plus polynomials. Adv. Comput. Math. 29, 147–177 (2008)

    Article  MathSciNet  Google Scholar 

  26. 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 

  27. Zuppa, C.: Error estimates for moving least square approximations. Bull. Braz. Math. Soc. (N.S) 34(2), 231–249 (2003)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We are grateful to the referees for the careful reading and for their valuable suggestions that improved the manuscript.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Str. M Kogălniceanu Nr. 1, RO-400084 , Cluj-Napoca, Romania

    Teodora Cătinaş & Andra Malina

  2. Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

    Andra Malina

Authors
  1. Teodora Cătinaş
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  2. Andra Malina
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Correspondence to Teodora Cătinaş.

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Cătinaş, T., Malina, A. Spherical interpolation of scattered data using least squares thin-plate spline and inverse multiquadric functions. Numer Algor 97, 1397–1414 (2024). https://doi.org/10.1007/s11075-024-01755-6

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  • DOI: https://doi.org/10.1007/s11075-024-01755-6

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