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International Journal of Wavelets Multiresolution and Information Processing

Quasi-Monte Carlo tractability of integration problem in function spaces defined over products of balls

Jie Zhang, Yongping Liu

Published May 12, 2019• 3 citations
Publisher

Abstract

Recently, quasi-Monte Carlo (QMC) rules for multivariate integration in some weighted Sobolev spaces of functions defined over unit cubes [Formula: see text], products of [Formula: see text] copies of the simplex [Formula: see text] and products of [Formula: see text] copies of the unit sphere [Formula: see text] have been well-studied in the literature. In this paper, we consider QMC tractability of integrals of functions defined over product of [Formula: see text] copies of the ball [Formula: see text]. The space is a tensor product of [Formula: see text] reproducing kernel Hilbert spaces defined by positive and uniformly bounded weights [Formula: see text] for [Formula: see text]. We obtain matching necessary and sufficient conditions in terms of this weights for various notions of QMC tractability, including strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniformly weak tractability and [Formula: see text]-weak tractability.

DOI
10.1142/s0219691319500437

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Recently, quasi-Monte Carlo (QMC) rules for multivariate integration in some weighted Sobolev spaces of functions defined over unit cubes [Formula: see text], products of [Formula: see text] copies of the simplex [Formula: see text] and products of [Formula: see text] copies of the unit sphere [Formula: see text] have been well-studied in the literature. In this paper, we consider QMC tractability of integrals of functions defined over product of [Formula: see text] copies of the ball [Formula: see text]. The space is a tensor product of [Formula: see text] reproducing kernel Hilbert spaces defined by positive and uniformly bounded weights [Formula: see text] for [Formula: see text]. We obtain matching necessary and sufficient conditions in terms of this weights for various notions of QMC tractability, including strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniformly weak tractability and [Formula: see text]-weak tractability.
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 })
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We study the randomized worst-case error and the randomized error of scrambled quasi–Monte Carlo (QMC) quadrature as proposed by Owen. The function spaces considered in this article are the weighted Hilbert spaces generated by Haar-like wavelets and the weighted Sobolev-Hilbert spaces. Conditions are found under which multivariate integration is strongly tractable in the randomized worst-case setting and the randomized setting , respectively. The ε \varepsilon -exponents of strong tractability are found for the scrambled Niederreiter nets and sequences. The sufficient conditions for strong tractability for Sobolev spaces are more lenient for scrambled QMC quadratures than those for deterministic QMC net quadratures.
This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s[0,1]^s[0,1]s. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost 1/N1/N1/N, where NNN is the number of points, independently of dimension) to so-called “product and order dependent” (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets. doi:10.1017/S1446181112000077
This paper is devoted to the study of tractability of the $ L_2 $-approximation and integration from weighted Korobov spaces of increasing smoothness in the worst-case setting. The considered algorithms use information from the class $ \Lambda^{\rm all} $, including all continuous linear functionals, and from the class $ \Lambda^{\rm std} $, including function evaluations. Necessary and sufficient conditions on the weights of the function space for strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniform weak tractability, weak tractability and $ (\sigma, \tau) $-weak tractability, are provided. Our results give a comprehensive picture of the weight conditions for all standard notions of algebraic tractability. It may be helpful to study the tractability of nonhomogeneous tensor product spaces.