Analysis of Discrete L2 Projection on Polynomial Spaces with Random Evaluations

We analyse the problem of approximating a multivariate function by discrete least-squares projection on a polynomial space starting from random, noise-free observations. An area of possible application of such technique is Uncertainty Quantification (UQ) for computational models. We prove an optimal convergence estimate, up to a logarithmic factor, in the monovariate case, when the observation points are sampled in a bounded domain from a probability density function bounded away from zero, provided the number of samples scales quadratically with the dimension of the polynomial space. Several numerical tests are presented both in the monovariate and multivariate case, confirming our theoretical estimates. The numerical tests also clarify how the convergence rate depends on the number of sampling points, on the polynomial degree, and on the smoothness of the target function