Convergence of subdivision schemes on Riemannian manifolds with nonpositive sectional curvature

Svenja Hüning, Johannes Wallner

Research output: Contribution to journalArticleResearchpeer-review

Abstract

This paper studies well-definedness and convergence of subdivision schemes which operate on Riemannian manifolds with nonpositive sectional curvature. These schemes are constructed from linear ones by replacing affine averages by the Riemannian centre of mass. In contrast to previous work, we consider schemes without any sign restriction on the mask, and our results apply to all input data. We also analyse the Hölder continuity of the resulting limit curves. Our main result states that if the norm of the derived scheme (resp. iterated derived scheme) is smaller than the corresponding dilation factor then the adapted scheme converges. In this way, we establish that convergence of a linear subdivision scheme is almost equivalent to convergence of its nonlinear manifold counterpart.
Original languageEnglish
Pages (from-to)1689–1709
Number of pages21
JournalAdvances in Computational Mathematics
Volume45
Issue number3
DOIs
Publication statusPublished - Jun 2019

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Nonpositive Curvature
Subdivision Scheme
Sectional Curvature
Riemannian Manifold
Masks
Barycentre
Dilation
Mask
Restriction
Converge
Norm
Curve

Cite this

Convergence of subdivision schemes on Riemannian manifolds with nonpositive sectional curvature. / Hüning, Svenja; Wallner, Johannes.

In: Advances in Computational Mathematics, Vol. 45, No. 3, 06.2019, p. 1689–1709 .

Research output: Contribution to journalArticleResearchpeer-review

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