Weighted Monte Carlo Integration

Abstract

This work considers Monte Carlo methods for approximating the integral of any twice-differentiable function f over a hypercube. Whereas earlier Monte Carlo schemes have yilded on $O({1 / n})$ convergence rate for the expected square error, we show that by allowing nonlinear operations on the random samples $\{ (U_i ,f(U_i ))\} _{i = 1}^n $, much more rapid convergence can be achieved. Specifically, we give a rule which attains rates of $O({1 / {n^4 }})$ and $O({1 \mathord{\left/ {\vphantom {1 {n^4 }}} \right. \kern-\nulldelimiterspace} {n^2 }})$ in one and two dimensions respectively. Analysis shows that our algorithms become worse than the usual Monte Carlo method as the dimension of the domain increases, and these findings point to the possibility that “crude” Monte Carlo has an asymptotic optimality property among all Monte Carlo rules, linear and otherwise.