Numerical Performance of Optimized Frolov Lattices in Tensor Product Reproducing Kernel Sobolev Spaces

Foundations of Computational Mathematics - In this paper, we deal with several aspects of the universal Frolov cubature method, which is known to achieve optimal asymptotic convergence rates in a...     

On the orthogonality of the Chebyshev–Frolov lattice and applications

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. We prove that these lattices are orthogonal and possess a lattice representation matrix with orthogonal columns and entries not larger than 2 (in modulus). In particular, we clarify the existence of orthogonal admissible lattices in higher dimensions. The orthogonality property allows for an efficient enumeration of these lattices in axis parallel boxes. Hence they serve for a practical implementation of the Frolov cubature formulas, which recently drew attention due to their optimal convergence rates in a broad range of Besov–Lizorkin–Triebel spaces. As an application, we efficiently enumerate the Frolov cubature nodes in the d-cube \([-1/2,1/2]^d\) up to dimension \(d=16\).