We propose and analyse numerical algorithms based on weighted least squares for the approximation of a bounded real-valued function on a general bounded domain |$\varOmega \subset \mathbb{R}^d$|⁠. Given any |$n$|-dimensional approximation space |$V_n \subset L^2(\varOmega )$|⁠, the analysis in Cohen and Migliorati (2017, Optimal weighted least-squares methods. SMAI J. Comput. Math., 3, 181–203) shows the existence of stable and optimally converging weighted least-squares estimators, using a number of function evaluations |$m$| of the order |$n \ln n$|⁠. When an |$L^2(\varOmega )$|-orthonormal basis of |$V_n$| is available in analytic form, such estimators can be constructed using the algorithms described in Cohen and Migliorati (2017, Optimal weighted least-squares methods. SMAI J. Comput. Math., 3, 181–203, 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 |$\varOmega $| is an irregular domain such that the analytic form of an |$L^2(\varOmega )$|-orthonormal basis is not available, stable and quasi-optimally weighted least-squares estimators can still be constructed from |$V_n$|⁠, again with |$m$| of the order |$n \ln n$|⁠, but using a suitable surrogate basis of |$V_n$| orthonormal in a discrete sense. The computational cost for the calculation of the surrogate basis depends on the Christoffel function of |$\varOmega $| and |$V_n$|⁠. Numerical results validating our analysis are presented.

中文翻译：

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通过加权最小二乘方法对不规则域上的函数进行多元逼近

我们提出并分析基于加权最小二乘的数值算法，以逼近一般有界域| $ \ varOmega \ subset \ mathbb {R} ^ d $ |⁠上的有界实值函数。给定任何| $ n $ | 维近似空间| $ V_n \子集L ^ 2（\ varOmega）$ |⁠，在Cohen和Migliorati分析（2017年，最优加权的最小二乘法。SMAI J. COMPUT数学，3，181-203）使用大量函数评估| $ m $ |，表明存在稳定且最优收敛的加权最小二乘估计量 订单的$ N \ LN N $ | |⁠。当| $ L ^ 2（\ varOmega）$ | | $ V_n $ |的正交基在解析形式可用，例如估计器可以使用Cohen和Migliorati描述的算法来构建（2017年，最优加权的最小二乘法。SMAI J. COMPUT。数学式，3，181-203，第5章）。如果基础也具有乘积形式，则这些算法的计算复杂度在| $ d $ |中呈线性 和| $ m $ |⁠。在本文中，我们表明当| $ \ varOmega $ | 是一个不规则域，使得| $ L ^ 2（\ varOmega）$ |的解析形式 -正交的基础不可用，仍然可以通过| $ V_n $ |⁠，再次使用| $ m $ |构造稳定和准最佳加权的最小二乘估计量。订单| $ n \ ln n $ |⁠，但使用| $ V_n $ |的适当替代基准 离散意义上的正交。计算代理基础的计算成本取决于| $ \ varOmega $ |的Christoffel函数。和| $ V_n $ |⁠。数值结果验证了我们的分析。