We propose and analyse numerical algorithms based on weighted least squares for the approximation of a real-valued function on a general bounded domain $\Omega \subset \mathbb{R}^d$. Given any $n$-dimensional approximation space $V_n \subset L^2(\Omega)$, the analysis in [6] shows the existence of stable and optimally converging weighted least-squares estimators, using a number of function evaluations $m$ of the order $n \log n$. When an $L^2(\Omega)$-orthonormal basis of $V_n$ is available in analytic form, such estimators can be constructed using the algorithms described in [6,Section 5]. If the basis also has product form, then these algorithms have computational complexity linear in $d$ and $m$. In this paper we show that, when $\Omega$ is an irregular domain such that the analytic form of an $L^2(\Omega)$-orthonormal basis is not available, stable and quasi-optimally weighted least-squares estimators can still be constructed from $V_n$, again with $m$ of the order $n \log n$, but using a suitable surrogate basis of $V_n$ orthonormal in a discrete sense. The computational cost for the calculation of the surrogate basis depends on the Christoffel function of $\Omega$ and $V_n$. Numerical results validating our analysis are presented.

中文翻译：

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通过加权最小二乘法对不规则域上的函数进行多元逼近

我们提出并分析了基于加权最小二乘法的数值算法，用于在一般有界域 $\Omega \subset \mathbb{R}^d$ 上逼近实值函数。给定任何 $n$ 维近似空间 $V_n \subset L^2(\Omega)$，[6] 中的分析表明存在稳定且最优收敛的加权最小二乘估计量，使用多个函数评估 $m订单 $n \log n$ 的 $。当 $V_n$ 的 $L^2(\Omega)$-标准正交基以解析形式可用时，可以使用 [6，第 5 节] 中描述的算法构建此类估计量。如果基也有乘积形式，那么这些算法的计算复杂度为 $d$ 和 $m$。在本文中，我们表明，当 $\Omega$ 是一个不规则域，使得 $L^2(\Omega)$-正交基的解析形式不可用时，稳定的和准最优加权最小二乘估计量仍然可以从 $V_n$ 构建，同样使用 $m$ 的阶数为 $n \log n$，但使用离散意义上的 $V_n$ 标准正交的合适替代基础。计算代理基的计算成本取决于 $\Omega$ 和 $V_n$ 的 Christoffel 函数。提供了验证我们分析的数值结果。