Semi-regular Interpolatory RBF-based Subdivision Schemes

We present a MATLAB function to compute the subdivision matrices of semi-regular univariate interpolatory RBF-based binary subdivision schemes. The construction is the adaptation of the one presented in "Stationary binary subdivision schemes using radial basis function interpolation", B.-G. Lee, Y. J. Lee, J. Yoon (Adv. Comput. Math, 2006) and "Analysis of stationary subdivision schemes for curve design based on radial basis function interpolation", Y. J. Lee, J. Yoon (Appl. Math. Comput., 2010), to the semi-regular case, i.e. when the starting mesh is formed by two different uniform mesh that meet eachother at 0.
The main function, RBFs_semi.m, given the stepsizes of the two uniform mesh, the family of radial basis function, the number of points used for the local computation, the required polynomial reproduction and, eventually, further parameters, determines the subdivision matrix of the scheme in the form of the regular mask on the left, the regular mask on the right and the irregular part of the matrix around 0. The supported families of RBFs are (inverse) multi-quadric, Gaussian, Wendland's functions, Wu's functions, Buhmann's functions, polyharmonic functions and Euclid's hat functions (see e.g. "Meshfree approximation methods with MATLAB", G. E. Fasshauer). For further information about how to choose the parameters for each family see the files in the Aux folder.

本项研究提出了一种基于 MATLAB 的函数，用于计算半规则一元插值径向基函数 (RBF) 基二叉分形方案的细分矩阵。该构建方法是对 B.-G. 李、Y. J. 李、J. Yoon 在《基于径向基函数插值的定常二叉分形方案》（Adv. Comput. Math，2006）以及 Y. J. 李、J. Yoon 在《基于径向基函数插值的曲线设计定常细分方案分析》（Appl. Math. Comput.，2010）中提出的方案在半规则情形下的适应，即当起始网格由两个不同均匀网格在 0 点相交形成时的情况。主要函数 RBFs_semi.m，在给定两个均匀网格的步长、径向基函数族、局部计算使用的点数、所需的插值多项式以及可能的额外参数的情况下，确定方案的细分矩阵，该矩阵以左边的规则掩码、右边的规则掩码以及围绕 0 的不规则矩阵部分的形式呈现。支持的径向基函数族包括（逆）多二次、高斯、Wendland 函数、Wu 函数、Buhmann 函数、多调和函数以及欧几里得帽函数（参见例如 G. E. Fasshauer 的《带有 MATLAB 的无网格近似方法》）。关于如何为每个函数族选择参数的更多信息，请参阅辅助文件夹中的文件。