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.