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

Mou Cai, Josef Dick, Takashi Goda

TL;DR

This work introduces a single, shifted rank‑1 lattice reconstruction framework for approximating smooth periodic functions in weighted Korobov spaces. By partitioning the Fourier index set into fibers and solving small least‑squares problems using multiple shifts per fiber, the method mitigates aliasing without resorting to multiple lattices, while preserving lattice simplicity. The authors prove optimal deterministic $L_{\infty}$ convergence $\mathcal{O}(N^{-\alpha+1/2+\varepsilon})$ and, with randomized shifts, optimal $L_2$ convergence $\mathcal{O}(N^{-\alpha+\varepsilon})$, supported by rigorous error analyses. They also establish existence and practical construction of suitable generating vectors and shifts, and provide numerical experiments that corroborate the theoretical guarantees and demonstrate aliasing reduction and accurate approximation. The results offer a scalable, provably optimal approach for high‑dimensional function approximation in Korobov spaces with applications to sampling and reconstruction in numerical analysis.

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 $\mathcal{O}(N^{-α+1/2+\varepsilon})$ 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 $\mathcal{O}(N^{-α+\varepsilon})$. Numerical experiments are presented to support the theoretical results.

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

TL;DR

This work introduces a single, shifted rank‑1 lattice reconstruction framework for approximating smooth periodic functions in weighted Korobov spaces. By partitioning the Fourier index set into fibers and solving small least‑squares problems using multiple shifts per fiber, the method mitigates aliasing without resorting to multiple lattices, while preserving lattice simplicity. The authors prove optimal deterministic convergence and, with randomized shifts, optimal convergence , supported by rigorous error analyses. They also establish existence and practical construction of suitable generating vectors and shifts, and provide numerical experiments that corroborate the theoretical guarantees and demonstrate aliasing reduction and accurate approximation. The results offer a scalable, provably optimal approach for high‑dimensional function approximation in Korobov spaces with applications to sampling and reconstruction in numerical analysis.

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 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 -approximation error in the worst-case setting, namely for arbitrarily small . Moreover, by incorporating random shifts, the algorithm attains the optimal rate for the -approximation error in the randomized setting, which is . Numerical experiments are presented to support the theoretical results.

Paper Structure

This paper contains 19 sections, 10 theorems, 130 equations, 1 figure, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review
  3. Notation and background
  4. Weighted Korobov space
  5. Rank-1 lattice point set
  6. The approximation algorithms and the construction of shifted lattice point sets
  7. The deterministic approximation algorithm
  8. The randomized algorithm
  9. Existence of good generating vectors
  10. Construction of generating vectors with large figure of merit
  11. A bound on the fiber length
  12. Existence of good shifts
  13. Error analysis
  14. Convergence analysis for the worst case error in $L_\infty$ norm
  15. Convergence analysis for the random case error
  16. ...and 4 more sections

Key Result

Lemma 2.1

For $d\in \mathbb{N}$, $\alpha>1/2$, positive weights $\boldsymbol{\gamma}=(\gamma_1,\gamma_2,\ldots)$, and $M>0$, let $A_{\alpha,\boldsymbol{\gamma},M}$ be defined by eq:Def_hyperbolic_set. Then, for any $\lambda>1/ \alpha$, we have where $\zeta(s) = \sum_{k=1}^\infty k^{-s}$ denotes the Riemann zeta function.

Figures (1)

  • Figure 1: Histogram of $R_{1, \bm{1}, M}(1,g)$ for $10^5$ random choices of $g \in \{1, \ldots, N-1\}$, where $N = 999983$ (the largest prime number $\le 10^6$), $M = 22580$, $\alpha =1$, $\gamma_1 = \gamma_2 = 1$, $|A_{1, \bm{1}, M}| = 1009757 > N$. The median $R_{1, \bm{1}, M}(1,g)$ is $5$ and the minimum is $3$.

Theorems & Definitions (20)

  • Lemma 2.1
  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Lemma \ref{['lem_rho']}
  • Lemma 3.3
  • proof
  • Remark 3.1
  • Lemma 3.4
  • ...and 10 more