Quasi-Monte Carlo tractability of high dimensional integration over products of simplices

Quasi-Monte Carlo (QMC) methods for high dimensional integrals have been well studied for the unit cube. Sloan and Woźniakowski [18] partially solve the question of why they are significantly more efficient than Monte carlo methods. Kuo and Sloan [9] prove similar results for integration over product of spheres. We study the QMC tractability of integrals of functions defined over the product of m copies of the simplex T d ⊂ R. As with Kuo and Sloan [9] the domain is a tensor product of m reproducing kernel Hilbert spaces defined by ‘weights’ γm,j , for j = 1, 2, . . . , m. As with the product of unit cubes and product of spheres, we prove that the strong QMC tractability holds iff lim supm→∞ ∑m j=1 γm,j < ∞ and QMC tractability holds iff lim supm→∞ ∑m j=1 γm,j/ log(m+ 1) < ∞.