Title Participants Abstract "Variations on Component-by-Component Construction Algorithms of Lattice Rules" "Adrian Ebert" "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." "Constructing lattice points for numerical integration by a reduced fast successive coordinate search algorithm" "Adrian Ebert" "In this paper, we study an efficient algorithm for constructing node sets of high-quality quasi-Monte Carlo integration rules for weighted Korobov, Walsh, and Sobolev spaces. The algorithm presented is a reduced fast successive coordinate search (SCS) algorithm, which is adapted to situations where the weights in the function space show a sufficiently fast decay. The new SCS algorithm is designed to work for the construction of lattice points, and, in a modified version, for polynomial lattice points, and the corresponding integration rules can be used to treat functions in different kinds of function spaces. We show that the integration rules constructed by our algorithms satisfy error bounds of optimal convergence order. Furthermore, we give details on efficient implementation such that we obtain a considerable speed-up of previously known SCS algorithms. This improvement is illustrated by numerical results. The speed-up obtained by our results may be of particular interest in the context of QMC for PDEs with random coefficients, where both the dimension and the required number of points are usually very large. Furthermore, our main theorems yield previously unknown generalizations of earlier results."