Institute of Geometry

Subdivision schemes in nonlinear geometries are constructed from linear schemes by expressing the linear schemes in terms of basic geometric constructions, and then finding an equivalent to these constructions in the nonlinear geometry under consideration. We employ projection-type mappings, the exponential mapping in Riemannian geometry as well as in Lie groups, and others. The analysis of the nonlinear schemes which arise in this way is based on their proximity to the linear schemes which they are derived from.

Our work on smoothness analysis of nonlinear schemes has been supported by FWF grant P18575 from 2006-2007 (q.v.). The work on multivariate subdivision processes is supported by FWF grant P19870.

The new results we have been obtaining concern lazy wavelet transforms based on nonlinar subdivision rules, stability properties, smoothness of subdivision near extraordinary vertices, and others. Analysis mainly employs the method of `proximity inequalities'.

• O. Ebner. Nonlinear Markov semigroups and refinement schemes on metric spaces. Geometry Preprint 2011/02, TU Graz, November 2011. [arxiv:1112.6003].
• A. Weinmann. Subdivision schemes with general dilation in the geometric and nonlinear setting. J. Approx. Theory 164 (2012), 105-137. [doi].
• O. Ebner. Convergence of iterative schemes in metric spaces. Proc. American Math. Soc (2011), to appear.
• A. Weinmann. Interpolatory multiscale representation for functions between manifolds. SIAM J. Math. Analysis (2012), to appear.
• P. Grohs and J. Wallner. Definability and stability of multiscale decompositions for manifold-valued data. J. Franklin Institute (2011), to appear. [doi], [arxiv:1001.1517].
• A. Weinmann. Analysis of nonlinear geometric subdivision schemes on polyhedral meshes. PhD thesis, Technische Universität Graz, 2010.
• P. Grohs. Approximation order from stability of nonlinear subdivision schemes. J. Approx. Theory 162 (2010), 1085-1094. [MR].
• P. Grohs. Stability of manifold-valued subdivision and multiscale transforms. Constr. Approx 32 (2010), 569-596. [doi].
• A. Weinmann. Nonlinear subdivision schemes on irregular meshes. Constr. Approx 31 (2010), 395-415. [MR], [doi].
• P. Grohs. A general proximity analysis of nonlinear subdivision schemes. SIAM J. Math. Anal. 42/2 (2010), 729-750. [MR].
• N. Dyn, P. Grohs, and J. Wallner. Approximation order of interpolatory nonlinear subdivision schemes. J. Comput. Appl. Math. 223 (2010), 1697-1703. [MR], [doi].
• J. Wallner, E. Nava Yazdani, and A. Weinmann. Convergence and smoothness analysis of subdivision rules in Riemannian and symmetric spaces. Adv. Comp. Math 34 (2011), 201-218. [MR], [doi].
• P. Grohs. Higher order smoothness of interpolatory multivariate subdivision in Lie groups. IMA J. Numer. Anal. 27 (2009), 760-772. [doi].
• P. Grohs and J. Wallner. Interpolatory wavelets for manifold-valued data. Appl. Comput. Harmon. Anal. 27 (2009), 325-333. [Zbl], [MR], [doi].
• P. Grohs and J. Wallner. Log-exponential analogues of univariate subdivision schemes in Lie groups and their smoothness properties. In M. Neamtu and L. L. Schumaker, editors, Approximation Theory XII: San Antonio 2007, pages 181-190. Nashboro Press, 2008, ISBN 978-0-9728482-9-9. [Zbl], [MR].
• P. Grohs. Smoothness analysis of subdivision schemes on regular grids by proximity. SIAM J. Numerical Analysis 46 (2008), 2169-2182. [MR].