On the orthogonality of the Chebyshev–Frolov lattice and applications

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 ∈ N, the resulting lattice is an admissible lattice in the sense of Skriganov [12]. These are related to the Frolov cubature formulas, which recently drew attention due to their optimal convergence rates [18] in a broad range of Besov-Lizorkin-Triebel spaces. 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. *

Lattice rules with random n achieve nearly the optimal O(n−α−1<

“…Note that the coefficient of n −λ−1/2+δ in the latter term is bounded under the assumptions we made, i.e., α > λ > 1/2 and 0 < δ < λ − 1/2, as well as (9). This completes the proof.…”

“…The advantage over previous related works lies in the potential for implementation. Other known algorithms which achieve the optimal error bounds, such as those based on Frolov's method (see, e.g., [6,9,11,28,29,30]), are very difficult to implement especially in high dimensions. For our algorithm, a simple probabilistic approach can be used to obtain suitable generating vectors, see Remark 10 below.…”

“…However, the algorithm in [29] is quite impractical in high dimensions, in contrast to the algorithm that will be analyzed in the following. This mainly comes from the fact that the involved point sets, i.e., certain subsets of irrational lattices in R d , seem to be very hard to implement, see, e.g., [9,30]. Moreover, the presented upper bound in [29] is at least exponential in d and probably not improvable, see [29,Remark 3.2].…”

Change of Variable in Spaces of Mixed Smoothness and Numerical Integration of Multivariate Functions on the Unit Cube

“…To gain this, we use a reduction of the problem to theÅ s p,θ setting via tailored transformations in connection with the Frolov cubature formulae [12,36], which recently attracted significant interest [19,21,[44][45][46]. It has been already proved by Bykovskii [3] …”

“…We have 19) where the sum on the right-hand side is taken over all k ∈ N d 0 such that 0 ≤ k i ≤ m if i ∈ e 1 ( j, ) and k i = 0 otherwise. In addition, we put (k · h) = (k 1 h 1 , .…”