A family of non-oscillatory 6-point interpolatory subdivision schemes
- PDF / 1,393,772 Bytes
- 35 Pages / 439.642 x 666.49 pts Page_size
- 89 Downloads / 165 Views
A family of non-oscillatory 6-point interpolatory subdivision schemes ˜ 1· Rosa Donat1 · Sergio L´opez-Urena 2 Maria Sant´agueda
Received: 17 May 2016 / Accepted: 18 December 2016 © Springer Science+Business Media New York 2017
Abstract In this paper we propose and analyze a new family of nonlinear subdivision schemes which can be considered non-oscillatory versions of the 6-point Deslauries-Dubuc (DD) interpolatory scheme, just as the Powerp schemes are considered nonlinear non-oscillatory versions of the 4-point DD interpolatory scheme. Their design principle may be related to that of the Powerp schemes and it is based on a weighted analog of the Powerp mean. We prove that the new schemes reproduce exactly polynomials of degree three and stay ’close’ to the 6-point DD scheme in smooth regions. In addition, we prove that the first and second difference schemes are well defined for each member of the family, which allows us to give a simple proof of the uniform convergence of these schemes and also to study their stability as in [19, 22]. However our theoretical study of stability is not conclusive and we perform a series of numerical experiments that seem to point out that only a few members of the new family of schemes are stable. On the other hand, extensive numerical testing reveals that, for smooth data, the approximation order and the regularity of the limit function may be similar to that of the 6-point DD scheme and larger than what is obtained with the Powerp schemes. Communicated by: Yuesheng Xu Rosa Donat
[email protected] Sergio L´opez-Ure˜na [email protected] Maria Sant´agueda [email protected] 1
Departament de Matem`atiques, Universitat de Val`encia, Doctor Moliner Street 50, 46100 Burjassot, Valencia, Spain
2
Departament d’Educaci´o, Universitat Jaume I de Castell´o de la Plana, Castell´o de la Plana, Spain
R. Donat et al.
Keywords Nonlinear subdivision schemes · Convergence · Stability · Approximation order · Non-oscillatory
1 Introduction Subdivision schemes are recursive processes used for the fast generation of curves and surfaces in computer-aided geometric design, as well as an essential ingredient in many multiscale algorithms used in data compression. In some applications, the given data need to be retained at each step of the refinement process, which requires the use of interpolatory subdivision schemes. The so-called Deslauries-Dubuc (DD henceforth) subdivision schemes [6] are a well known family of interpolatory subdivision schemes which can be interpreted as a recursive application of a piecewise polynomial interpolatory tool [10, 11]. A general setting by which a piecewise polynomial interpolation technique can be used to provide the set of local rules that defines a subdivision scheme has been described in [10]: Assuming that χ l ⊂ χ l+1 are two nested grids on Rm , f l is a set of known data associated to the grid χ l and I [x, ·] is a piecewise polynomial reconstruction technique, new data associated to the grid χ l+1 can be generated as follows fil+1 = I [xil+1 , f l
Data Loading...
