A Monte Carlo method for integration of multivariate smooth functions
This work provides a theoretical breakthrough for Monte Carlo integration, showing that optimal convergence rates can be dimension-independent for smooth functions, which is a significant improvement over deterministic methods.
The paper presents a Monte Carlo integration method using randomly shifted and dilated lattice points, achieving optimal convergence rates for functions in Sobolev spaces with isotropic, anisotropic, or mixed smoothness, independent of dimension. The mean squared error is bounded by n^{-1/2} times the L2-norm of the Fourier transform outside a region, yielding an order of n^{-s-1/2} for p≥2.
We study a Monte Carlo algorithm that is based on a specific (randomly shifted and dilated) lattice point set. The main result of this paper is that the mean squared error for a given compactly supported, square-integrable function is bounded by $n^{-1/2}$ times the $L_2$-norm of the Fourier transform outside a region around the origin, where $n$ is the expected number of function evaluations. As corollaries we obtain the optimal order of convergence for functions from the Sobolev spaces $H^s_p$ with isotropic, anisotropic, or mixed smoothness with given compact support for all values of the parameters. If the region of integration is the unit cube, we obtain the same optimal orders for functions without boundary conditions. This proves, in particular, that the optimal order of convergence in the latter case is $n^{-s-1/2}$ for $p\ge2$, which is, in contrast to the case of deterministic algorithms, independent of the dimension. This shows that Monte Carlo algorithms can improve the order by more than $n^{-1/2}$ for a whole class of natural function spaces.