On the orthogonality of the Chebyshev-Frolov lattice and applications
Provides a practical computational tool for high-dimensional numerical integration in function spaces with mixed smoothness.
The paper proves that Chebyshev-Frolov lattices are orthogonal and have a representation matrix with entries bounded by 2, enabling efficient enumeration of Frolov cubature nodes up to dimension 16.
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 $d=16$.