论文标题

径向基函数的缩放

Scaling of Radial Basis Functions

论文作者

Larsson, Elisabeth, Schaback, Robert

论文摘要

本文研究了缩放对径向基函数插值行为的影响。它专注于某些中心方面,但并不试图详尽。最重要的问题是:基于内核的interpolant的误差如何随所选择的内核的规模而变化?标准误差如何变化?而且,由于固定功能可能在允许尺度(如全局sobolev空格)的空间中,是否存在与函数最匹配的空间尺度?最后一个问题是在Sobolev空间的肯定中回答的,但是所需的规模可能很难估计。事实证明,函数的可伸缩性受到分析内核生成的空格的限制,除非功能是限制的。与其他论文相反,通过独立的计算,在实验检查量表时,多项式和多种元素作为平坦限制。数值结果表明,如果用户从一开始就包括固定限制案例,那么对近牌量表的搜索是值得怀疑的。如果没有足够的数据直接评估错误,则可以改变标准误差绑定的尺度,以替换interpolant范围的未知函数的范围。这遵循质量上实际错误的行为,但对于估计误差最佳尺度的值只有有限的值。对于具有无限平滑度的内核和功能,事实证明,给定的插值数据不足以确定有用的量表。

This paper studies the influence of scaling on the behavior of Radial Basis Function interpolation. It focuses on certain central aspects, but does not try to be exhaustive. The most important questions are: How does the error of a kernel-based interpolant vary with the scale of the kernel chosen? How does the standard error bound vary? And since fixed functions may be in spaces that allow scalings, like global Sobolev spaces, is there a scale of the space that matches the function best? The last question is answered in the affirmative for Sobolev spaces, but the required scale may be hard to estimate. Scalability of functions turns out to be restricted for spaces generated by analytic kernels, unless the functions are band-limited. In contrast to other papers, polynomials and polyharmonics are included as flat limits when checking scales experimentally, with an independent computation. The numerical results show that the hunt for near-flat scales is questionable, if users include the flat limit cases right from the start. When there are not enough data to evaluate errors directly, the scale of the standard error bound can be varied, up to replacing the norm of the unknown function by the norm of the interpolant. This follows the behavior of the actual error qualitatively well, but is only of limited value for estimating error-optimal scales. For kernels and functions with unlimited smoothness, the given interpolation data are proven to be insufficient for determining useful scales.

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