Enumeration of the Chebyshev-Frolov lattice points in axis-parallel boxes

TL;DR

This paper introduces an improved algorithm for enumerating Chebyshev-Frolov lattice points in axis-parallel boxes, extending efficiency to higher dimensions relevant for numerical integration.

Contribution

The authors develop a new enumeration algorithm for Chebyshev-Frolov lattice points that is efficient up to dimension 32, surpassing previous methods limited to dimension 16.

Findings

Efficient enumeration algorithm for dimensions up to 32.

Enhanced computational performance over previous methods.

Applicable to high-dimensional cubature formulas.

Abstract

For a positive integer $d$, the $d$-dimensional Chebyshev-Frolov lattice is the $\mathbb{Z}$-lattice in $\mathbb{R}^d$ generated by the Vandermonde matrix associated to the roots of the $d$-dimensional Chebyshev polynomial. It is important to enumerate the points from the Chebyshev-Frolov lattices in axis-parallel boxes when $d = 2^n$ for a non-negative integer $n$, since the points are used for the nodes of Frolov's cubature formula, which achieves the optimal rate of convergence for many spaces of functions with bounded mixed derivatives and compact support. The existing enumeration algorithm for such points by Kacwin, Oettershagen and Ullrich is efficient up to dimension $d=16$. In this paper we suggest a new enumeration algorithm of such points for $d=2^n$, efficient up to $d=32$.