Discrete least-squares approximations over optimized downward closed polynomial spaces in arbitrary dimension

We analyze the accuracy of the discrete least-squares approximation of a function $u$ in multivariate polynomial spaces $\mathbb{P}_\Lambda:={\rm span} \{y\mapsto y^\nu \,: \, \nu\in \Lambda\}$ with $\Lambda\subset \mathbb{N}_0^d$ over the domain $\Gamma:=[-1,1]^d$, based on the sampling of this function at points $y^1,\dots,y^m \in \Gamma$. The samples are independently drawn according to a given probability density $\rho$ belonging to the class of multivariate beta densities, which includes the uniform and Chebyshev densities as particular cases. We restrict our attention to polynomial spaces associated with \emph{downward closed} sets $\Lambda$ of \emph{prescribed} cardinality $n$, 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 $L^2(\Gamma,d\rho)$ with the best achievable polynomial approximation error when using downward closed sets of cardinality $n$. We establish conditions between the dimension $n$ and the size $m$ of the sample, under which these two errors are proven to be comparable. Our main finding is that the dimension $d$ enters only moderately in the resulting trade-off between $m$ and $n$, in terms of a logarithmic factor $\ln(d)$, and is even absent when the optimization is restricted to a relevant subclass of downward closed sets, named {\it anchored} sets. In principle, this allows one to use these methods in arbitrarily high or even infinite dimension. Our analysis builds upon [2] which considered fixed and nonoptimized downward closed multi-index sets. Potential applications of the proposed results are found in the development and analysis of numerical methods for computing the solution to high-dimensional parametric or stochastic PDEs.