Control variate Monte Carlo estimators based on sparse polynomial chaos expansions

We introduce two control variate Monte Carlo estimators where the control is based on the truncated sparse polynomial chaos expansion of the function in hand. We use the control variate estimators to estimate the lower and upper Sobol' indices in some applications, and compare them numerically with some of the best Monte Carlo estimators in the literature. The results suggest that in computationally expensive problems where a low-order polynomial chaos expansion is not an accurate approximation of the model but highly correlated with it, the control variate estimators are either the best or among the best in terms of efficiency.