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A deterministic multiple-shift lattice algorithm for function approximation in Korobov and half-period Cosine spaces

Jiarui Du, Josef Dick

Abstract

Approximating multivariate periodic functions in weighted Korobov spaces via rank-1 lattices is fundamentally limited by frequency aliasing. Existing optimal-rate methods rely on randomized constructions or large pre-computations. We propose a fully deterministic multiple-shift lattice algorithm without pre-computation. First, we develop a simplified multiple shift framework for aliased frequency fibers that reduces sampling costs. Second, leveraging the Chinese Remainder Theorem and the Weil bound, we introduce an adaptive hybrid construction that algebraically guarantees the full rank and bounded condition number of the reconstruction matrix. We rigorously prove that this deterministic method maintains the optimal convergence rate in the worst-case setting. Furthermore, we extend this framework to non-periodic, half-period cosine spaces via the tent transformation. By establishing a strict projection equivalence, we prove that the algorithm attains optimal $L_2$ and $L_\infty$ approximation orders in the half-period cosine space, successfully resolving an open theoretical problem posed by Suryanarayana et al. (2016). This mathematically also validates the proposed algorithm as a generic meshless spectral solver for high-dimensional boundary value problems, such as the Poisson equation with Neumann conditions. Numerical experiments corroborate the theoretical bounds, demonstrating an order-of-magnitude reduction in sampling complexity over probabilistic baselines while ensuring absolute deterministic stability.

A deterministic multiple-shift lattice algorithm for function approximation in Korobov and half-period Cosine spaces

Abstract

Approximating multivariate periodic functions in weighted Korobov spaces via rank-1 lattices is fundamentally limited by frequency aliasing. Existing optimal-rate methods rely on randomized constructions or large pre-computations. We propose a fully deterministic multiple-shift lattice algorithm without pre-computation. First, we develop a simplified multiple shift framework for aliased frequency fibers that reduces sampling costs. Second, leveraging the Chinese Remainder Theorem and the Weil bound, we introduce an adaptive hybrid construction that algebraically guarantees the full rank and bounded condition number of the reconstruction matrix. We rigorously prove that this deterministic method maintains the optimal convergence rate in the worst-case setting. Furthermore, we extend this framework to non-periodic, half-period cosine spaces via the tent transformation. By establishing a strict projection equivalence, we prove that the algorithm attains optimal and approximation orders in the half-period cosine space, successfully resolving an open theoretical problem posed by Suryanarayana et al. (2016). This mathematically also validates the proposed algorithm as a generic meshless spectral solver for high-dimensional boundary value problems, such as the Poisson equation with Neumann conditions. Numerical experiments corroborate the theoretical bounds, demonstrating an order-of-magnitude reduction in sampling complexity over probabilistic baselines while ensuring absolute deterministic stability.

Paper Structure

This paper contains 22 sections, 14 theorems, 88 equations, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weighted Korobov Space
  4. Rank-1 Lattices and Fibers
  5. Construction of Generating Vector
  6. Review of the Multiple Shift Algorithm
  7. A Simplified Multiple Shift Algorithm
  8. Existence of Good Shifts: A Probabilistic Baseline
  9. The deterministic construction of multiple shifts
  10. Deterministic Construction in Low Dimensions
  11. Deterministic Construction via the Union of Rank-1 Lattices with a Common Generator
  12. Adaptive Deterministic Strategy
  13. Theoretical analysis
  14. Approximation on Non-periodic functions and Applications to Non-periodic Neumann Problems
  15. The Half-Period Cosine Space and Tent Transformation
  16. ...and 7 more sections

Key Result

Lemma 1

Let $N > 2$ be a prime number and $\gamma_j \in (0,1]$. Let $\boldsymbol{g}$ be constructed by Algorithm alg:CBCg and $M$ be chosen as the lower bound of $\rho(\boldsymbol{g})$. Let $m \in \mathbb{Z}$ satisfy $2^{m-1} < M^{1/\alpha} \le 2^m$. Then $|A_{\alpha,\boldsymbol{\gamma},M}| \le N/2$, $\rho( $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure 1: The total number of shifts $S$ evaluated across different strategies.
  • Figure 2: Sampling complexity and numerical stability comparison against state-of-the-art probabilistic methods.
  • Figure 3: Empirical growth of the required number of shifts $S$ with respect to $\log_2(N)$.
  • Figure 4: The $L_2$ and $L_{\infty}$ errors for the periodic test function $f_1$ across dimensions $d=2$ and $d=4$.
  • Figure 5: The required number of shifts $S$ for the meshless PDE solver across dimensions $d=2$ and $d=4$.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Definition 1
  • Lemma 1
  • Remark 1
  • Remark 2: Algorithmic Advantage via FFT
  • Lemma 2
  • proof
  • Remark 3: Algorithmic Improvement via Uniform Shifts
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 24 more