发表机构
TU Bergakademie Freiberg（弗莱贝格工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于移动最小二乘的调和逼近方法（HAMLS），将散乱数据过渡到求积网格，避免全局病态求解，在分布不良数据下实现快速稳定的全局调和逼近。
AI 中文摘要
我们考虑在环面 $\mathbb{T}^d$、球面 $\mathcal{S}^d$ 和旋转群 $\mathrm{SO}(3)$ 上的调和逼近。虽然已有几种全局方法可用于从散乱数据计算调和展开，但它们的稳定性和尤其是运行时间强烈依赖于节点的几何形状，特别是依赖于足够小的填充距离 $h$。即使数据中存在一个大的空洞，也可能导致不稳定和长时间运行。我们提出了通过移动最小二乘（HAMLS）的调和逼近，这是一种全局调和逼近方案，避免了病态的全局求解，并且在超定和欠定设置下都能良好工作。主要思想是从散乱数据到求积网格的中间过渡，通过移动最小二乘（MLS）实现。这将一个大的全局问题替换为许多可以单独正则化的小型局部问题。然后通过使用快速傅里叶变换的求积获得全局调和逼近。这使得全局阶段快速、稳定且非迭代。对于在分布良好且填充距离 $h$ 较小的节点上采样的足够光滑函数，我们将所得的 $\mathrm{L}^2$ 误差界定为从求积网格上的精确值获得的调和逼近误差加上 $h^{K+1}$ 阶的项，其中 $K$ 是 MLS 步骤中使用的多项式次数。在 $\mathcal{S}^2$ 上使用分布不良节点的数值实验表明，HAMLS 实现了与全局最小二乘逼近相当的误差，同时速度显著更快，尤其是对于较大的带宽。
英文摘要
We consider harmonic approximation on the torus $\mathbb{T}^d$, the sphere $\mathcal{S}^d$, and the rotation group $\mathrm{SO}(3)$. While several global approaches are available for computing a harmonic expansion from scattered data, their stability and, especially, their runtime depend strongly on the geometry of the nodes, in particular on a sufficiently small fill distance. Even a single large hole in the data may cause instability and long runtimes. We propose harmonic approximation via moving least squares (HAMLS), a global harmonic approximation scheme that avoids the ill-conditioned global solve and works well in both the overdetermined and the underdetermined settings. The main idea is an intermediate transition from the scattered data to a quadrature grid, which is realized via moving least squares (MLS). This replaces one large global problem by many small local ones that can be regularized individually. The global harmonic approximation is then obtained via quadrature using a fast Fourier transform. This makes the global stage fast, stable and non-iterative. For sufficiently smooth functions sampled at well-distributed nodes with small fill distance $h$, we bound the resulting $\mathrm{L}^2$-error by the error of the harmonic approximation obtained from exact values on the quadrature grid, plus a term of order $h^{K+1}$, where $K$ is the polynomial degree employed in the MLS step. Numerical experiments on $\mathcal{S}^2$ with ill-distributed nodes show that HAMLS achieves errors comparable to those of global least-squares approximation, while being significantly faster, especially for larger bandwidths.
Comments20 pages, 5 figures, 1 table; code at https://github.com/mtex-toolbox/mtex-paper/tree/master/HarmonicApproximationViaMovingLeastSquares
