A note on approximation in weighted Korobov spaces via multiple rank-1 lattices

TL;DR

This paper extends the analysis of multivariate approximation in weighted Korobov spaces using multiple rank-1 lattices to lower smoothness levels and general weights, and introduces randomized algorithms with strong convergence guarantees.

Contribution

It broadens the applicability of lattice-based approximation methods to functions with lower smoothness and general weights, and demonstrates the effectiveness of randomized algorithms.

Findings

Achieves optimal convergence rates for $1/2<lpha\u2264 1$ with general weights.

Provides a summability condition for strong polynomial tractability.

Randomized algorithms attain near-optimal convergence in root mean squared $L_2$ error.

Abstract

This paper studies the multivariate approximation of functions in weighted Korobov spaces using multiple rank-1 lattice rules. It has been shown by K\"{a}mmerer and Volkmer (2019) that algorithms based on multiple rank-1 lattices achieve the optimal convergence rate for the $L_{\infty}$ error in Wiener-type spaces, up to logarithmic factors. While this result was translated to weighted Korobov spaces in the recent monograph by Dick, Kritzer, and Pillichshammer (2022), the analysis requires the smoothness parameter $\alpha$ to be greater than $1$ and is restricted to product weights. In this paper, we extend this result for multiple rank-1 lattice-based algorithms to the case where $1/2<\alpha\le 1$ and for general weights, covering a broader range of periodic functions with low smoothness and general relative importance of variables. We also provide a summability condition on the weights to ensure strong polynomial tractability for any $\alpha>1/2$. Furthermore, by incorporating random shifts into multiple rank-1 lattice-based algorithms, we prove that the resulting randomized algorithm achieves a nearly optimal convergence rate in terms of the worst-case root mean squared $L_2$ error, while retaining the same tractability property.