Advertisement

BIT Numerical Mathematics

, Volume 56, Issue 4, pp 1165–1188 | Cite as

Bivariate hierarchical Hermite spline quasi-interpolation

  • Cesare Bracco
  • Carlotta Giannelli
  • Francesca Mazzia
  • Alessandra Sestini
Article

Abstract

Spline quasi-interpolation (QI) is a general and powerful approach for the construction of low cost and accurate approximations of a given function. In order to provide an efficient adaptive approximation scheme in the bivariate setting, we consider quasi-interpolation in hierarchical spline spaces. In particular, we study and experiment the features of the hierarchical extension of the tensor-product formulation of the Hermite BS quasi-interpolation scheme. The convergence properties of this hierarchical operator, suitably defined in terms of truncated hierarchical B-spline bases, are analyzed. A selection of numerical examples is presented to compare the performances of the hierarchical and tensor-product versions of the scheme.

Keywords

B-splines Hermite quasi-interpolation Hierarchical spaces Truncated hierarchical B-splines 

Mathematics Subject Classification

65D07 65D15 

Notes

Acknowledgments

This work was supported by the programs “Finanziamento Giovani Ricercatori 2014” and “Progetti di Ricerca 2015” (Gruppo Nazionale per il Calcolo Scientifico of the Istituto Nazionale di Alta Matematica Francesco Severi, GNCS - INdAM) and by the project DREAMS (MIUR Futuro in Ricerca RBFR13FBI3).

References

  1. 1.
    Berdinsky, D., Kim, T., Cho, D., Bracco, C., Kiatpanichgij, S.: Bases of T-meshes and the refinement of hierarchical B-splines. Comput. Methods Appl. Mech. Eng. 283, 841–855 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    de Boor, C., Fix, M.G.: Spline approximation by quasi-interpolants. J. Approx. Theory 8, 19–45 (1973)CrossRefMATHGoogle Scholar
  3. 3.
    Buffa, A., Giannelli, C.: Adaptive isogeometric methods with hierarchical splines: error estimator and convergence Math. Models Methods Appl. Sci. 26, 1–25 (2016)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dagnino, C., Remogna, S., Sablonniere, P.: Error bounds on the approximation of functions and partial derivatives by quadratic spline quasi-interpolants on non-uniform criss-cross triangulations of a rectangular domain. BIT 53, 87–109 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dokken, T., Lyche, T., Pettersen, K.F.: Polynomial splines over locally refined box-partitions. Comput. Aided Geom. Design 30, 331–356 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Giannelli, C., Jüttler, B.: Bases and dimensions of bivariate hierarchical tensor-product splines. J. Comput. Appl. Math. 239, 162–178 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Giannelli, C., Jüttler, B., Speleers, H.: THB-splines: the truncated basis for hierarchical splines. Comput. Aided Geom. Design 29, 485–498 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Giannelli, C., Jüttler, B., Speleers, H.: Strongly stable bases for adaptively refined multilevel spline spaces. Adv. Comp. Math. 40, 459–490 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Iurino, A.: BS Hermite Quasi-Interpolation Methods for Curves and Surfaces. PhD thesis, Università di Bari (2014)Google Scholar
  10. 10.
    Iurino, A., Mazzia, F.: The C library QIBSH for Hermite Quasi-Interpolation of Curves and Surfaces. Dipartimento di Matematica, Università degli Studi di Bari, Report 11/2013 (2013)Google Scholar
  11. 11.
    Kraft, R.: Adaptive and linearly independent multilevel B-splines. In: Le Méhauté, A., Rabut, C., Schumaker, L.L. (eds.) Surface Fitting and Multiresolution Methods, pp. 209–218. Vanderbilt University Press, Nashville (1997)Google Scholar
  12. 12.
    Lee, B.G., Lyche, T., Mørken, K.: Some examples of quasi-interpolants constructed from local spline projectors. In: Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces: Oslo 2000, pp. 243–252. Vanderbilt University Press, Nashville (2001)Google Scholar
  13. 13.
    Li, X., Deng, J., Chen, F.: Polynomial splines over general T-meshes. Visual Comput. 26, 277–286 (2010)CrossRefGoogle Scholar
  14. 14.
    Lyche, T., Schumaker, L.L.: Local spline approximation. J. Approx. Theory 15, 294–325 (1975)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Mazzia, F., Sestini, A.: The BS class of Hermite spline quasi-interpolants on nonuniform knot distributions. BIT 49, 611–628 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Mazzia, F., Sestini, A.: Quadrature formulas descending from BS Hermite spline quasi-interpolation. J. Comput. Appl. Math. 236, 4105–4118 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Mokriš, D., Jüttler, B., Giannelli, C.: On the completeness of hierarchical tensor-product Bsplines. J. Comput. Appl. Math. 271, 53–70 (2014)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Speleers, H., Manni, C.: Effortless quasi-interpolation in hierarchical spaces. Numer. Math. 132, 155–184 (2016)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Sablonniere, P.: Recent progress on univariate and multivariate polynomial and spline quasi–interpolants, trends and applications in constructive approximation. In: de Bruin, M.G., Mache, D.H., Szabados, J. (eds.) International Series of Numerical Mathematics, vol. 151, pp. 229–245 Birkhauser Verlag, Basel (2005)Google Scholar
  20. 20.
    Sederberg, T.W., Cardon, D.L., Finnigan, G.T., North, N.S., Zheng, J., Lyche, T.: T-spline simplification and local refinement. ACM Trans. Graph. 23, 276–283 (2004)CrossRefGoogle Scholar
  21. 21.
    Schumaker, L.L., Wang, L.: Approximation power of polynomial splines on T-meshes. Comput. Aided Geom. Design 29, 599–612 (2012)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Vuong, A.V., Giannelli, C., Jüttler, B., Simeon, B.: A hierarchical approach to adaptive local refinement in isogeometric analysis. Comput. Methods Appl. Mech. Engrg. 200, 3554–3567 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Cesare Bracco
    • 1
  • Carlotta Giannelli
    • 1
  • Francesca Mazzia
    • 2
  • Alessandra Sestini
    • 1
  1. 1.Dipartimento di Matematica e Informatica “U. Dini”Università di FirenzeFirenzeItaly
  2. 2.Dipartimento di MatematicaUniversità di BariBariItaly

Personalised recommendations