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arXiv 1205.3255math.NAcs.NAmath.CA
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单位球面上核空间中的局部化基

Localized bases for kernel spaces on the unit sphere

High Point University（海普特大学）；University of Hawaii（夏威夷大学）；Texas A&M University（德克萨斯农工大学）；Boise State University（博伊西州立大学）

E. Fuselier, T. Hangelbroek, F. J. Narcowich, J. D. Ward, G. B. Wright

备注：
This article supersedes arXiv:1111.1013 "Better bases for kernel spaces," which proved existence of better bases for various kernel spaces. This article treats a smaller class of kernels, but presents an algorithm for constructing better bases and demonstrates its effectiveness with more elaborate examples. A quasi-interpolation scheme is introduced that provides optimal linear convergence rates
发表信息：
SIAM J. Numer. Anal. 51-5 (2013), pp. 2538-2562
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AI总结：

针对球面上由正定或条件正定核生成的逼近/插值空间，提出一种基于仅O((log N)^2)个核组合的易计算、空间局部化、小足迹且鲁棒的基，并证明其L_p稳定性和多项式衰减估计，实现最优L_p逼近阶。

摘要：

从正定或条件正定核空间进行逼近/插值是分析和综合散乱数据日益流行的工具，并且是许多无网格方法的核心。对于一组$N$个散乱点，此类空间的标准基使用$N$个全局支持的核；对于大$N$，计算成本过高。易计算、良好局部化且具有“小足迹”基元素（即仅使用少量核的元素）的基一直不可用。在球面$\mathbb{S}^2$上工作，专注于受限表面样条核（例如，限制在球面上的薄板样条），我们构建了易计算、空间良好局部化、小足迹、鲁棒的基，用于相关的核空间。我们的理论预测，每个局部基元素仅通过组合$\mathcal{O}((\log N)^2)$个核构建，这使得构建计算成本低廉。我们证明了新基是$L_p$稳定的，并满足相对于数据点密度平稳的多项式衰减估计，并提出了一个准插值方案，提供最优$L_p$逼近阶。尽管我们专注于$\mathbb{S}^2$，但大部分理论适用于其他流形——$\mathbb{S}^d$、旋转群等。最后，我们构建算法实现这些方案，并用于数值实验，验证了我们在$\mathbb{S}^2$上涉及超过十五万个数据点的插值问题的理论。

英文摘要：

Approximation/interpolation from spaces of positive definite or conditionally positive definite kernels is an increasingly popular tool for the analysis and synthesis of scattered data, and is central to many meshless methods. For a set of $N$ scattered sites, the standard basis for such a space utilizes $N$ \emph{globally} supported kernels; computing with it is prohibitively expensive for large $N$. Easily computable, well-localized bases, with "small-footprint" basis elements - i.e., elements using only a small number of kernels -- have been unavailable. Working on $\sphere$, with focus on the restricted surface spline kernels (e.g. the thin-plate splines restricted to the sphere), we construct easily computable, spatially well-localized, small-footprint, robust bases for the associated kernel spaces. Our theory predicts that each element of the local basis is constructed by using a combination of only $\mathcal{O}((\log N)^2)$ kernels, which makes the construction computationally cheap. We prove that the new basis is $L_p$ stable and satisfies polynomial decay estimates that are stationary with respect to the density of the data sites, and we present a quasi-interpolation scheme that provides optimal $L_p$ approximation orders. Although our focus is on $\mathbb{S}^2$, much of the theory applies to other manifolds - $\mathbb{S}^d$, the rotation group, and so on. Finally, we construct algorithms to implement these schemes and use them to conduct numerical experiments, which validate our theory for interpolation problems on $\mathbb{S}^2$ involving over one hundred fifty thousand data sites.

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