In this paper we derive local $L_2$ error estimates for penalized least-squares approximation on the $d$-dimensional unit sphere ${\mathbb{S}}^d\subseteq {\mathbb{R}}^{d+1}$, given noisy, scattered, local data representing an underlying function from a Sobolev space of order $s>d∕2$ defined on a non-empty connected open set $\Omega \subseteq {\mathbb{S}}^d$ with Lipschitz-continuous boundary. The quadratic regularization functional has two terms, one measuring the squared pointwise ${\ell }_2$-discrepancy from the local data, the other containing the squared native space norm of a radial basis function (RBF), multiplied by a regularization parameter. The RBF is chosen so that its native space is equivalent to the (global) Sobolev space of order $s$ on ${\mathbb{S}}^d$. While both the data and the approximated function are local, we minimize the quadratic functional over all functions in the native space of the RBF, and obtain as exact minimizer a (global) radial basis function approximation. By choosing the RBF to be a Wendland function the resulting linear system has a sparse matrix which is easily computed. We consider three different strategies for choosing the smoothing parameter, namely Morozov’s discrepancy principle and two a priori strategies, and derive $L_2\left(\Omega \right)$ error estimates for each strategy. As auxiliary tools for proving the local $L_2$ error estimates we develop both a local $L_2$ sampling inequality and a suitable Sobolev extension theorem. The paper concludes with numerical experiments.

在本文中，我们推导了局部 ${\mathrm{大号}}_2$ 罚最小二乘近似的误差估计 $d$维单位球 ${\mathbb{小号}}^d\subseteq {\mathbb{[R}}^{d+\mathrm{1个}}$，给出嘈杂，分散的局部数据，这些数据表示Sobolev阶空间中的基础函数 $s>d∕2$ 在非空连接的开放集上定义 $\Omega \subseteq {\mathbb{小号}}^d$与Lipschitz连续边界。二次正则化函数具有两项，一项测量点平方的平方${\ell }_2$-与本地数据的差异，另一个包含径向基函数（RBF）的平方本机空间范数乘以正则化参数。选择RBF的目的是使其本机空间等于有序的（全局）Sobolev空间$s$ 上 ${\mathbb{小号}}^d$。虽然数据和近似函数都是局部函数，但我们将RBF原始空间中所有函数的二次函数最小化，并获得（全局）径向基函数近似作为精确的最小化子。通过将RBF选择为Wendland函数，所得线性系统具有易于计算的稀疏矩阵。我们考虑选择平滑参数的三种不同策略，即莫罗佐夫的差异原理和两种先验策略，然后得出${\mathrm{大号}}_2（\Omega ）$每个策略的错误估计。作为证明当地情况的辅助工具${\mathrm{大号}}_2$ 误差估计我们都开发了本地 ${\mathrm{大号}}_2$采样不等式和合适的Sobolev扩展定理。本文以数值实验作为结束。