Strong tractability of multivariate integration using quasi-Monte Carlo algorithms

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 depends on ε−1 and the dimension s. Strong tractability means that it does not depend on s and is bounded by a polynomial in ε−1. The least possible value of the power of ε−1 is called the ε-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-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-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)-sequences and Sobol sequences achieve the optimal convergence order O(N−1+δ) for any δ > 0 independent of the dimension with a worst case deterministic guarantee (where N is the number of function evaluations). This implies that strong tractability with the best ε-exponent can be achieved in appropriate weighted Sobolev spaces by using Niederreiter’s (t, s)-sequences and Sobol sequences.