A note on Korobov lattice rules for integration of analytic functions

TL;DR

This paper demonstrates that Korobov lattice rules can effectively achieve exponential-weak tractability in high-dimensional integration of analytic periodic functions with exponentially decaying Fourier coefficients.

Contribution

The paper fills a gap by proving that Korobov lattice rules are suitable algorithms for exponential-weak tractability in weighted Korobov spaces of analytic functions.

Findings

Korobov lattice rules achieve exponential-weak tractability

Explicit algorithms for this tractability are provided

The results apply to high-dimensional integration of analytic functions

Abstract

We study numerical integration for a weighted Korobov space of analytic periodic functions for which the Fourier coefficients decay exponentially fast. In particular, we are interested in how the error depends on the dimension $d$. Many recent papers deal with this problem or similar problems and provide matching necessary and sufficient conditions for various notions of tractability. In most cases even simple algorithms are known which allow to achieve these notions of tractability. However, there is a gap in the literature: while for the notion of exponential-weak tractability one knows matching necessary and sufficient conditions, so far no explicit algorithm has been known which yields the desired result. In this paper we close this gap and prove that Korobov lattice rules are suitable algorithms in order to achieve exponential-weak tractability for integration in weighted Korobov spaces of analytic periodic functions.