Monte Carlo integration with adaptive variance reduction: an asymptotic analysis

The crude Monte Carlo approximates the integral $$S(f)=\int_a^b f(x)\,\mathrm dx$$ with expected error (deviation) $\sigma(f)N^{-1/2},$ where $\sigma(f)^2$ is the variance of $f$ and $N$ is the number of random samples. If $f\in C^r$ then special variance reduction techniques can lower this error to the level $N^{-(r+1/2)}.$ In this paper, we consider methods of the form $$\overline M_{N,r}(f)=S(L_{m,r}f)+M_n(f-L_{m,r}f),$$ where $L_{m,r}$ is the piecewise polynomial interpolation of $f$ of degree $r-1$ using a partition of the interval $[a,b]$ into $m$ subintervals, $M_n$ is a Monte Carlo approximation using $n$ samples of $f,$ and $N$ is the total number of function evaluations used. We derive asymptotic error formulas for the methods $\overline M_{N,r}$ that use nonadaptive as well as adaptive partitions. Although the convergence rate $N^{-(r+1/2)}$ cannot be beaten, the asymptotic constants make a huge difference. For example, for $\int_0^1(x+d)^{-1}\mathrm dx$ and $r=4$ the best adaptive methods overcome the nonadaptive ones roughly $10^{12}$ times if $d=10^{-4},$ and $10^{29}$ times if $d=10^{-8}.$ In addition, the proposed adaptive methods are easily implementable and can be well used for automatic integration. We believe that the obtained results can be generalized to multivariate integration.