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Modified moving least squares with polynomial bases for scattered data approximation

Author

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  • Joldes, Grand Roman
  • Chowdhury, Habibullah Amin
  • Wittek, Adam
  • Doyle, Barry
  • Miller, Karol

Abstract

One common problem encountered in many fields is the generation of surfaces based on values at irregularly distributed nodes. To tackle such problems, we present a modified, robust moving least squares (MLS) method for scattered data smoothing and approximation. The error functional used in the derivation of the classical MLS approximation is augmented with additional terms based on the coefficients of the polynomial base functions. This allows quadratic base functions to be used with the same size of the support domain as linear base functions, resulting in better approximation capability. The increased robustness of the modified MLS method to irregular nodal distributions makes it suitable for use across many fields. The analysis is supported by several univariate and bivariate examples.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:266:y:2015:i:c:p:893-902
    DOI: 10.1016/j.amc.2015.05.150
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    References listed on IDEAS

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    1. Xia Jin & Grand Roman Joldes & Karol Miller & King H. Yang & Adam Wittek, 2014. "Meshless algorithm for soft tissue cutting in surgical simulation," Computer Methods in Biomechanics and Biomedical Engineering, Taylor & Francis Journals, vol. 17(7), pages 800-811, May.
    2. Hui Zou & Trevor Hastie, 2005. "Addendum: Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 768-768, November.
    3. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 301-320, April.
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    1. Wang, Qiao & Zhou, Wei & Cheng, Yonggang & Ma, Gang & Chang, Xiaolin & Miao, Yu & Chen, E, 2018. "Regularized moving least-square method and regularized improved interpolating moving least-square method with nonsingular moment matrices," Applied Mathematics and Computation, Elsevier, vol. 325(C), pages 120-145.
    2. Bauer, Benedikt & Devroye, Luc & Kohler, Michael & Krzyżak, Adam & Walk, Harro, 2017. "Nonparametric estimation of a function from noiseless observations at random points," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 93-104.
    3. Francomano, Elisa & Hilker, Frank M. & Paliaga, Marta & Venturino, Ezio, 2018. "Separatrix reconstruction to identify tipping points in an eco-epidemiological model," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 80-91.
    4. Hongtao Yang & Hao Wang & Bo Li, 2024. "Analysis of Meshfree Galerkin Methods Based on Moving Least Squares and Local Maximum-Entropy Approximation Schemes," Mathematics, MDPI, vol. 12(3), pages 1-20, February.
    5. 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.
    6. Wang, Qiao & Zhou, Wei & Feng, Y.T. & Ma, Gang & Cheng, Yonggang & Chang, Xiaolin, 2019. "An adaptive orthogonal improved interpolating moving least-square method and a new boundary element-free method," Applied Mathematics and Computation, Elsevier, vol. 353(C), pages 347-370.

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