Advertisement

Advances in Computational Mathematics

, Volume 43, Issue 5, pp 1059–1074 | Cite as

Hermite subdivision on manifolds via parallel transport

  • Caroline MoosmüllerEmail author
Open Access
Article

Abstract

We propose a new adaption of linear Hermite subdivision schemes to the manifold setting. Our construction is intrinsic, as it is based solely on geodesics and on the parallel transport operator of the manifold. The resulting nonlinear Hermite subdivision schemes are analyzed with respect to convergence and C 1 smoothness. Similar to previous work on manifold-valued subdivision, this analysis is carried out by proving that a so-called proximity condition is fulfilled. This condition allows to conclude convergence and smoothness properties of the manifold-valued scheme from its linear counterpart, provided that the input data are dense enough. Therefore the main part of this paper is concerned with showing that our nonlinear Hermite scheme is “close enough”, i.e., in proximity, to the linear scheme it is derived from.

Keywords

Hermite subdivision Manifolds subdivision C1 analysis Proximity 

Mathematics Subject Classification (2010)

41A25 65D17 53A99 

Notes

Acknowledgments

Open access funding provided by Graz University of Technology. The author would like to thank Johannes Wallner for helpful discussions on earlier versions of this paper and gratefully acknowledges the suggestions of the anonymous reviewers.

References

  1. 1.
    do Carmo, M.P.: Riemannian Geometry. Birkhäuser Verlag (1992)Google Scholar
  2. 2.
    Conti, C., Merrien, J.L., Romani, L.: Dual Hermite subdivision schemes of de Rham-type. BIT Numer. Math. 54, 955–977 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Dubuc, S.: Scalar and Hermite subdivision schemes. Appl. Comput. Harmon. Anal. 21(3), 376–394 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Dubuc, S., Merrien, J.L.: Convergent vector and Hermite subdivision schemes. Constr. Approx. 23(1), 1–22 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Dubuc, S., Merrien, J.L. Approximation Theory XII. In: Neamtu, M., Schumaker, L.L. (eds.): De Rham transform of a Hermite subdivision scheme, pp. 121–132. Nashboro Press, Nashville, TN (2008)Google Scholar
  6. 6.
    Dubuc, S., Merrien, J.L.: Hermite subdivision schemes and Taylor polynomials. Constr. Approx. 29(2), 219–245 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Dyn, N., Levin, D.: Approximation Theory VIII. Vol 2: Wavelets and Multilevel Approximation. In: Chui, C.K., Schumaker, L.L. (eds.): Analysis of hermite-type subdivision schemes, pp. 117–124. World Sci., River Edge, NJ (1995)Google Scholar
  8. 8.
    Dyn, N., Levin, D.: Spline Functions and the Theory of Wavelets. In: Dubuc, S., Deslauriers, G (eds.) : Analysis of Hermite-interpolatory subdivision schemes, pp. 105–113. Amer. Math. Soc., Providence, RI (1999)Google Scholar
  9. 9.
    Grohs, P.: A general proximity analysis of nonlinear subdivision schemes. SIAM J. Math. Anal. 42(2), 729–750 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Han, B., Yu, T., Xue, Y.: Noninterpolatory Hermite subdivision schemes. Math. Comput. 74(251), 1345–1367 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press (1979)Google Scholar
  12. 12.
    Kobayashi, S., Nomizu, K.: Foundations of differential geometry, vol. 2. Wiley (1969)Google Scholar
  13. 13.
    Merrien, J.L.: A family of Hermite interpolants by bisection algorithms. Numer. Algorithms 2(2), 187–200 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Merrien, J.L., Sauer, T.: From Hermite to stationary subdivision schemes in one and several variables. Adv. Comput. Math. 36(4), 547–579 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Moosmüller, C.: C 1 analysis of Hermite subdivision schemes on manifolds. SIAM J. Numer. Anal. 54(5), 3003–3031 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Onishchik, A.L., Vinberg, E.B.: Lie Groups and Lie Algebras I: Foundations of Lie Theory. Springer (1993)Google Scholar
  17. 17.
    Postnikov, M.M.: Geometry VI: Riemannian Geometry. Springer (2001)Google Scholar
  18. 18.
    Wallner, J.: Smoothness analysis of subdivision schemes by proximity. Constr. Approx. 24, 289–318 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Wallner, J., Dyn, N.: Convergence and C 1 analysis of subdivision schemes on manifolds by proximity. Comput. Aided Geom. Des. 22(7), 593–622 (2005)CrossRefzbMATHGoogle Scholar
  20. 20.
    Wallner, J., Nava Yazdani, E., Weinmann, A.: Convergence and smoothness analysis of subdivision rules in Riemannian and symmetric spaces. Adv. Comput. Math. 34(2), 201–218 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Xie, G., Yu, T.: Smoothness equivalence properties of general manifold-valued data subdivision schemes. SIAM Journal on Multiscale Modeling & Simulation 7(3), 1073–1100 (2009)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institut für GeometrieTU GrazGrazAustria

Personalised recommendations