A note on approximation in weighted Korobov spaces via multiple rank-1 lattices
Mou Cai, Takashi Goda
TL;DR
The paper tackles high-dimensional approximation of periodic functions in the weighted Korobov space $H_{d,α,\boldsymbol{γ}}$ by using a union of rank-1 lattice rules. By introducing a reconstruction strategy based on multiple lattices and, in a randomized variant, random shifts, it achieves near-optimal convergence rates for both $L_∞$ (deterministic) and $L_2$ (randomized) errors across the full range $α>1/2$ with general weights. The authors derive explicit error bounds, establish strong polynomial tractability under summability conditions on the weights, and show that their approach improves upon single-lattice limits, especially for low smoothness ($1/2<α≤1$) and non-product weights. These results broaden applicability to high-dimensional problems with complex variable importance, while maintaining feasible computational costs through aliasing-free lattice unions and a probabilistic lattice construction. Overall, the work extends the optimality of multiple-rank-1 lattice methods to broader function spaces and weight structures, with clear implications for tractable high-dimensional approximation and PDE applications with random coefficients.
Abstract
This paper studies the multivariate approximation of functions in weighted Korobov spaces using multiple rank-1 lattice rules. It has been shown by Kämmerer 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 $α$ 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 $1/2<α\le 1$ 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 $α>1/2$. 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.