Strong tractability of multivariate integration using quasi–Monte Carlo algorithms

Abstract

Abstract. 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.

Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients

“…We do not consider these QMC rules in this paper because they do not presently give results that are as good as randomly shifted lattice rules. (We remark that the scaling of γ u in our definition (2.1) is consistent with [31,34]. However, all results in [16,28,8] will be consistent with the notation in the present paper upon the substitution γ u → γ 1/2 u .)…”

“…The analysis in [34] is based on the so-called Niederreiter and Sobol sequences which are low-discrepancy sequences that can be generated explicitly, and which are extensible in both s and N . The lattice rules in [16,28] are constructed using a different error criterion in a non-Hilbert space setting.…”

“…For the lattice rules of [16,28] we get O(N −1+δ ) convergence, but again with p ≤ 1/2. Niederreiter and Sobol sequences give O(N −1+δ ) convergence, see [34], but they require an even stronger condition that p ≤ 1/3. Our best results are therefore for randomly shifted lattice rules with POD weights (6.5): they provide the convergence rate O(N −1+δ ) under the weakest summability requirements on ψ j L ∞ (D) ; specifically, when the summability exponent p is between 2/3 and 1, the QMC rates correspond exactly to the bounds for (nonlinear) best N -term approximation results obtained in [5].…”

Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond

“…(We remark that our scaling with weights γ 1/2 u is consistent with that of Sloan and Woźniakowski [53] and Wang [59], but the scaling γ u was used by Dick and Pillichshammer [6] and Joe [26].) Unlike the Lfor computing the weighted star discrepancy for a given point set (except when the dimensionality s is as low as 2 or 3), and one must work with some form of upper bound.…”

“…Two QMC constructions related to this formula are given by Sloan et al [49,50]. The other important approach uses the non-Hilbert setting of q = r = 1 (and thus q = r = ∞) and so studies the weighted star discrepancy [6,22,26,53,59]…”