Vandermonde matrices are usually exponentially ill-conditioned and often result in unstable approximations. In this paper, we introduce and analyze the \textit{multivariate Vandermonde with Arnoldi (V+A) method}, which is based on least-squares approximation together with a Stieltjes orthogonalization process, for approximating continuous, multivariate functions on $d$-dimensional irregular domains. The V+A method addresses the ill-conditioning of the Vandermonde approximation by creating a set of discrete orthogonal basis with respect to a discrete measure. The V+A method is simple and general. It relies only on the sample points from the domain and requires no prior knowledge of the domain. In this paper, we first analyze the sample complexity of the V+A approximation. In particular, we show that, for a large class of domains, the V+A method gives a well-conditioned and near-optimal $N$-dimensional least-squares approximation using $M=\mathcal{O}(N^2)$ equispaced sample points or $M=\mathcal{O}(N^2\log N)$ random sample points, independently of $d$. We also give a comprehensive analysis of the error estimates and rate of convergence of the V+A approximation. Based on the multivariate V+A approximation, we propose a new variant of the weighted V+A least-squares algorithm that uses only $M=\mathcal{O}(N\log N)$ sample points to give a near-optimal approximation. Our numerical results confirm that the (weighted) V+A method gives a more accurate approximation than the standard orthogonalization method for high-degree approximation using the Vandermonde matrix.

翻译：Vandermonde矩阵通常呈指数级病态，常导致不稳定的逼近结果。本文引入并分析了一种基于最小二乘逼近与Stieltjes正交化过程的"多元Vandermonde与Arnoldi方法"（V+A方法），用于在$d$维不规则区域上逼近连续多元函数。V+A方法通过构建关于离散测度的离散正交基，解决了Vandermonde逼近的病态问题。该方法简单且通用，仅依赖于区域内的采样点，无需域的先验知识。首先，本文分析了V+A逼近的样本复杂度：对于一大类区域，V+A方法能利用$M=\mathcal{O}(N^2)$个等距采样点或$M=\mathcal{O}(N^2\log N)$个随机采样点，独立于$d$，实现良态且近最优的$N$维最小二乘逼近。此外，我们还给出了V+A逼近的误差估计与收敛速率的全面分析。基于多元V+A逼近，我们提出了一种加权V+A最小二乘算法的新变体，仅需$M=\mathcal{O}(N\log N)$个采样点即可实现近最优逼近。数值实验证实，对于基于Vandermonde矩阵的高阶逼近，（加权）V+A方法比标准正交化方法具有更高的逼近精度。