Multiscale radial kernels with high-order generalized Strang-Fix conditions
Multiscale radial kernels with high-order generalized Strang-Fix conditions
复制标题
具有高阶广义 Strang-Fix 条件的多尺度径向核
DOI：
10.1007/s11075-019-00820-9
复制
发表时间：
2017
影响因子：
2.1
通讯作者：
Zhou Xuan
中科院分区：
文献类型：
--
作者：
Gao Wenwu;Zhou Xuan
本文提供了一种通用且简单的方法，用于从给定的单变量生成器显式构造具有高阶广义 Strang-Fix 条件的多尺度径向核。生成的内核是通过对生成器的关于尺度变量的scaledf形式采用线性函数来构造的。还导出了核的等价除差形式；在此基础上，提出了一种快速稳定计算多尺度径向核的类金字塔算法。此外，还给出了核在空间域和频域上的表征，表明广义Strang-Fix条件、矩条件和卷积意义上的多项式再现条件是彼此等价的。因此，作为副产品，本文提供了这三个经典概念的统一观点。这些内核可用于构造具有高近似精度的准插值以及构造具有高近似阶数的卷积算子等等。作为一个例子，我们为不规则间隔的数据构建了一个准插值方案，并得出了其误差估计和多尺度径向核尺度参数的选择。本文末尾给出了使用我们的准插值法逼近二元 Franke 函数的数值结果。理论和数值结果都表明，满足高阶广义 Strang-Fix 条件的多尺度径向核的准插值通常提供高近似阶数。
The paper provides a general and simple approach for explicitly constructing multiscale radial kernels with high-order generalized Strang-Fix conditions from a given univariate generator. The resulting kernels are constructed by taking a linear functional to the scaledf-form of the generator with respect to the scale variable. Equivalent divided difference forms of the kernels are also derived; based on which, a pyramid-like algorithm for fast and stable computation of multiscale radial kernels is proposed. In addition, characterizations of the kernels in both the spatial and frequency domains are given, which show that the generalized Strang-Fix condition, the moment condition, and the condition of polynomial reproduction in the convolution sense are equivalent to each other. Hence, as a byproduct, the paper provides a unified view of these three classical concepts. These kernels can be used to construct quasi-interpolation with high approximation accuracy and construct convolution operators with high approximation orders, to name a few. As an example, we construct a quasi-interpolation scheme for irregularly spaced data and derived its error estimates and choices of scale parameters of multiscale radial kernels. Numerical results of approximating a bivariate Franke function using our quasi-interpolation are presented at the end of the paper. Both theoretical and numerical results show that quasi-interpolation with multiscale radial kernels satisfying high-order generalized Strang-Fix conditions usually provides high approximation orders.