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.
Similar content being viewed by others
References
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)
de Boor, C., Fix, M.G.: Spline approximation by quasi-interpolants. J. Approx. Theory 8, 19–45 (1973)
Buffa, A., Giannelli, C.: Adaptive isogeometric methods with hierarchical splines: error estimator and convergence Math. Models Methods Appl. Sci. 26, 1–25 (2016)
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)
Dokken, T., Lyche, T., Pettersen, K.F.: Polynomial splines over locally refined box-partitions. Comput. Aided Geom. Design 30, 331–356 (2013)
Giannelli, C., Jüttler, B.: Bases and dimensions of bivariate hierarchical tensor-product splines. J. Comput. Appl. Math. 239, 162–178 (2013)
Giannelli, C., Jüttler, B., Speleers, H.: THB-splines: the truncated basis for hierarchical splines. Comput. Aided Geom. Design 29, 485–498 (2012)
Giannelli, C., Jüttler, B., Speleers, H.: Strongly stable bases for adaptively refined multilevel spline spaces. Adv. Comp. Math. 40, 459–490 (2014)
Iurino, A.: BS Hermite Quasi-Interpolation Methods for Curves and Surfaces. PhD thesis, Università di Bari (2014)
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)
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)
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)
Li, X., Deng, J., Chen, F.: Polynomial splines over general T-meshes. Visual Comput. 26, 277–286 (2010)
Lyche, T., Schumaker, L.L.: Local spline approximation. J. Approx. Theory 15, 294–325 (1975)
Mazzia, F., Sestini, A.: The BS class of Hermite spline quasi-interpolants on nonuniform knot distributions. BIT 49, 611–628 (2009)
Mazzia, F., Sestini, A.: Quadrature formulas descending from BS Hermite spline quasi-interpolation. J. Comput. Appl. Math. 236, 4105–4118 (2012)
Mokriš, D., Jüttler, B., Giannelli, C.: On the completeness of hierarchical tensor-product Bsplines. J. Comput. Appl. Math. 271, 53–70 (2014)
Speleers, H., Manni, C.: Effortless quasi-interpolation in hierarchical spaces. Numer. Math. 132, 155–184 (2016)
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)
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)
Schumaker, L.L., Wang, L.: Approximation power of polynomial splines on T-meshes. Comput. Aided Geom. Design 29, 599–612 (2012)
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)
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Tom Lyche.
Rights and permissions
About this article
Cite this article
Bracco, C., Giannelli, C., Mazzia, F. et al. Bivariate hierarchical Hermite spline quasi-interpolation. Bit Numer Math 56, 1165–1188 (2016). https://doi.org/10.1007/s10543-016-0603-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10543-016-0603-3