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 longterm 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 of subdivision schemes on Riemannian manifolds with nonpositive
sectional curvature.
Adv. Comput. Math (2019), to appear.
[arxiv:1710.08621].

C. Moosmüller and N. Dyn.
Increasing the smoothness of vector and Hermite subdivision schemes.
IMA J. Num. Analysis (2018).
[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 (2017), 10591074.
[doi].

C. Moosmüller.
C^{1} 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), 473486.
[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), 717734.
[doi], [arxiv:1112.6003].

O. Ebner.
Convergence of
iterative schemes in metric spaces.
Proc. American Math. Soc 141 (2013), 677686.
[Zbl], [MR], [doi].

A. Weinmann.
Interpolatory
multiscale representation for functions between manifolds.
SIAM J. Math. Analysis 44 (2012), 172191.
[doi].

A. Weinmann.
Subdivision schemes with general dilation in the geometric and nonlinear
setting.
J. Approx. Theory 164 (2012), 105137.
[doi].

P. Grohs and J. Wallner.
Definability and
stability of multiscale decompositions for manifoldvalued data.
J. Franklin Institute 349 (2012), 16481664.
[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 24June 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), 201218.
[Zbl], [MR], [doi].

P. Grohs.
Stability of
manifoldvalued subdivision and multiscale transforms.
Constr. Approx 32 (2010), 569596.
[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), 729750.
[MR], [doi].

P. Grohs.
Approximation order
from stability of nonlinear subdivision schemes.
J. Approx. Theory 162 (2010), 10851094.
[MR].

N. Dyn, P. Grohs, and J. Wallner.
Approximation order
of interpolatory nonlinear subdivision schemes.
J. Comput. Appl. Math. 223 (2010), 16971703.
[MR], [doi].

A. Weinmann.
Nonlinear
subdivision schemes on irregular meshes.
Constr. Approx 31 (2010), 395415.
[MR], [doi].

P. Grohs and J. Wallner.
Interpolatory
wavelets for manifoldvalued data.
Appl. Comput. Harmon. Anal. 27 (2009), 325333.
[Zbl], [MR], [doi].

P. Grohs.
Smoothness equivalence
properties of univariate subdivision schemes and their projection
analogues.
Num. Math. 113 (2009), 163180.
[doi].

P. Grohs.
Smoothness
analysis of subdivision schemes on regular grids by proximity.
SIAM J. Numerical Analysis 46 (2008), 21692182.
[MR], [doi].

P. Grohs.
Smoothness of
interpolatory multivariate subdivision in Lie groups.
IMA J. Numer. Anal. 27 (2009), 760772.
[doi].

P. Grohs and J. Wallner.
Logexponential
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 181190. Nashboro Press, 2008, ISBN
9780972848299.
[Zbl], [MR].

J. Wallner, E. Nava Yazdani, and P. Grohs.
Smoothness
properties of Lie group subdivision schemes.
Multiscale Modeling and Simulation 6 (2007), 493505.
[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), 356374.
[doi].

J. Wallner.
Smoothness analysis
of subdivision schemes by proximity.
Constr. Approx. 24/3 (2006), 289318.
[Zbl], [MR], [doi].

J. Wallner and N. Dyn.
Convergence and
C^{1} analysis of subdivision schemes on manifolds by proximity.
Comput. Aided Geom. Design 22/7 (2005), 593622.
[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 235244. Kluwer Academic Publ., 2002.
