SIAM Review, Volume 62, Issue 2, Page 483-508, January 2020.
Polynomial approximations constructed using a least-squares approach form a ubiquitous technique in numerical computation. One of the simplest ways to generate data for least-squares problems is with random sampling of a function. We discuss theory and algorithms for stability of the least-squares problem using random samples. The main lesson from our discussion is that the intuitively straightforward (``standard") density for sampling frequently yields suboptimal approximations, whereas sampling from a non-standard density, called the induced distribution, yields near-optimal approximations. We present a recent theory that demonstrates why sampling from the induced distribution is optimal and provide several numerical experiments that support the theory. Software is also provided that reproduces the figures in this paper.

中文翻译：

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构造最小二乘多项式逼近

SIAM评论，第62卷，第2期，第483-508页，2020年1月。
使用最小二乘法构造的多项式逼近形成了数值计算中的普遍技术。为最小二乘问题生成数据的最简单方法之一是对函数进行随机采样。我们讨论使用随机样本的最小二乘问题稳定性的理论和算法。从我们的讨论中得出的主要教训是，用于采样的直观直观（标准）密度经常会产生次优的近似值，而从称为诱导分布的非标准密度进行采样会产生近优的近似值。这证明了为什么从感应分布中采样是最佳的，并提供了支持该理论的几个数值实验，还提供了可复制本文中的数据的软件。