Successive coordinate search and component-by-component construction of rank-1 lattice rules

Abstract

The (fast) component-by-component (CBC) algorithm is an efficient tool for the construction of generating vectors for quasi-Monte Carlo rank-1 lattice rules in weighted reproducing kernel Hilbert spaces. We consider product weights, which assign a weight to each dimension. These weights encode the effect a certain variable (or a group of variables by the product of the individual weights) has. Smaller weights indicate less importance. Kuo (J Complex 19:301–320, 2003 [3]) proved that CBC constructions achieve the optimal rate of convergence in the respective function spaces, but this does not imply the algorithm will find the generating vector with the smallest worst-case error. In fact it does not. We investigate a generalization of the component-by-component construction that allows for a general successive coordinate search (SCS), based on an initial generating vector, and with the aim of getting closer to the smallest worst-case error. The proposed method admits the same type of worst-case error bounds as the CBC algorithm, independent of the choice of the initial vector. Under the same summability conditions on the weights as in (Kuo J Complex 19:301–320, 2003 [3]) the error bound of the algorithm can be made independent of the dimension d and we achieve the same optimal order of convergence for the function spaces from (Kuo, J Complex 19:301–320, 2003 [3]). Moreover, a fast version of our method, based on the fast CBC algorithm as in Nuyens and Cools (Math Comput 75:903–920, 2006, [5]), is available, reducing the computational cost of the algorithm to O(dnlog(n)) operations, where n denotes the number of function evaluations. Numerical experiments seeded by a Korobov-type generating vector show that the new SCS algorithm will find better choices than the CBC algorithm and the effect is better for slowly decaying weights.