On the orthogonality of the Chebyshev-Frolov lattice and applications

TL;DR

This paper proves the orthogonality of Chebyshev-Frolov lattices in dimensions that are powers of two, enabling efficient node enumeration for Frolov cubature formulas with applications in high-dimensional numerical integration.

Contribution

It establishes the orthogonality of Chebyshev-Frolov lattices and provides a lattice representation with bounded entries, facilitating efficient node enumeration in high dimensions.

Findings

Lattices are orthogonal in dimensions that are powers of two.

Lattice representation matrices have entries not larger than 2.

Efficient enumeration of cubature nodes up to dimension 16.

Abstract

We deal with lattices that are generated by the Vandermonde matrices associated to the roots of Chebyshev-polynomials. If the dimension $d$ of the lattice is a power of two, i.e. $d=2^m, m \in \mathbb{N}$, the resulting lattice is an admissible lattice in the sense of Skriganov. Those are related to the Frolov cubature formulas, which recently drew attention due to their optimal convergence rates in a broad range of function spaces with mixed smoothness. We prove that the resulting lattices are orthogonal and possess a lattice representation matrix with entries not larger than $2$ (in modulus). This allows for an efficient enumeration of the Frolov cubature nodes in the $d$-cube $[-1/2,1/2]^d$ up to dimension $d=16$.