Advertisement

Basis Functions for Scattered Data Quasi-Interpolation

  • Nira GrubergerEmail author
  • David Levin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9213)

Abstract

Given scattered data of a smooth function in \({I\!R}^d\), we consider quasi-interpolation operators for approximating the function. In order to use these operators for the derivation of useful schemes for PDE solvers, we would like the quasi-interpolation operators to be of compact support and of high approximation power. The quasi-interpolation operators are generated through known quasi-interpolation operators on uniform grids, and the resulting basis functions are represented by finite combinations of box-splines. A special attention is given to point-sets of varying density. We construct basis functions with support sizes and approximation power related to the local density of the data points. These basis functions can be used in Finite Elements and in Isogeometric Analysis in cases where a non-uniform mesh is required, the same as T-splines are being used as basis functions for introducing local refinements in flow problems.

Keywords

Scattered Data Quasi-Interpolation Box-splines 

References

  1. 1.
    Buhmann, M.D., Dyn, N., Levin, D.: On quasi-interploation by radial basis functions with scattered centers. Constr. Approx. 11(2), 239–254 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    de Boor, C., Hollig, K.: B-splines from parallelepipeds. J. d’Analyse Math. 42, 99–115 (1982–1983)Google Scholar
  3. 3.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar
  4. 4.
    Risler, J.J.: Mathematical Methods for CAD. Cambridge University Press, Cambridge (1993)Google Scholar
  5. 5.
    de Boor, C., Hollig, K., Riemenschneider, S.: Box Splines. Springer, New York (1993)CrossRefGoogle Scholar
  6. 6.
    Levin, D.: The approximation power of moving least-squares. Math. Comp. 67(224), 1517–1531 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bazilevs, Y., Calo, V.M., Cottrell, J.A., Evans, J.A., Hughes, T.J.R., Lipton, S., Scott, M.A., Sederberg, T.W.: Isogeometric analysis using T-splines. Comput. Meth. Appl. Mech. Eng. 199(5), 229–263 (2010). ElsevierzbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Ruppin Academic CenterEmek HeferIsrael
  2. 2.School of Mathematical SciencesTel-Aviv UniversityTel Aviv-yafoIsrael

Personalised recommendations