Skip to main content
SpringerLink
Log in

Hierarchical spline spaces: quasi-interpolants and local approximation estimates

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

A local approximation study is presented for hierarchical spline spaces. Such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement in any dimensionality. We provide approximation estimates for general hierarchical quasi-interpolants expressed in terms of the truncated hierarchical basis. Under some mild assumptions, we prove that such hierarchical quasi-interpolants and their derivatives possess optimal local approximation power in the general q-norm with \(1\leq q\leq \infty \). In addition, we detail a specific family of hierarchical quasi-interpolants defined on uniform hierarchical meshes in any dimensionality. The construction is based on cardinal B-splines of degree p and central factorial numbers of the first kind. It guarantees polynomial reproduction of degree p and it requires only function evaluations at grid points (odd p) or half-grid points (even p). This results in good approximation properties at a very low cost, and is illustrated with some numerical experiments.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. de Boor, C.: Quasi interpolants and approximation power of multivariate splines. In: Dahmen, W., Gasca, M., Micchelli, C.A. (eds.) Computation of Curves and Surfaces, pp 313–345. Kluwer (1990)

  2. de Boor, C.: A Practical Guide to Splines. Revised Edition. Springer (2001)

  3. de Boor, C., Fix, G.J.: Spline approximation by quasi-interpolants. J. Approx. Theory 8, 19–45 (1973)

    Article  MATH  Google Scholar 

  4. Bracco, C., Giannelli, C., Mazzia, F., Sestini, A.: Bivariate hierarchical Hermite spline quasi-interpolation. Preprint (2015)

  5. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 2nd Edition. Springer-Verlag (2002)

  6. Buffa, A., Giannelli, C.: Adaptive isogeometric methods with hierarchical splines: Error estimator and convergence. Math. Models Methods Appl. Sci. 26, 1–25 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Butzer, P.L., Schmidt, K., Stark, E.L., Vogt, L.: Central factorial numbers; their main properties and some applications. Numer. Funct. Anal. Optim. 10, 419–488 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Davydov, O.: On local refinement of smooth finite elements and splines. Matem. Mod. 14, 109–116 (2002)

    MathSciNet  MATH  Google Scholar 

  9. Dyn, N., Levin, D., Rippa, S.: Data dependent triangulations for piecewise linear interpolation. IMA J. Numer. Anal. 10, 137–154 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Forsey, D.R., Bartels, R.H.: Hierarchical B-spline refinement. Comput. Graph. 22, 205–212 (1988)

    Article  Google Scholar 

  11. Giannelli, C., Jüttler, B., Speleers, H.: THB-splines: The truncated basis for hierarchical splines. Comput. Aided Geom. Des. 29, 485–498 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Giannelli, C., Jüttler, B., Speleers, H.: Strongly stable bases for adaptively refined multilevel spline spaces. Adv. Comp. Math. 40, 459–490 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Greiner, G., Hormann, K.: Interpolating and approximating scattered 3D-data with hierarchical tensor product B-splines. In: Le Méhauté, A., Rabut, C., Schumaker, L.L. (eds.) Surface Fitting and Multiresolution Methods, pp 163–172. Vanderbilt University Press (1997)

  14. 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)

  15. Kraft, R.: Adaptive und linear unabhängige Multilevel B-Splines und ihre Anwendungen. Ph.D. thesis, Universität Stuttgart (1998)

    MATH  Google Scholar 

  16. Kroó, A., Révész, S.: On Bernstein and Markov-type inequalities for multivariate polynomials on convex bodies. J. Approx. Theory 99, 134–152 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kroó, A.: On Bernstein–Markov-type inequalities for multivariate polynomials in L q -norm. J. Approx. Theory 159, 85–96 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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 (2001)

  19. Lyche, T., Schumaker, L.L.: Local spline approximation methods. J. Approx. Theory 15, 294–325 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rabut, C.: Locally tensor product functions. Numer. Algor. 39, 329–348 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sablonnière, P.: Recent progress on univariate and multivariate polynomial or spline quasi-interpolants. In: de Brujn, M.G., Mache, D.H., Szabadoz, J. (eds.) Trends and Applications in Constructive Approximation, ISNM, vol. 151, pp 229–245. Birhäuser Verlag Basel (2005)

  22. Schumaker, L.L.: Spline Functions: Basic Theory, 3rd Edition. Cambridge University Press (2007)

  23. Speleers, H., Dierckx, P., Vandewalle, S.: Quasi-hierarchical Powell–Sabin B-splines. Comput. Aided Geom. Des. 26, 174–191 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Speleers, H., Dierckx, P., Vandewalle, S.: On the local approximation power of quasi-hierarchical Powell–Sabin splines. In: Dæhlen, M., Floater, M.S., Lyche, T., Merrien, J.L., Mørken, K., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces, Lecture Notes in Computer Science, vol. 5862, pp 419–433 (2010)

  25. Speleers, H., Manni, C.: Effortless quasi-interpolation in hierarchical spaces. Numer. Math. 132, 155–184 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Vuong, A.-V., Giannelli, C., Jüttler, B., Simeon, B.: A hierarchical approach to adaptive local refinement in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 200, 3554–3567 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Rome ‘Tor Vergata’, Via della Ricerca Scientifica, 00133, Rome, Italy

    Hendrik Speleers

Authors
  1. Hendrik Speleers
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hendrik Speleers.

Additional information

Communicated by: Larry L. Schumaker

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Speleers, H. Hierarchical spline spaces: quasi-interpolants and local approximation estimates. Adv Comput Math 43, 235–255 (2017). https://doi.org/10.1007/s10444-016-9483-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-016-9483-y

Keywords

Mathematics Subject Classification (2010)

Search

Navigation