Three Families of Nonlinear Subdivision Schemes

Nira Dyn*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

7 Scopus citations


Three families of nonlinear subdivision schemes, derived from linear schemes, are reviewed. The nonlinearity is introduced into the linear schemes by adapting the schemes to the data. The first family, derived from the four-point interpolatory linear subdivision scheme, consists of geometrically controlled schemes, which are either shape preserving or artifact-free. The second family of schemes is designed for the functional setting, to be used in constructions of multiscale representations of piecewise smooth functions. The schemes are extensions of the Dubuc-Deslauriers 2N-point interpolatory schemes, with the classical local interpolation replaced by ENO or WENO local interpolation. The third family consists of subdivision schemes on smooth manifolds. These schemes are derived from converging linear schemes, represented in terms of repeated binary averages. The analysis of the nonlinear schemes is done either by proximity to the linear schemes from which they are derived, or by methods adapted from methods for linear schemes.

Original languageEnglish
Title of host publicationStudies in Computational Mathematics
Number of pages16
StatePublished - 2006

Publication series

NameStudies in Computational Mathematics
ISSN (Print)1570-579X


  • 2000 MSC 65D05
  • 65D07
  • 65D10
  • 65D15
  • 65D17
  • 65U05
  • 65U07
  • ENO interpolation
  • adaptive tension parameter
  • convexity preserving scheme
  • data dependent scheme
  • geodesics
  • linear and nonlinear subdivision scheme
  • projection onto a manifold
  • refinement on a manifold
  • repeated binary averages


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