Analysis of Discrete \(L^2\) Projection on Polynomial Spaces with Random Evaluations
We analyze 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 for computational models. We prove an optimal convergence estimate, up to a logarithmic factor, in the univariate case, when the observation points are sampled in a bounded domain from a probability density function bounded away from zero and bounded from above, provided the number of samples scales quadratically with the dimension of the polynomial space. Optimality is meant in the sense that the weighted \(L^2\) norm of the error committed by the random discrete projection is bounded with high probability from above by the best \(L^\infty \) error achievable in the given polynomial space, up to logarithmic factors. Several numerical tests are presented in both the univariate and multivariate cases, 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.
approximation theory error analysis multivariate polynomial approximation nonparametric regression noise-free data generalized polynomial chaos point collocation 41a10 41a25 65n12 65n15 65n35