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

TL;DR

This paper introduces a fully deterministic multiple-shift lattice algorithm for high-dimensional function approximation in Korobov and half-period Cosine spaces, achieving optimal convergence and reduced sampling complexity.

Contribution

It develops a novel deterministic framework using multiple shifts, algebraic guarantees, and the Chinese Remainder Theorem, extending to non-periodic spaces and solving an open theoretical problem.

Findings

Achieves optimal convergence rates in worst-case settings.

Reduces sampling complexity by an order of magnitude compared to probabilistic methods.

Validates the algorithm as a meshless spectral solver for boundary value problems.

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.