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We analyze the accuracy of the discrete least-squares approximation of a function  in multivariate polynomial spaces  with  over the domain , based on the sampling of this function at points . The samples are independently drawn according to a given probability density  belonging to the class of multivariate beta densities, which includes the uniform and Chebyshev densities as particular cases. Motivated by recent results on high-dimensional parametric and stochastic PDEs, we restrict our attention to polynomial spaces associated with  sets  of  cardinality , and we optimize the choice of the space for the given sample. This implies, in particular, that the selected polynomial space depends on the sample. We are interested in comparing the error of this least-squares approximation measured in  with the best achievable polynomial approximation error when using downward closed sets of cardinality . We establish conditions between the dimension  and the size  of the sample, under which these two errors are proved to be comparable. Our main finding is that the dimension  enters only moderately in the resulting trade-off between  and , in terms of a logarithmic factor , and is even absent when the optimization is restricted to a relevant subclass of downward closed sets, named  sets. In principle, this allows one to use these methods in arbitrarily high or even infinite dimension. Our analysis builds upon (Chkifa et al. in ESAIM Math Model Numer Anal 49(3):815–837, ), which considered fixed and nonoptimized downward closed multi-index sets. Potential applications of the proposed results are found in the development and analysis of efficient numerical methods for computing the solution to high-dimensional parametric or stochastic PDEs, but are not limited to this area.

Discrete Least-Squares Approximations over Optimized Downward Closed Polynomial Spaces in Arbitrary Dimension