A Necessary and Sufficient Proximity Condition for Smoothness Equivalence of Nonlinear Subdivision Schemes
Tóm tắt
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.
Tài liệu tham khảo
A. S. Cavaretta, W. Dahmen, and C. A. Micchelli. Stationary subdivision. Mem. Amer. Math. Soc., 453, 1991. American Math. Soc, Providence.
I. Daubechies and J. Lagarias. Two-scale difference equations I. existence and global regularity of solutions. SIAM J. Math. Anal., 22(5):1388–1410, 1991.
R. A. DeVore and G. G. Lorentz. Constructive Approximation. Springer-Verlag, 1993.
Z. Ditzian. Moduli of smoothness using discrete data. J. Approx. Theory, 49:115–129, 1987.
T. Duchamp, G. Xie, and T. P.-Y. Yu. Single basepoint subdivision schemes for manifold-valued data: Time-symmetry without space-symmetry. Foundations of Computational Mathematics, 13(5):693–728, 2013.
T. Duchamp, G. Xie, and T. P.-Y. Yu. On a new proximition condition for manifold-valued subdivision schemes. In Gregory E. Fasshauer and Larry L. Schumaker, editors, Approximation Theory XIV: San Antonio 2013, volume 83 of Springer Proceedings in Mathematics & Statistics, pages 65–79. Springer, Cham, 2014.
N. Dyn. Subdivision Schemes in Computer-Aided Geometric Design, pages 36–104. Advances in Numerical Analysis II, Wavelets Subdivision Algorithms and Radial Basis Functions. Clarendon Press, Oxford, 1992.
N. Dyn and R. Goldman. Convergence and smoothness of nonlinear lane-riesenfeld algorithms in the functional setting. Foundations of Computational Math., 11:79–94, 2011.
P. Grohs. Smoothness analysis of subdivision schemes on regular grids by proximity. SIAM Journal on Numerical Analysis, 46(4):2169–2182, 2008.
P. Grohs. Smoothness equivalence properties of univariate subdivision schemes and their projection analogues. Numer. Math., 113(2):163–180, 2009.
P. Grohs. Smoothness of interpolatory multivariate subdivision in Lie groups. IMA Journal of Numerical Analysis, 29(3):760–772, 2009.
P. Grohs. A general proximity analysis of nonlinear subdivision schemes. SIAM Journal on Mathematical Analysis, 42(2):729–750, 2010.
P. Grohs. Stability of manifold-valued subdivision schemes and multiscale transformations. Constructive Approximation, 32(3):569–596, 2010.
P. Grohs. Finite elements of arbitrary order and quasiinterpolation for Riemannian data. IMA Journal of Numerical Analysis, 33(3):849–874, 2013.
M. W. Hirsch, C. C. Pugh, and M. Shub. Invariant manifolds. Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin, 1977.
R. A. Horn and C. R. Johnson. Topics in matrix analysis. Cambridge University Press, 1991.
Y. Meyer. Wavelets and operators, volume 37 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1992. Translated from the 1990 French original by D. H. Salinger.
P. Oswald and T. Shingel. Commutator estimate for nonlinear subdivision. In Michael Floater, Tom Lyche, Marie-Laurence Mazure, Knut Mrken, and LarryL. Schumaker, editors, Mathematical Methods for Curves and Surfaces, volume 8177 of Lecture Notes in Computer Science, pages 383–402. Springer Berlin Heidelberg, 2014.
I. Ur Rahman, I. Drori, V. C. Stodden, D. L. Donoho, and P. Schröder. Multiscale representations for manifold-valued data. Multiscale Modeling and Simulation, 4(4):1201–1232, 2005.
O. Rioul. Simple regularity criteria for subdivision schemes. SIAM J. Math. Anal., 23(6):1544–1576, November 1992.
J. Wallner. Smoothness analysis of subdivision schemes by proximity. Constructive Approximation, 24(3):289–318, 2006.
J. Wallner and N. Dyn. Convergence and \(C^1\) analysis of subdivision schemes on manifolds by proximity. Computer Aided Geometric Design, 22(7):593–622, 2005.
J. Wallner, E. Nava Yazdani, and P. Grohs. Smoothness properties of Lie group subdivision schemes. Multiscale Modeling and Simulation, 6(2):493–505, 2007.
J. Wallner, E. Nava Yazdani, and A. Weinmann. Convergence and smoothness analysis of subdivision rules in Riemannian and symmetric spaces. Advances in Computational Mathematics, 34(2):201–218, 2011.
A. Weinmann. Nonlinear subdivision schemes on irregular meshes. Constructive Approximation, 31(3):395–415, 2010.
G. Xie and T. P.-P. Yu. An improved proximity\(\Rightarrow \)smoothness theorem. http://www.math.drexel.edu/~tyu/Papers/WeakPimpliesS.pdf, 2015.
G. Xie and T. P.-Y. Yu. Smoothness equivalence properties of manifold-valued data subdivision schemes based on the projection approach. SIAM Journal on Numerical Analysis, 45(3):1200–1225, 2007.
G. Xie and T. P.-Y. Yu. Smoothness equivalence properties of general manifold-valued data subdivision schemes. Multiscale Modeling and Simulation, 7(3):1073–1100, 2008.
G. Xie and T. P.-Y. Yu. Smoothness equivalence properties of interpolatory Lie group subdivision schemes. IMA Journal of Numerical Analysis, 30(3):731–750, 2009.
G. Xie and T. P.-Y. Yu. Approximation order equivalence properties of manifold-valued data subdivision schemes. IMA Journal of Numerical Analysis, 32(2):687–700, 2012.
G. Xie and T. P.-Y. Yu. Invariance property of the proximity condition in nonlinear subdivision. Journal of Approximation Theory, 164(8):1097–1110, 2012.
E. Nava Yazdani and T. P.-Y. Yu. On Donoho’s Log-Exp subdivision scheme: Choice of retraction and time-symmetry. Multiscale Modeling and Simulation, 9(4):1801–1828, 2011.
A. Zygmund. Smooth functions. Duke Math. J., 12:47–76, 1945.