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Hermite–Birkhoff interpolation on scattered data on the sphere and other manifolds

Author

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  • Allasia, Giampietro
  • Cavoretto, Roberto
  • De Rossi, Alessandra

Abstract

The Hermite–Birkhoff interpolation problem of a function given on arbitrarily distributed points on the sphere and other manifolds is considered. Each proposed interpolant is expressed as a linear combination of basis functions, the combination coefficients being incomplete Taylor expansions of the interpolated function at the interpolation points. The basis functions depend on the geodesic distance, are orthonormal with respect to the point-evaluation functionals, and have all derivatives equal zero up to a certain order at the interpolation points. A remarkable feature of such interpolants, which belong to the class of partition of unity methods, is that their construction does not require solving linear systems. Numerical tests are given to show the interpolation performance.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:318:y:2018:i:c:p:35-50
    DOI: 10.1016/j.amc.2017.05.018
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    References listed on IDEAS

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    1. Yao, Guangming & Duo, Jia & Chen, C.S. & Shen, L.H., 2015. "Implicit local radial basis function interpolations based on function values," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 91-102.
    2. Li, Ming & Cao, Feilong, 2015. "Multiscale interpolation on the sphere: Convergence rate and inverse theorem," Applied Mathematics and Computation, Elsevier, vol. 263(C), pages 134-150.
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    Cited by:

    1. Samir, Chafik & Adouani, Ines, 2019. "C1 interpolating Bézier path on Riemannian manifolds, with applications to 3D shape space," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 371-384.
    2. Xia, Peng & Lei, Na & Dong, Tian, 2023. "On the linearization methods for univariate Birkhoff rational interpolation," Applied Mathematics and Computation, Elsevier, vol. 445(C).
    3. Samir, Chafik & Huang, Wen, 2021. "Coordinate descent optimization for one-to-one correspondence and supervised classification of 3D shapes," Applied Mathematics and Computation, Elsevier, vol. 388(C).
    4. 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.
    5. Cavoretto, R. & De Rossi, A. & Perracchione, E., 2023. "Learning with Partition of Unity-based Kriging Estimators," Applied Mathematics and Computation, Elsevier, vol. 448(C).

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