Multivariate approximation of functions on irregular domains by weighted least-squares methods

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.