Tractability results for integration in subspaces of the Wiener algebra

TL;DR

This paper investigates the computational complexity of multivariate integration within subspaces of the Wiener algebra, revealing conditions for tractability and intractability in both deterministic and randomized settings.

Contribution

It provides new intractability results for the standard Wiener algebra and demonstrates strong polynomial tractability in certain subspaces under randomized algorithms.

Findings

Intractability in the standard Wiener algebra for deterministic integration.

Polynomial tractability in a subspace of the Wiener algebra with randomized methods.

Improved $oldsymbol{ extit{ ext{ε}}}$-exponent for randomized integration over certain subspaces.

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 $\varepsilon$-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.