A Tool for Custom Construction of QMC and RQMC Point Sets

Abstract

We present LatNet Builder, a software tool to find good parameters for lattice rules, polynomial lattice rules, and digital nets in base 2, for quasi-Monte Carlo (QMC) and randomized quasi-Monte Carlo (RQMC) sampling over the sdimensional unit hypercube. The selection criteria are figures of merit that give different weights to different subsets of coordinates. They are upper bounds on the worst-case error (for QMC) or variance (for RQMC) for integrands rescaled to have a norm of at most one in certain Hilbert spaces of functions. Various Hilbert spaces, figures of merit, types of constructions, and search methods are covered by the tool. We provide simple illustrations of what it can do.

Construction-free median quasi-Monte Carlo rules for function spaces with unspecified smoothness and general weights

“…On the other hand, there are effective search algorithms which, for a given α and a given selection of weights γ u , can construct good rank-1 lattice or polynomial lattice point sets for the function spaces determined by these parameters, and for which the worstcase error converges as O(N −α+ ) for any > 0 [10,20]. Software that implement such algorithms is also freely available [25,27,35]. These algorithms typically use a greedy component-by-component (CBC) construction approach proposed originally by [19], then re-introduced and popularized by [41].…”

“…Their combination gives the product and order-dependent (POD) weights, for which γ u = Γ |u| j∈u γ j [31]. The main reason why the most popular choices of weights have this form is that the existing search algorithms are truly efficient for large s only when the weights have this specific POD form [11,22,25,27,31,36]. Then, by using a fast-CBC approach that speeds up the search by exploiting a fast Fourier transform [27,36], one can find a generating vector that gives a worst-case error of O(N −α+ ) in O(sN log N ) operations for ordinary lattices and O(αsN log N ) for polynomial lattices (with interlacing).…”