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

Quasi-Monte Carlo (QMC) methods for high dimensional integrals over unit cubes and products of spheres are well-studied in literature. We study QMC tractability of integrals of functions defined over the product of $m$ copies of the simplex $T^d\subset {\mathbb{R}}^d$. The domain is a tensor product of $m$ reproducing kernel Hilbert spaces defined by `weights’ ${\gamma }_{m,j}$, for $j=1,2,\dots ,m$. Similar to the results on the unit cube in $m$ dimensions, and the product of $m$ copies of the $d$-dimensional sphere, we prove that strong polynomial tractability holds iff ${\mathrm{lim\hspace{0.17em}sup}}_{m\to \mathrm{\infty }}{\sum }_{j=1}^m{\gamma }_{m,j}<\mathrm{\infty }$ and polynomial tractability holds iff ${\mathrm{lim\hspace{0.17em}sup}}_{m\to \mathrm{\infty }}\frac{{\sum }_{j=1}^m{\gamma }_{m,j}}{\log (m+1)}<\mathrm{\infty }$. We also show that weak tractability holds iff ${\lim }_{m\to \mathrm{\infty }}\frac{{\sum }_{j=1}^m{\gamma }_{m,j}}{m}=0$. The proofs employ Sobolev space techniques and weighted reproducing kernel Hilbert space techniques for the simplex and products of simplices as domain. Properties of orthogonal polynomials on a simplex are also used extensively.