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Tractability results for integration in subspaces of the Wiener algebra

Josef Dick, Takashi Goda, Kosuke Suzuki

TL;DR

The paper analyzes the tractability of multivariate integration on subspaces of the Wiener algebra on $[0,1)^d$ under deterministic and randomized algorithms. It proves intractability for the standard Wiener algebra in the deterministic setting, and establishes strong or polynomial tractability for several weighted subspaces $F_{d,r}$ (notably $r_2$, $r_3$, and $r_4$) using explicit quasi-Monte Carlo constructions and an existence argument. In the randomized setting, it shows strong tractability with an $oldsymbol{\varepsilon}$-exponent of $1$ for $r \\ge r_1$ via a constructive randomized QMC rule based on rank-1 lattices, improving upon Monte Carlo rates; it also yields a tight bound in terms of dimension for certain product-form weights. Collectively, the results illuminate the tractability landscape for Wiener-algebra-based integration, furnish concrete integration schemes (Korobov p-sets and rank-1 lattices), and connect to Hoeffding-type probabilistic bounds to compare deterministic and randomized performance.

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.

Tractability results for integration in subspaces of the Wiener algebra

TL;DR

The paper analyzes the tractability of multivariate integration on subspaces of the Wiener algebra on under deterministic and randomized algorithms. It proves intractability for the standard Wiener algebra in the deterministic setting, and establishes strong or polynomial tractability for several weighted subspaces (notably , , and ) using explicit quasi-Monte Carlo constructions and an existence argument. In the randomized setting, it shows strong tractability with an -exponent of for via a constructive randomized QMC rule based on rank-1 lattices, improving upon Monte Carlo rates; it also yields a tight bound in terms of dimension for certain product-form weights. Collectively, the results illuminate the tractability landscape for Wiener-algebra-based integration, furnish concrete integration schemes (Korobov p-sets and rank-1 lattices), and connect to Hoeffding-type probabilistic bounds to compare deterministic and randomized performance.

Abstract

In this paper, we present some new (in-)tractability results related to the integration problem in subspaces of the Wiener algebra over the -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 -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.

Paper Structure

This paper contains 10 sections, 7 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Integration and tractability
  3. Subspaces of Wiener algebra and known results
  4. Summary of our results
  5. Lower bound
  6. (Strong) Tractability in the deterministic setting
  7. Numerical integration in $F_{d,r_2}$
  8. Numerical integration in the space $F_{d,r_3}$
  9. Existence result
  10. Strong tractability in the randomized setting

Key Result

Theorem 2.1

Consider the standard Wiener algebra, i.e., the space eq:subspace_wiener with $r(\boldsymbol{k})= 1$ for all $\boldsymbol{k}\in \mathbb{Z}^d$. The minimal worst-case error for general algorithms of the form eq:general_quadrature equals $1/2$ for any $n,d\geq 1$, which implies that the information co

Theorems & Definitions (17)

  • Definition 1.1
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • proof : Proof of Theorem \ref{['thm:main2']}
  • Theorem 3.4
  • ...and 7 more