Volume 11, Issue 1
Rational Quasi-Interpolation Approximation of Scattered Data in $\mathbb{R}^3$

Renzhong Feng & Lifang Song

Numer. Math. Theor. Meth. Appl., 11 (2018), pp. 169-186.

Published online: 2018-11

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  • Abstract

This paper is concerned with a piecewise smooth rational quasi-interpolation with algebraic accuracy of degree $(n+1)$ to approximate the scattered data in $\mathbb{R}^3$. We firstly use the modified Taylor expansion to expand the mean value coordinates interpolation with algebraic accuracy of degree one to one with algebraic accuracy of degree $(n+1)$. Then, based on the triangulation of the scattered nodes in $\mathbb{R}^2$, on each triangle a rational quasi-interpolation function is constructed. The constructed rational quasi-interpolation is a linear combination of three different expanded mean value coordinates interpolations and it has algebraic accuracy of degree $(n+1)$. By comparing accuracy, stability, and efficiency with the $C^1$-Tri-interpolation method of Goodman [16] and the MQ Shepard method, it is observed that our method has some computational advantages.

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@Article{NMTMA-11-169, author = {}, title = {Rational Quasi-Interpolation Approximation of Scattered Data in $\mathbb{R}^3$}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {11}, number = {1}, pages = {169--186}, abstract = {

This paper is concerned with a piecewise smooth rational quasi-interpolation with algebraic accuracy of degree $(n+1)$ to approximate the scattered data in $\mathbb{R}^3$. We firstly use the modified Taylor expansion to expand the mean value coordinates interpolation with algebraic accuracy of degree one to one with algebraic accuracy of degree $(n+1)$. Then, based on the triangulation of the scattered nodes in $\mathbb{R}^2$, on each triangle a rational quasi-interpolation function is constructed. The constructed rational quasi-interpolation is a linear combination of three different expanded mean value coordinates interpolations and it has algebraic accuracy of degree $(n+1)$. By comparing accuracy, stability, and efficiency with the $C^1$-Tri-interpolation method of Goodman [16] and the MQ Shepard method, it is observed that our method has some computational advantages.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2017-0019}, url = {http://global-sci.org/intro/article_detail/nmtma/10649.html} }
TY - JOUR T1 - Rational Quasi-Interpolation Approximation of Scattered Data in $\mathbb{R}^3$ JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 169 EP - 186 PY - 2018 DA - 2018/11 SN - 11 DO - http://doi.org/10.4208/nmtma.OA-2017-0019 UR - https://global-sci.org/intro/article_detail/nmtma/10649.html KW - AB -

This paper is concerned with a piecewise smooth rational quasi-interpolation with algebraic accuracy of degree $(n+1)$ to approximate the scattered data in $\mathbb{R}^3$. We firstly use the modified Taylor expansion to expand the mean value coordinates interpolation with algebraic accuracy of degree one to one with algebraic accuracy of degree $(n+1)$. Then, based on the triangulation of the scattered nodes in $\mathbb{R}^2$, on each triangle a rational quasi-interpolation function is constructed. The constructed rational quasi-interpolation is a linear combination of three different expanded mean value coordinates interpolations and it has algebraic accuracy of degree $(n+1)$. By comparing accuracy, stability, and efficiency with the $C^1$-Tri-interpolation method of Goodman [16] and the MQ Shepard method, it is observed that our method has some computational advantages.

Renzhong Feng & Lifang Song. (2020). Rational Quasi-Interpolation Approximation of Scattered Data in $\mathbb{R}^3$. Numerical Mathematics: Theory, Methods and Applications. 11 (1). 169-186. doi:10.4208/nmtma.OA-2017-0019
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