Error bounds for quasi-Monte Carlo integration with nets

Abstract:

We analyze the error introduced by approximately calculating the $s$-dimensional Lebesgue measure of a Jordan-measurable subset of $I^s=[0,1)^s$. We give an upper bound for the error of a method using a $(t,m,s)$-net, which is a set with a very regular distribution behavior. When the subset of $I^s$ is defined by some function of bounded variation on ${\bar I}^{s-1}$, the error is estimated by means of the variation of the function and the discrepancy of the point set which is used. A sharper error bound is established when a $(t,m,s)$-net is used. Finally a lower bound of the error is given, for a method using a $(0,m,s)$-net. The special case of the 2-dimensional Hammersley point set is discussed.