A higher order lattice-based method for high-dimensional numerical integration using periodization

Abstract: We present a novel numerical integration method for non-periodic functions over the high-dimensional unit cube $[0,1]<sup>s$ by a specially crafted (unequally-)weighted quadrature rule using transformed lattice points, with the additional option of subsampling. The method is designed for integrands whose mixed derivatives up to order $α$ are square-integrable with respect to a product Chebyshev density. The ingredients are (i) periodizing the integrand, (ii) approximating the smooth periodic function by a kernel interpolant at lattice points, (iii) integrating exactly the product of the kernel interpolant and the nonsmooth density, together with subsampling alternatives for (ii) and~(iii). With $|J|$ corresponding to the number of function evaluations, we achieve the convergence rate close to the order $|J|<sup>{-(α-1/2)}$ and $|J|<sup>{-α}\,($1.110721$)<sup>s$, where the implied constants are independent of $s$ under favorable conditions. We can interpolate these two results to trade between the convergence rate and the growth in $s$. We provide numerical experiments to demonstrate our theory.