A Parallel Search for Korobov Lattice Rules.pdf

A Parallel Search for Korobov Lattice Rules Karl Entacher and Bernhard Hechenleitner∗ Abstract. We present results from an extensive parallel search for Korobov lattice rules using the LLL-spectral test with a new normalization strategy. The resulting lattice parameters are distributed via a web-server [9] which provides general information on the spectral test, a database for lat- tice rule parameters, software for spectral test calculations and related applications, efficient on-line parameter searches, scientists working in the field of MCQMC Methods, and further links and ref- erences. 1. Introduction The method of good lattice points (GLP) also called Korobov lattice rules is a central technique from the fields of Monte Carlo (MC) and quasi-Monte Carlo (QMC) methods. Good lattice points are classical node sets for QMC integration, defined by the Russian mathematician Korobov [15, 16, 17]. s For y ∈ R let {y} = y − y be the fractional part of y. Consider a vector a ∈ Z , s ≥ 2. A Korobov lattice rule is defined by the set Pm := { xn : 0 ≤ n m}, with xn := n·a . (1) m In the following we will use the term Korobov lattice rule Pm only for special vectors a defined by a 2 s−1 parameter a with 1 a m, m ∈ N, and a := (1, a, a , . . . , a ), s ≥ 2, see [16]. The set Pm can be seen as the intersection of the s-dimensional unit cube Is := [0, 1)s with the lattice s s L (a, m) = k b : k ∈ Z ,