Stability of kernel-based interpolation
Stability of kernel-based interpolation
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DOI：
10.1007/s10444-008-9093-4
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发表时间：
2010-02
影响因子：
1.7
通讯作者：
S. Marchi;R. Schaback
中科院分区：
文献类型：
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作者：
S. Marchi;R. Schaback
由于核矩阵的条件过大，基于径向基函数或非径向核函数平移的插值算法在数值上是不稳定的。但如果在不考虑特殊基的情况下，在函数空间中进行稳定性评估，本文证明了基于核的插值是稳定的。只要数据不是太分散，插值的L2或L ∞范数可以由数据的离散的L2和L2 ∞范数上界.此外，拉格朗日基函数是一致有界的，勒贝格常数的增长最多像数据点的数量的平方根。然而，这种分析仅适用于有限光滑度的内核。数值例子支持我们的界限，但也表明，无限光滑的内核的情况下，必须导致更坏的界限在未来的工作中，而所观察到的勒贝格常数的内核有限的光滑度，甚至似乎是独立的样本大小和填充距离。
It is often observed that interpolation based on translates of radial basis functions or non-radial kernels is numerically unstable due to exceedingly large condition of the kernel matrix. But if stability is assessed in function space without considering special bases, this paper proves that kernel-based interpolation is stable. Provided that the data are not too wildly scattered, theL2orL∞norms of interpolants can be bounded above by discrete ℓ2and ℓ∞norms of the data. Furthermore, Lagrange basis functions are uniformly bounded and Lebesgue constants grow at most like the square root of the number of data points. However, this analysis applies only to kernels of limited smoothness. Numerical examples support our bounds, but also show that the case of infinitely smooth kernels must lead to worse bounds in future work, while the observed Lebesgue constants for kernels with limited smoothness even seem to be independent of the sample size and the fill distance.