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Families of non-linear subdivision schemes for scattered data fitting and their non-tensor product extensions

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  • Mustafa, Ghulam
  • Hameed, Rabia

Abstract

In this article, families of non-linear subdivision schemes are presented that are based on univariate polynomials up to degree three. These families of schemes are constructed by using dynamic iterative re-weighed least squares method. These schemes are suitable for fitting noisy data with outliers. The codes are designed in a Python environment to numerically fit the given data points. Although these schemes are non-interpolatory, but have the ability to preserve the shape of the initial polygon in case of non-noisy initial data. The numerical examples illustrate that the schemes constructed by non-linear polynomials give better performance than the schemes that are constructed by linear polynomials (Mustafa et al., 2015). Moreover, the numerical examples show that these schemes have the ability to reproduce polynomials and do not cause over and under fitting of the data. Furthermore, families of non-linear bivariate subdivision schemes are also presented that are based on linear and non-linear bivariate polynomials.

Suggested Citation

  • Mustafa, Ghulam & Hameed, Rabia, 2019. "Families of non-linear subdivision schemes for scattered data fitting and their non-tensor product extensions," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 214-240.
    • Handle: RePEc:eee:apmaco:v:359:y:2019:i:c:p:214-240
    DOI: 10.1016/j.amc.2018.12.075
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    References listed on IDEAS

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    1. Joldes, Grand Roman & Chowdhury, Habibullah Amin & Wittek, Adam & Doyle, Barry & Miller, Karol, 2015. "Modified moving least squares with polynomial bases for scattered data approximation," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 893-902.
    2. Dell’Accio, Francesco & Di Tommaso, Filomena & Hormann, Kai, 2018. "Reconstruction of a function from Hermite–Birkhoff data," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 51-69.
    3. Li, Xin & Chang, Yubo, 2018. "Non-uniform interpolatory subdivision surface," Applied Mathematics and Computation, Elsevier, vol. 324(C), pages 239-253.
    4. Amirfakhrian, M. & Mafikandi, H., 2016. "Approximation of parametric curves by Moving Least Squares method," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 290-298.
    5. Hameed, Rabia & Mustafa, Ghulam, 2017. "Family of a-point b-ary subdivision schemes with bell-shaped mask," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 289-302.
    6. Allasia, Giampietro & Cavoretto, Roberto & De Rossi, Alessandra, 2018. "Hermite–Birkhoff interpolation on scattered data on the sphere and other manifolds," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 35-50.
    7. Amat, S. & Choutri, A. & Ruiz, J. & Zouaoui, S., 2018. "On a nonlinear 4-point ternary and non-interpolatory subdivision scheme eliminating the Gibbs phenomenon," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 16-26.
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