SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2698-2718, December 2024.
Abstract. The computation of global radial basis function (RBF) approximations requires the solution of a linear system which, depending on the choice of RBF parameters, may be ill-conditioned. We study the stability and accuracy of approximation methods using the Gaussian RBF in all scaling regimes of the associated shape parameter. The approximation is based on discrete least squares with function samples on a bounded domain, using RBF centers both inside and outside the domain. This results in a rectangular linear system. We show for one-dimensional approximations that linear scaling of the shape parameter with the degrees of freedom is optimal, resulting in constant overlap between neighboring RBF’s regardless of their number, and we propose an explicit suitable choice of the proportionality constant. We show numerically that highly accurate approximations to smooth functions can also be obtained on bounded domains in several dimensions, using a linear scaling with the degrees of freedom per dimension. We extend the least squares approach to a collocation-based method for the solution of elliptic boundary value problems and illustrate that the combination of centers outside the domain, oversampling, and optimal scaling can result in accuracy close to machine precision in spite of having to solve very ill-conditioned linear systems.

中文翻译：

---

有界域上稳定且准确的最小二乘径向基函数近似


SIAM 数值分析杂志，第 62 卷，第 6 期，第 2698-2718 页，2024 年 12 月。

抽象。全局径向基函数 （RBF） 近似的计算需要一个线性系统的解，根据 RBF 参数的选择，该方程组可能是病态的。我们研究了使用高斯 RBF 的近似方法在相关形状参数的所有缩放机制中的稳定性和准确性。近似基于离散最小二乘法，函数样本位于有界域上，在域内和域外都使用 RBF 中心。这会产生一个矩形线性系统。我们表明，对于一维近似，形状参数与自由度的线性缩放是最佳的，导致相邻 RBF 之间不断重叠，而不管它们的数量如何，我们提出了一个明确的比例常数的合适选择。我们用数值证明，使用每个维度自由度的线性缩放，也可以在多个维度的有界域上获得对平滑函数的高精度近似。我们将最小二乘法扩展为基于搭配的方法来解决椭圆边值问题，并说明域外中心、过采样和最优缩放的组合可以导致接近机器精度的精度，尽管必须求解条件非常恶劣的线性系统。