Smoothness analysis of linear and nonlinear Hermite subdivision schemes

Caroline Moosmüller

Research output: ThesisDoctoral ThesisResearch

Abstract

Hermite subdivision schemes are iterative methods for refining discrete point-vector 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 manifold-valued data using intrinsic constructions such as geodesics and parallel transport. In the case of submanifolds of R^n, we also consider manifold-valued 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 structure-preserving. 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 so-called 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.
Original languageEnglish
Awarding Institution
  • Institute of Geometry (5070)
Supervisors/Advisors
  • Wallner, Johannes, Supervisor
Publication statusPublished - 12 Dec 2016

Fingerprint

Subdivision Scheme
Hermite
Smoothness
Iterative methods
Refining
Derivatives
Regularity
Submanifolds
Geometry
Projection
Orthogonal Group
Subdivision
Proximity
Geodesic
Smoothing
Scalar
Iteration
Derivative
Invariant

ASJC Scopus subject areas

  • Numerical Analysis

Fields of Expertise

  • Information, Communication & Computing

Cite this

Smoothness analysis of linear and nonlinear Hermite subdivision schemes. / Moosmüller, Caroline.

2016. 109 p.

Research output: ThesisDoctoral ThesisResearch

Moosmüller, C 2016, 'Smoothness analysis of linear and nonlinear Hermite subdivision schemes', Institute of Geometry (5070).
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N2 - Hermite subdivision schemes are iterative methods for refining discrete point-vector 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 manifold-valued data using intrinsic constructions such as geodesics and parallel transport. In the case of submanifolds of R^n, we also consider manifold-valued 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 structure-preserving. 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 so-called 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 point-vector 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 manifold-valued data using intrinsic constructions such as geodesics and parallel transport. In the case of submanifolds of R^n, we also consider manifold-valued 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 structure-preserving. 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 so-called 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

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