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arXiv:2410.09060 · Journal of Complexity

Tractability results for integration in subspaces of the Wiener algebra

Josef Dick, Takashi Goda, Kosuke Suzuki

Published Apr 16, 2025• 4 citations
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Abstract

In this paper, we present some new (in-)tractability results related to the integration problem in subspaces of the Wiener algebra over the d -dimensional unit cube. We show that intractability holds for multivariate integration in the standard Wiener algebra in the deterministic setting, in contrast to polynomial tractability in an unweighted subspace of the Wiener algebra recently shown by Goda (2023). Moreover, we prove that multivariate integration in the subspace of the Wiener algebra introduced by Goda is strongly polynomially tractable if we switch to the randomized setting, where we obtain a better ε -exponent than the one implied by the standard Monte Carlo method . We also identify subspaces in which multivariate integration in the deterministic setting are (strongly) polynomially tractable and we compare these results with the bound which can be obtained via Hoeffding's inequality.

DOI
10.1016/j.jco.2025.101948

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It has long been known that the multivariate integration problem for the unit ball in Cr([0,1]d)C^r([0,1]^d)Cr([0,1]d) is intractable for fixed finite rrr. H. Woźniakowski has recently conjectured that this is true even if r=∞r=\inftyr=∞. This paper establishes a partial result in this direction. We prove that the multivariate integration problem, for infinitely differential functions all of whose variables are bounded by one, is not strongly tractable.
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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In the theory of tractability of multivariate problems one usually studies problems with finite smoothness. Then we want to know which sss-variate problems can be approximated to within ε\varepsilonε by using, say, polynomially many in sss and ε−1\varepsilon^{-1}ε−1 function values or arbitrary linear functionals. There is a recent stream of work for multivariate analytic problems for which we want to answer the usual tractability questions with ε−1\varepsilon^{-1}ε−1 replaced by 1+log⁡ε−11+\log \varepsilon^{-1}1+logε−1. In this vein of research, multivariate integration and approximation have been studied over Korobov spaces with exponentially fast decaying Fourier coefficients. This is work of J. Dick, G. Larcher, and the authors. There is a natural need to analyze more general analytic problems defined over more general spaces and obtain tractability results in terms of sss and 1+log⁡ε−11+\log \varepsilon^{-1}1+logε−1. The goal of this paper is to survey the existing results, present some new results, and propose further questions for the study of tractability of multivariate analytic questions.