We study the problem of constructing rank-1 lattice rules which have good bounds on the " weighted star discrepancy ". Here the non-negative weights are general weights rather than the product weights considered in most earlier works. In order to show the existence of such good lattice rules, we use an averaging argument, and a similar argument is used… (More)
We examine the question of constructing shifted lattice rules of rank one with an arbitrary number of points n, an arbitrary shift, and small weighted star discrepancy. An upper bound on the weighted star discrepancy, that depends on the lattice parameters and is easily computable, serves as a figure of merit. It is known that there are lattice rules for… (More)
Rank-1 lattice rules based on a weighted star discrepancy with weights of a product form have been previously constructed under the assumption that the number of points is prime. Here, we extend these results to the non-prime case. We show that if the weights are summable, there exist lattice rules whose weighted star discrepancy is O(n −1+δ), for any δ >… (More)
We study the problem of constructing shifted rank-1 lattice rules for the approximation of high-dimensional integrals with a low weighted star discrepancy , for classes of functions having bounded weighted variation, where the weighted variation is defined as the weighted sum of Hardy-Krause variations over all lower dimensional projections of the… (More)
We study the convergence of the variance for randomly shifted lattice rules for numerical multiple integration over the unit hypercube in an arbitrary number of dimensions. We consider integrands that are square integrable but whose Fourier series are not necessarily absolutely convergent. For such integrands, a bound on the variance is expressed through a… (More)
We study the problem of constructing shifted rank-1 lattice rules for the approximation of high-dimensional integrals with a low worst-case error, for classes of functions having bounded weighted variation, where the weighted variation is defined as the weighted sum of Hardy-Krause variations over all lower dimensional projections of the integrand. Under… (More)
We prove a number of algebraic relations that appear in Ramanujan's Lost Notebook .