Reducing variance using iterated control variates

Abstract

In this paper we describe a new variance reduction method for Monte Carlo integration based on an iterated computation of L^{2} approximations using control variates. This computation leads to non linear unbiased estimates for each of the coefficients of the truncated L^{2} expansion. We give estimations of the variance of these estimates without further hypotheses on the approximation basis. We study especially the convergence of our algorithm in the case of a polynomial decay of these coefficients. As an application, regular monodimensional functions will be approximated using a Fourier basis on periodised functions, Legendre and Tchebychef polynomial L^2 approximations. The order of our method will appear to be almost optimal in this case. Numerical examples will be given as a comparison with standard Monte Carlo estimates.