Abstract
Original language  English 

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Publication status  Published  12 Dec 2016 
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ASJC Scopus subject areas
 Numerical Analysis
Fields of Expertise
 Information, Communication & Computing
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Smoothness analysis of linear and nonlinear Hermite subdivision schemes. / Moosmüller, Caroline.
2016. 109 p.Research output: Thesis › Doctoral Thesis › Research
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TY  THES
T1  Smoothness analysis of linear and nonlinear Hermite subdivision schemes
AU  Moosmüller, Caroline
PY  2016/12/12
Y1  2016/12/12
N2  Hermite subdivision schemes are iterative methods for refining discrete pointvector data in order to obtain, in the limit, a function together with its derivatives. In this thesis we study the convergence behavior of such subdivision schemes as well as the regularity of the functions which arise as their limits. Furthermore, we establish properties of Hermite schemes in nonlinear situations, especially of schemes whose definition is solely via intrinsic properties of the geometry the data are contained in. The first part of this thesis addresses Hermite subdivision schemes in the setting of manifolds. We present two adaptations of linear schemes to operate on manifoldvalued data using intrinsic constructions such as geodesics and parallel transport. In the case of submanifolds of R^n, we also consider manifoldvalued schemes which are defined from linear ones by applying a projection. This approach is not intrinsic, since projections depend on the submanifold's embedding in R^n. If the submanifold in question is invariant with respect to certain transformations, however, the projection approach can be structurepreserving. For example, this is the case for the special orthogonal group SO_3. Furthermore, we present a framework for analyzing nonlinear Hermite schemes with respect to convergence and C^1 smoothness. This is based on a socalled proximity condition, which allows us to conclude convergence and smoothness properties of a nonlinear scheme from its linear counterpart. In the second part we present a method for constructing both vector and Hermite subdivision schemes with limits of high regularity. This is inspired by a similar method in scalar subdivision and works by manipulating symbols. Via the iterated application of the smoothing procedure we developed, an Hermite scheme with limits of regularity at least C^1 can be transformed to a new scheme of arbitrarily high regularity. In particular, this method gives rise to new linear Hermite schemes.
AB  Hermite subdivision schemes are iterative methods for refining discrete pointvector data in order to obtain, in the limit, a function together with its derivatives. In this thesis we study the convergence behavior of such subdivision schemes as well as the regularity of the functions which arise as their limits. Furthermore, we establish properties of Hermite schemes in nonlinear situations, especially of schemes whose definition is solely via intrinsic properties of the geometry the data are contained in. The first part of this thesis addresses Hermite subdivision schemes in the setting of manifolds. We present two adaptations of linear schemes to operate on manifoldvalued data using intrinsic constructions such as geodesics and parallel transport. In the case of submanifolds of R^n, we also consider manifoldvalued schemes which are defined from linear ones by applying a projection. This approach is not intrinsic, since projections depend on the submanifold's embedding in R^n. If the submanifold in question is invariant with respect to certain transformations, however, the projection approach can be structurepreserving. For example, this is the case for the special orthogonal group SO_3. Furthermore, we present a framework for analyzing nonlinear Hermite schemes with respect to convergence and C^1 smoothness. This is based on a socalled proximity condition, which allows us to conclude convergence and smoothness properties of a nonlinear scheme from its linear counterpart. In the second part we present a method for constructing both vector and Hermite subdivision schemes with limits of high regularity. This is inspired by a similar method in scalar subdivision and works by manipulating symbols. Via the iterated application of the smoothing procedure we developed, an Hermite scheme with limits of regularity at least C^1 can be transformed to a new scheme of arbitrarily high regularity. In particular, this method gives rise to new linear Hermite schemes.
M3  Doctoral Thesis
ER 