Institute of Geometry

This project has been funded by the Austrian Science Fund (FWF) under grants P18575, P19870, and the doctoral programme W1230. In this way FWF supported Ph.D. students and Postdocs Philipp Grohs (2006—2010), Esfandiar Navayazdani (2006—2007), Andreas Weinmann (2007—2011), Oliver Ebner (2009—2012), Caroline Moomüller (2013—2017), and Svenja Hüning (2016—).

Subdivision schemes in vector spaces are well understood with respect to their convergence, smoothness of limits, capability of irregular combinatorics, approximation power, and their relation to multiresolution transforms. They are prominently used in computer graphics. It is the long-term purpose of this research project to establish a similar body of knowledge for subdivision processes in nonlinear geometries. The latter occur in all places where the data under consideration do not naturally live in a vector space, but rather in a Lie group (like position data) or a symmetric space (like diffusion tensor image data) or in Euclidean space minus obstacles.

The main idea in dealing with the nonlinear situation is to interpret the basic linear operations basic to subdivision (namely, affine combinations and convex combinations) in ways which permit the transfer to more general situations. This can be done by means of exponential mappings, in a variational way, or by means of a probabilistic interpretation. For instance, a convex combination has an interpretation as the expected value of a random variable. One of the major technical tools in investigating nonlinar subdision processes are proximity inequalities which relate them to their linear counterparts. We have achieved a long list of results. Key ones concern the fact that the limits of certain manifold schemes are as smooth as their linear counterparts (including irregular cases), or that the probabilistic interpretation leads to schemes in CH metric spaces which converge if and only if the linear ones do. We have also successfully investigated stability, approximation power, and the extension to Hermite schemes.

• S. Hüning and J. Wallner. Convergence analysis of subdivision processes on the sphere. IMA J. Num. Analysis 42 (2022), 698-711. [doi], [arxiv: 2001.09426].
• S. Hüning. Geometric and algebraic analysis of subdivision processes. PhD thesis, TU Graz, 2019.
• J. Wallner. Geometric subdivision and multiscale transforms. In P. Grohs, M. Holler, and A. Weinmann, editors, Handbook of Variational Methods for Nonlinear Geometric Data, pages 121-152. Springer, 2020. [doi], [arxiv: 1907.07550].
• S. Hüning and J. Wallner. Convergence of subdivision schemes on Riemannian manifolds with nonpositive sectional curvature. Adv. Comput. Math 45 (2019), 1689-1709. [MR], [doi], [arxiv: 1710.08621].
• C. Moosmüller and N. Dyn. Increasing the smoothness of vector and Hermite subdivision schemes. IMA J. Num. Analysis 39 (2019), 579-606. [doi], [arxiv: 1710.06560].
• C. Moosmüller. Smoothness analysis of linear and nonlinear Hermite subdivision schemes. PhD thesis, TU Graz, 2016.
• C. Moosmüller. Hermite subdivision on manifolds via parallel transport. Adv. Computat. Mathematics 43 (2017), 1059-1074. [MR], [doi].
• C. Moosmüller. C¹ analysis of Hermite subdivision schemes on manifolds. SIAM J. Numerical Analysis 54 (2016), 3003–3031. [MR], [doi].
• J. Wallner. On convergent interpolatory subdivision schemes in Riemannian geometry. Constr. Approx. 40 (2014), 473-486. [Zbl], [MR], [doi].
• O. Ebner. Stochastic aspects of refinement schemes on metric spaces. PhD thesis, TU Graz, 2012.
• O. Ebner. Stochastic aspects of refinement schemes on metric spaces. SIAM J. Num. Anal 52 (2014), 717-734. [doi], [arxiv: 1112.6003].
• O. Ebner. Convergence of iterative schemes in metric spaces. Proc. American Math. Soc 141 (2013), 677-686. [Zbl], [MR], [doi].
• A. Weinmann. Interpolatory multiscale representation for functions between manifolds. SIAM J. Math. Analysis 44 (2012), 172-191. [doi].
• A. Weinmann. Subdivision schemes with general dilation in the geometric and nonlinear setting. J. Approx. Theory 164 (2012), 105-137. [doi].
• P. Grohs and J. Wallner. Definability and stability of multiscale decompositions for manifold-valued data. J. Franklin Institute 349 (2012), 1648-1664. [Zbl], [MR], [doi], [arxiv: 1001.1517].
• O. Ebner. Stochastic aspects of refinement schemes on metric spaces. PhD thesis, Technische Universität Graz, 2012.
• O. Ebner. Stochastic aspects of nonlinear refinement algorithms. In K. Jetter, S. Smale, and D.-X. Zhou, editors, Learning Theory and Approximation, volume 31/2012 of Oberwolfach Reports, page 49. 2012. Abstracts from the workshop held June 24--June 30th, 2012. [doi].
• A. Weinmann. Analysis of nonlinear geometric subdivision schemes on polyhedral meshes. PhD thesis, Technische Universität Graz, 2010.
• 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. [Zbl], [MR], [doi].
• P. Grohs. Stability of manifold-valued subdivision and multiscale transforms. Constr. Approx 32 (2010), 569-596. [doi].
• P. Grohs. Approximation theory in manifolds, 2010. Habilitation thesis, TU Graz.
• P. Grohs. A general proximity analysis of nonlinear subdivision schemes. SIAM J. Math. Anal. 42/2 (2010), 729-750. [MR], [doi].
• P. Grohs. Approximation order from stability of nonlinear subdivision schemes. J. Approx. Theory 162 (2010), 1085-1094. [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].
• A. Weinmann. Nonlinear subdivision schemes on irregular meshes. Constr. Approx 31 (2010), 395-415. [MR], [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. Smoothness equivalence properties of univariate subdivision schemes and their projection analogues. Num. Math. 113 (2009), 163-180. [doi].
• P. Grohs. Smoothness analysis of subdivision schemes on regular grids by proximity. SIAM J. Numerical Analysis 46 (2008), 2169-2182. [MR], [doi].
• P. Grohs. Smoothness of interpolatory multivariate subdivision in Lie groups. IMA J. Numer. Anal. 27 (2009), 760-772. [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].
• J. Wallner, E. Nava Yazdani, and P. Grohs. Smoothness properties of Lie group subdivision schemes. Multiscale Modeling and Simulation 6 (2007), 493-505. [MR].
• P. Grohs. Smoothness analysis of nonlinear subdivision schemes on regular grids. PhD thesis, TU Wien, 2007.
• J. Wallner and H. Pottmann. Intrinsic subdivision with smooth limits for graphics and animation. ACM Trans. Graphics 25/2 (2006), 356-374. [doi].
• J. Wallner. Smoothness analysis of subdivision schemes by proximity. Constr. Approx. 24/3 (2006), 289-318. [Zbl], [MR], [doi].
• J. Wallner and N. Dyn. Convergence and C¹ analysis of subdivision schemes on manifolds by proximity. Comput. Aided Geom. Design 22/7 (2005), 593-622. [Zbl], [MR], [doi].
• M. Hofer, H. Pottmann, and B. Ravani. Subdivision algorithms for motion design based on homologous points. In J. Lenarčičand F. Thomas, editors, Advances in Robot Kinematics, pages 235-244. Kluwer Academic Publ., 2002.