Variations on Component-by-Component Construction Algorithms of Lattice Rules
In the conducted research we develop efficient algorithms for constructing node sets of high-quality quasi-Monte Carlo (QMC) methods which can be used for approximating high-dimensional integrals of multivariate functions. In particular, we study the construction of rank-1 lattice rules and polynomial lattice rules, which are both specified by a generating vector, for numerical integration in weighted function spaces such as Korobov, Sobolev and Walsh spaces. The obtained construction schemes are mainly greedy algorithms which generate QMC point sets that, as we demonstrate, achieve optimal error convergence rates in the respective function spaces. We show that under certain conditions on the weights, which are incorporated in the definitions of the considered function spaces, the obtained error estimates become independent of the dimension, and thus the integration problem becomes tractable. Furthermore, we derive fast implementations of the numerous construction algorithms and confirm our theoretical findings with numerical results and experiments. As an application, we investigate one of the considered algorithms for the construction of QMC finite element methods for a class of elliptic PDEs with random diffusion coefficients and carry out a combined error analysis.