Numerical integration in $\log$-Korobov and $\log$-cosine spaces

TL;DR

This paper develops endpoint estimates for quasi-Monte Carlo integration errors in log-Korobov and log-cosine spaces, especially near the critical smoothness threshold, using lattice rules and a log-scale analysis.

Contribution

It introduces a log-scale framework for analyzing integration errors near the smoothness boundary and refines tractability conditions for weighted Korobov spaces.

Findings

Achieves error bounds of order N^{-1/2} with log-scale factors

Provides dimension-independent error bounds under new summability conditions

Extends results to log-cosine spaces with tent-transformed lattice rules

Abstract

QMC rules are equal weight quadrature rules for approximating integrals over $[0,1]^s$. One line of research studies the integration error of functions in the unit ball of so-called Korobov spaces, which are Hilbert spaces of periodic functions on $[0,1]^s$ with square integrable partial mixed derivatives of order $\alpha$. Using Parseval's identity, this smoothness can be defined for all real numbers $\alpha > 1/2$. This condition is necessary as otherwise the Korobov space contains discontinuous functions for which function evaluation is not well defined. This paper is concerned with more precise endpoint estimates of the integration error using QMC rules for Korobov spaces with $\alpha$ arbitrarily close to $1/2$. To obtain such estimates we introduce a $\log$-scale for functions with smoothness close to $1/2$, which we call $\log$-Korobov spaces. We show that lattice rules can be used to obtain an integration error of order $\mathcal{O}(N^{-1/2} (\log N)^{-\mu(1-\lambda)/2})$ for any $1/\mu <\lambda \le 1$, where $\mu>1$ is a power in the $\log$-scale. We also consider tractability of numerical integration for weighted Korobov spaces with product weights $(\gamma_j)_{j \in \mathbb{N}}$. It is known that if $\sum_{j=1}^\infty \gamma_j^\tau < \infty$ for some $1/(2\alpha) < \tau \le 1$ one can obtain error bounds which are independent of the dimension. In this paper we give a more refined estimate for the case where $\tau$ is close to $1/(2 \alpha)$, namely we show dimension independent error bounds under the condition that $\sum_{j=1}^\infty \gamma_j \max\{1, \log \gamma_j^{-1}\}^{\mu(1-\lambda)} < \infty$ for some $1/\mu < \lambda \le 1$. The essential tool in our analysis is a $\log$-scale Jensen's inequality. The results described above also apply to integration in $\log$-cosine spaces using tent-transformed lattice rules.