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

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 $\mathcal{O}((\log N{)}^{2d-1})$ shifted copies of a single rank-1 lattice and recover each Fourier coefficient via a least-squares procedure. Writing $p$ for the total number of function evaluations, we show that the resulting approximation achieves the optimal convergence rate for the $L_{\infty }$-approximation error in the worst-case setting, namely $\mathcal{O}(p^{-\alpha +1/2+\epsilon })$ for arbitrarily small $\epsilon >0$. Moreover, by incorporating random shifts, the algorithm attains the optimal rate for the $L_2$-approximation error in the randomized setting, which is $\mathcal{O}(p^{-\alpha +\epsilon })$. Numerical experiments illustrate the practical performance of the algorithms and the qualitative behavior predicted by the theoretical analysis.