Two-stage approximation methods with extended B-splines

Davydov, O.; Prasiswa, J.; Reif, U.

doi:10.1090/S0025-5718-2013-02734-2

Two-stage approximation methods with extended B-splines
HTML articles powered by AMS MathViewer

by O. Davydov, J. Prasiswa and U. Reif PDF

Math. Comp. 83 (2014), 809-833 Request permission

Abstract:

We develop and analyze a framework for two-stage methods with EB-splines, applicable to continuous and discrete approximation problems. In particular, we propose a weighted discrete least squares fit that yields optimal convergence rates for sufficiently dense data on Lipschitz domains in $\mathbb {R}^d$.

References

Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
Carl de Boor, A practical guide to splines, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR 507062
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376, DOI 10.1007/978-1-4757-3658-8
Oleg Davydov. On the approximation power of local least squares polynomials. In I. J. Anderson J. Levesley and J. C. Mason, editors, Algorithms for Approximation IV, pages 346–353. University of Huddersfield, UK, 2002.
Oleg Davydov, Rossana Morandi, and Alessandra Sestini, Local hybrid approximation for scattered data fitting with bivariate splines, Comput. Aided Geom. Design 23 (2006), no. 9, 703–721. MR 2278767, DOI 10.1016/j.cagd.2006.04.001
Oleg Davydov, Alessandra Sestini, and Rossana Morandi, Local RBF approximation for scattered data fitting with bivariate splines, Trends and applications in constructive approximation, Internat. Ser. Numer. Math., vol. 151, Birkhäuser, Basel, 2005, pp. 91–102. MR 2147572, DOI 10.1007/3-7643-7356-3_{8}
Oleg Davydov and Frank Zeilfelder, Scattered data fitting by direct extension of local polynomials to bivariate splines, Adv. Comput. Math. 21 (2004), no. 3-4, 223–271. MR 2073142, DOI 10.1023/B:ACOM.0000032041.68678.fa
Paul Dierckx, Curve and surface fitting with splines, Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1218172
David R. Forsey and Richard H. Bartels. Hierarchical B-spline refinement. In SIGGRAPH ’88: Proceedings of the 15th annual conference on Computer graphics and interactive techniques, pages 205–212, New York, NY, USA, 1988. ACM.
M. von Golitschek, Norms of projectors onto spaces with Riesz bases, J. Comput. Appl. Math. 119 (2000), no. 1-2, 209–221. Dedicated to Professor Larry L. Schumaker on the occasion of his 60th birthday. MR 1774218, DOI 10.1016/S0377-0427(00)00379-4
Manfred von Golitschek and Larry L. Schumaker, Bounds on projections onto bivariate polynomial spline spaces with stable local bases, Constr. Approx. 18 (2002), no. 2, 241–254. MR 1890498, DOI 10.1007/s00365-001-0018-4
Jörg Haber, Frank Zeilfelder, Oleg Davydov, and Hans Peter Seidel. Smooth approximation and rendering of large scattered data sets. In Proceedings of the conference on Visualization ’01, VIS ’01, pages 341–348, Washington, DC, USA, 2001. IEEE Computer Society.
Klaus Höllig, Finite element methods with B-splines, Frontiers in Applied Mathematics, vol. 26, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. MR 1952348, DOI 10.1137/1.9780898717532
Klaus Höllig and Ulrich Reif, Nonuniform web-splines, Comput. Aided Geom. Design 20 (2003), no. 5, 277–294. MR 1988652, DOI 10.1016/S0167-8396(03)00045-1
Klaus Höllig, Ulrich Reif, and Joachim Wipper, Weighted extended B-spline approximation of Dirichlet problems, SIAM J. Numer. Anal. 39 (2001), no. 2, 442–462. MR 1860269, DOI 10.1137/S0036142900373208
Kurt Jetter, Joachim Stöckler, and Joseph D. Ward, Error estimates for scattered data interpolation on spheres, Math. Comp. 68 (1999), no. 226, 733–747. MR 1642746, DOI 10.1090/S0025-5718-99-01080-7
Ming-Jun Lai and Larry L. Schumaker, A domain decomposition method for computing bivariate spline fits of scattered data, SIAM J. Numer. Anal. 47 (2009), no. 2, 911–928. MR 2485438, DOI 10.1137/070710056
Byung-Gook Lee, Tom Lyche, and Knut Mørken, Some examples of quasi-interpolants constructed from local spline projectors, Mathematical methods for curves and surfaces (Oslo, 2000) Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 2001, pp. 243–252. MR 1858961
Yaron Lipman, Stable moving least-squares, J. Approx. Theory 161 (2009), no. 1, 371–384. MR 2558161, DOI 10.1016/j.jat.2008.10.011
Tom Lyche, Carla Manni, and Paul Sablonnière, Quasi-interpolation projectors for box splines, J. Comput. Appl. Math. 221 (2008), no. 2, 416–429. MR 2457674, DOI 10.1016/j.cam.2007.10.029
Bernhard Mößner and Ulrich Reif, Stability of tensor product B-splines on domains, J. Approx. Theory 154 (2008), no. 1, 1–19. MR 2460580, DOI 10.1016/j.jat.2008.02.001
Ulrich Reif. Polynomial approximation on domains bounded by diffeomorphic images of graphs. Journal of Approximation Theory, 2012. to appear.
Larry L. Schumaker, Two-stage spline methods for fitting surfaces, Approximation theory (Proc. Internat. Colloq., Inst. Angew. Math., Univ. Bonn, Bonn, 1976) Springer, Berlin, 1976, pp. 378–389. MR 0617766
Holger Wendland, Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17, Cambridge University Press, Cambridge, 2005. MR 2131724