A Component-by-Component Construction for the Trigonometric Degree

Abstract

We propose an alternative to the algorithm from Cools, Kuo, Nuyens (Computing, 2010) [5], for constructing lattice rules with good trigonometric degree. The original algorithm has construction cost O(|A d (m)| + dN log N) for an N-point lattice rule in d dimensions having trigonometric degree m, where the set A d (m) has exponential size in both d and m (in the "unweighted degree" case, which is what we consider here). We reduce the cost to O(dN(log N) 2 ) with an implicit constant governing the needed precision (which is dependent on N and d).

High-dimensional integration: The quasi-Monte Carlo way

“…Another branch of lattice rule analysis focuses on the trigonometric degree and similar quantities; see for example Cools and Lyness (2001), Lyness and Sørevik (2006), Cools and Nuyens (2008) and the references therein. Recent constructions in terms of trigonometric degrees include the works by Cools, Kuo and Nuyens (2010), Achtsis and Nuyens (2012) and Kämmerer, Kunis and Potts (2012).…”

The construction of good lattice rules and polynomial lattice rules

“…This naturally leads to the recently studied exponentially converging function spaces as in [15,32]. Similarly, the method in [1] is based on exponentially converging series to construct lattice rules with good trigonometric degree. It follows that, for 1 ≤ p ≤ ∞,…”

“…The worst-case error is then given by ρ(z, N ) −α . Such figures of merit are related to the classical concept of degree of precision which has been studied in, e.g., [1,4]. It can be seen that smaller p will shrink the unit ball on which the worst-case error (4) is defined, since |||f ||| r,α,γ ≥ |||f ||| r ,α,γ for any 1 ≤ r ≤ r ≤ ∞.…”