Skip to main content
SpringerLink
Log in

Effortless quasi-interpolation in hierarchical spaces

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We present a general and simple procedure to construct quasi-interpolants in hierarchical spaces. Such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement. The proposed hierarchical quasi-interpolants are described in terms of the so-called truncated hierarchical basis. Assuming a quasi-interpolant is selected for each space associated with a particular level in the hierarchy, the hierarchical quasi-interpolants are obtained without any additional manipulation. The main properties (like polynomial reproduction) of the quasi-interpolants selected at each level are locally preserved in the hierarchical construction. We show how to construct hierarchical local projectors, and the local approximation order of the underling hierarchical space is also investigated. The presentation is detailed for the truncated hierarchical B-spline basis, and we discuss its extension to a more general framework.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. Note that the truncated hierarchical basis is called quasi-hierarchical basis in [48, 49].

  2. Basically this means that the spline space contains constants and that an appropriate blossom can be constructed therein.

References

  1. Barrera, D., Ibáñez, M.J., Sablonnière, P., Sbibih, D.: Near-best quasi-interpolants associated with \(H\)-splines on a three-direction mesh. J. Comput. Appl. Math. 183, 133–152 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Beirão da Veiga, L., Buffa, A., Sangalli, G., Vázquez, R.: Analysis-suitable T-splines of arbitrary degree: definition, linear independence and approximation properties. Math. Models Methods Appl. Sci. 23, 1979–2003 (2013)

  3. Berdinsky, D., Kim, T.-W., Bracco, C., Cho, D., Mourrain, B., Oh, M.-J., Kiatpanichgij, S.: Dimensions and bases of hierarchical tensor-product splines. J. Comput. Appl. Math. 257, 86–104 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  4. 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)

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

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

    Article  MATH  Google Scholar 

  7. de Boor, C., Höllig, K., Riemenschneider, S.: Box Splines. Springer (1993)

  8. Carnicer, J.M., Mainar, E., Peña, J.M.: Critical length for design purposes and extended Chebyshev spaces. Constr. Approx. 20, 55–71 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chui, C.K.: Multivariate splines. CBMS-NSF Reg. Conf. Series Appl. Math., vol. 54. SIAM, Philadelphia (1988)

  10. Chui, C.K., Lai, M.-J.: A multivariate analog of Marsden’s identity and a quasi-interpolation scheme. Constr. Approx. 3, 111–122 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  11. Costantini, P., Lyche, T., Manni, C.: On a class of weak Tchebycheff systems. Numer. Math. 101, 333–354 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Costantini, P., Manni, C., Pelosi, F., Sampoli, M.L.: Quasi-interpolation in isogeometric analysis based on generalized B-splines. Comput. Aided Geom. Design 27, 656–668 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley (2009)

  14. Dahmen, W., Micchelli, C.A.: On the approximation order from certain multivariate spline spaces. J. Austral. Math. Soc. Ser. B. 26, 233–246 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dierckx, P.: On calculating normalized Powell-Sabin B-splines. Comput. Aided Geom. Design 15, 61–78 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dokken, T., Lyche, T., Pettersen, K.F.: Polynomial splines over locally refined box-partitions. Comput. Aided Geom. Design 30, 331–356 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  19. Giannelli, C., Jüttler, B.: Bases and dimensions of bivariate hierarchical tensor-product splines. J. Comput. Appl. Math. 239, 162–178 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  22. 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)

  23. Kim, M., Peters, J.: Symmetric box-splines on root lattices. J. Comput. Appl. Math. 235, 3972–3989 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. 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 

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

  26. Kvasov, B.I., Sattayatham, P.: GB-splines of arbitrary order. J. Comput. Appl. Math. 104, 63–88 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  27. Lai, M.J., Schumaker, L.L.: Spline Functions on Triangulations. Cambridge (2007)

  28. 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)

  29. Li, X., Deng, J., Chen, F.: Polynomial splines over general T-meshes. Vis. Comput. 26, 277–286 (2010)

    Article  MATH  Google Scholar 

  30. Li, X., Zheng, J., Sederberg, T.W., Hughes, T.J.R., Scott, M.A.: On linear independence of T-spline blending functions. Comput. Aided Geom. Design 29, 63–76 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lyche, T., Manni, C., Sablonnière, P.: Quasi-interpolation projectors for box splines. J. Comput. Appl. Math. 221, 416–429 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  33. Lyche, T., Schumaker, L.L., Stanley, S.: Quasi-interpolants based on trigonometric splines. J. Approx. Theory 95, 280–309 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  34. Manni, C., Pelosi, F., Speleers, H.: Local hierarchical \(h\)-refinements in IgA based on generalized B-splines. In: Floater, M.S., Lyche, T., Mazure, M.L., Mørken, K., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces. Lecture Notes in Computer Science vol. 8177, pp. 341–363 (2014)

  35. Manni, C., Sablonnière, P.: Quadratic spline quasi-interpolants on Powell-Sabin partitions. Adv. Comp. Math. 26, 283–304 (2007)

    Article  MATH  Google Scholar 

  36. Mazure, M.L.: How to build all Chebyshevian spline spaces good for geometric design? Numer. Math. 119, 517–556 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  37. Mazure, M.L.: On a new criterion to decide whether a spline space can be used for design. BIT Numer. Math. 52, 1009–1034 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  38. Mokris, D., Jüttler, B., Giannelli, C.: On the completeness of hierarchical tensor-product splines. J. Comput. Appl. Math. 271, 53–70 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  40. 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)

  41. Sbibih, D., Serghini, A., Tijini, A.: Polar forms and quadratic spline quasi-interpolants on Powell-Sabin partitions. Appl. Numer. Math. 59, 938–958 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  42. Schumaker, L.L.: Spline Functions: Basic Theory, Third edition. Cambridge (2007)

  43. Schumaker, L.L., Wang, L.: Approximation power of polynomial splines on T-meshes. Comput. Aided Geom. Design 29, 599–612 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  44. Sederberg, T.W., Zheng, J., Bakenov, A., Nasri, A.: T-splines and T-NURCCs. ACM Trans. Graphics 22, 477–484 (2003)

    Article  Google Scholar 

  45. Speleers, H.: A normalized basis for reduced Clough-Tocher splines. Comput. Aided Geom. Design 27, 700–712 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  46. Speleers, H.: Construction of normalized B-splines for a family of smooth spline spaces over Powell-Sabin triangulations. Constr. Approx. 37, 41–72 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  47. Speleers, H.: Multivariate normalized Powell-Sabin B-splines and quasi-interpolants. Comput. Aided Geom. Design 30, 2–19 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  49. 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)

  50. Vanraes, E., Windmolders, J., Bultheel, A., Dierckx, P.: Automatic construction of control triangles for subdivided Powel-Sabin splines. Comput. Aided Geom. Design 21, 671–682 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  51. 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  MATH  Google Scholar 

  52. Wang, G., Fang, M.: Unified and extended form of three types of splines. J. Comput. Appl. Math. 216, 498–508 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was partially supported by the Research Foundation Flanders, by the MIUR ‘Futuro in Ricerca 2013’ Programme through the project DREAMS, and by INdAM-GNCS Gruppo Nazionale per il Calcolo Scientifico. We would like to thank Bert Jüttler (Johannes Kepler University, Linz) for fruitful discussions and for pointing out the idea of the telescopic interpretation.

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Leuven, Celestijnenlaan 200A, 3001, Leuven, Belgium

    Hendrik Speleers

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

    Hendrik Speleers & Carla Manni

Authors
  1. Hendrik Speleers
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carla Manni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hendrik Speleers.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Speleers, H., Manni, C. Effortless quasi-interpolation in hierarchical spaces. Numer. Math. 132, 155–184 (2016). https://doi.org/10.1007/s00211-015-0711-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-015-0711-z

Mathematics Subject Classification

Search

Navigation