Strong tractability of multivariate integration using quasi-Monte Carlo algorithms

• Authors: Xiaoqun Wang
• Venue: Mathematics of Computation
• Year: 2003

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 Wozniakowski 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 Wozniakowski'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.