Variance reduction method based on sensitivity derivatives, Part 2

A previous paper introduced a sampling method (SDES) based on sensitivity derivatives to construct statistical moment estimates that are more efficient than standard Monte Carlo estimates. In this paper we sharpen previous theoretical results and introduce a criterion to guarantee that the variance of SDES estimates is smaller than the variance of the Monte Carlo estimate. Previous numerical experiments demonstrated, and here we prove analytically, that the first-order SDES and Monte Carlo estimates converge at the same rate. We illustrate the efficiency of the SDES method of order n, where n is fixed, to estimate statistical moments with a Korteweg–de Vries equation with uncertain initial conditions.