Optimal Monte Carlo Methods for $$L^2$$ L 2 -Approximation

Abstract

We construct Monte Carlo methods for the $$L^2$$ -approximation in Hilbert spaces of multivariate functions sampling not more than n function values of the target function. Their errors catch up with the rate of convergence and the preasymptotic behavior of the error of any algorithm sampling n pieces of arbitrary linear information, including function values.