Construction Algorithms for Generalized Polynomial Lattice Rules

Recently, generalized digital nets were introduced and shown to achieve almost optimal convergence rates when used in a quasi-Monte Carlo algorithm to approximate high-dimensional integrals over the unit cube. Since their inception, one method of constructing generalized digital nets from classical digital nets has been known. However, the desirable features of generalized digital nets motivate a quest for alternative, possibly better, constructions. Subsequently, a special case of generalized digital nets, generalized polynomial lattice rules, has been studied and encouraging numerical results were presented, showing that generalized polynomial lattice rules can improve on the original construction of generalized digital nets. However, it was not shown how to construct such generalized polynomial lattice rules avoiding an exhaustive search: This is the contribution of this paper. We show how to construct generalized polynomial lattice rules achieving optimal convergence rates for functions of arbitrary high smoothness using a component-by-component approach. In addition, we show how to combine a sieve-type algorithm with the component-by-component approach to construct generalized polynomial lattice rules adjusting themselves to the smoothness of the integrand up to a certain given degree; analogous results for Korobov polynomial lattice rules are also presented. In this way, we provide a direct approach to the construction of generalized digital nets no longer relying on classical digital nets.