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 e depends on e-1 and the dimension s. Strong tractability means that it does not depend on s and is bounded by a polynomial in e-1. The least possible value of the power of e-1 is called the e-exponent of strong tractability. Sloan and Wozniakowski established a necessary and sufficient condition of strong tractability in weighted Sobolev spaces, and showed that the e-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 e-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 e-exponent can be achieved in appropriate weighted Sobolev spaces by using Niederreiter's (t,s)-sequences and Sobol sequences.

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