Journal article
A Necessary and Sufficient Proximity Condition for Smoothness Equivalence of Nonlinear Subdivision Schemes
Foundations of computational mathematics, v 16(5), pp 1069-1114
01 Oct 2016
Abstract
In the recent literature on subdivision methods for approximation of manifold-valued data, a certain "proximity condition" comparing a nonlinear subdivision scheme to a linear subdivision scheme has proved to be a key analytic tool for analyzing regularity properties of the scheme. This proximity condition is now well known to be a sufficient condition for the nonlinear scheme to inherit the regularity of the corresponding linear scheme (this is called smoothness equivalence). Necessity, however, has remained an open problem. This paper introduces a smooth compatibility condition together with a new proximity condition (the differential proximity condition). The smooth compatibility condition makes precise the relation between nonlinear and linear subdivision schemes. It is shown that under the smooth compatibility condition, the differential proximity condition is both necessary and sufficient for smoothness equivalence. It is shown that the failure of the proximity condition corresponds to the presence of resonance terms in a certain discrete dynamical system derived from the nonlinear scheme. Such resonance terms are then shown to slow down the convergence rate relative to the convergence rate of the corresponding linear scheme. Finally, a super-convergence property of nonlinear subdivision schemes is used to conclude that the slowed decay causes a breakdown of smoothness. The proof of sufficiency relies on certain properties of the Taylor expansion of nonlinear subdivision schemes, which, in addition, explain why the differential proximity condition implies the proximity conditions that appear in previous work.
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Details
- Title
- A Necessary and Sufficient Proximity Condition for Smoothness Equivalence of Nonlinear Subdivision Schemes
- Creators
- Tom Duchamp - University of WashingtonGang Xie - East China University of Science and TechnologyThomas Yu - Drexel University
- Publication Details
- Foundations of computational mathematics, v 16(5), pp 1069-1114
- Publisher
- Springer Nature
- Number of pages
- 46
- Grant note
- 1115915 / Direct For Mathematical & Physical Scien; National Science Foundation (NSF); NSF - Directorate for Mathematical & Physical Sciences (MPS) Louis and Bessie Stein family Fundamental Research Funds for the Central Universities DMS 0915068; DMS 1115915 / National Science Foundation; National Science Foundation (NSF) 11101146 / National, Natural Science Foundation of China; National Natural Science Foundation of China (NSFC)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000385660400001
- Scopus ID
- 2-s2.0-84933575090
- Other Identifier
- 991021878115404721
InCites Highlights
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Computer Science, Theory & Methods
- Mathematics
- Mathematics, Applied