We study quasi–Monte Carlo algorithms based on low discrepancy sequences for multivariate integration. We consider the problem of how the minimal number of function evaluations needed to reduce the worst-case error from its initial error by a factor of ε \varepsilon depends on ε − 1 {\varepsilon }^{-1} and the dimension s s . Strong tractability means that it does not depend on s s and is bounded by a polynomial in ε − 1 {\varepsilon }^{-1} . The least possible value of the power of ε − 1 {\varepsilon }^{-1} is called the ε \varepsilon -exponent of strong tractability. Sloan and Woźniakowski established a necessary and sufficient condition of strong tractability in weighted Sobolev spaces, and showed that the ε \varepsilon -exponent of strong tractability is between 1 and 2. However, their proof is not constructive . In this paper we prove in a constructive way that multivariate integration in some weighted Sobolev spaces is strongly tractable with ε \varepsilon -exponent equal to 1, which is the best possible value under a stronger assumption than Sloan and Woźniakowski’s assumption. We show that quasi–Monte Carlo algorithms using Niederreiter’s ( t , s ) (t,s) -sequences and Sobol sequences achieve the optimal convergence order O ( N − 1 + δ ) O(N^{-1+\delta })

On the distribution of points in a cube and the approximate evaluation of integrals