A lattice algorithm with multiple shifts for function approximation in Korobov spaces

TL;DR

This paper introduces a lattice-based function approximation algorithm in Korobov spaces that uses multiple shifts and least-squares recovery to achieve optimal convergence rates in both worst-case and randomized settings.

Contribution

It presents a novel shifted rank-1 lattice algorithm with multiple shifts and least-squares recovery, achieving optimal approximation rates in Korobov spaces.

Findings

Achieves optimal $L_{ abla}$-approximation error rate $ ext{O}(N^{-eta})$ in worst-case setting.

Attains optimal randomized $L_{2}$-approximation rate $ ext{O}(N^{-eta})$ with random shifts.

Numerical experiments confirm theoretical convergence rates.

Abstract

In this paper, we propose a novel algorithm for function approximation in a weighted Korobov space based on shifted rank-1 lattice rules. To mitigate aliasing errors inherent in lattice-based Fourier coefficient estimation, we employ $\mathcal{O}((\log N)^{d} )$ good shifts and recover each Fourier coefficient via a least-squares procedure. We show that the resulting approximation achieves the optimal convergence rate for the $L_{\infty}$-approximation error in the worst-case setting, namely $\mathcal{O}(N^{-\alpha+1/2+\varepsilon})$ for arbitrarily small $\varepsilon>0$. Moreover, by incorporating random shifts, the algorithm attains the optimal rate for the $L_{2}$-approximation error in the randomized setting, which is $\mathcal{O}(N^{-\alpha+\varepsilon})$. Numerical experiments are presented to support the theoretical results.