Helene Laimer

Peter Kritzer, Helene Laimer, Friedrich Pillichshammer

We consider multivariate $\mathbb{L}_2$-approximation in reproducing kernel Hilbert spaces which are tensor products of weighted Walsh spaces and weighted Korobov spaces. We study the minimal worst-case error $e^{\mathbb{L}_2-\mathrm{app},Λ}(N,d)$ of all algorithms that use $N$ information evaluations from the class $Λ$ in the $d$-dimensional case. The two classes $Λ$ considered in this paper are the class $Λ^{\rm all}$ consisting of all linear functionals and the class $Λ^{\rm std}$ consisting only of function evaluations. The focus lies on the dependence of $e^{\mathbb{L}_2-\mathrm{app},Λ}(N,d)$ on the dimension $d$. The main results are conditions for weak, polynomial, and strong polynomial tractability.

Ralph Kritzinger, Helene Laimer, Mario Neumüller

The weighted star discrepancy is a quantitative measure for the performance of point sets in quasi-Monte Carlo algorithms for numerical integration. We consider polynomial lattice point sets, whose generating vectors can be obtained by a component-by-component construction to ensure a small weighted star discre-pancy. Our aim is to significantly reduce the construction cost of such generating vectors by restricting the size of the set of polynomials from which we select the components of the vectors. To gain this reduction we exploit the fact that the weights of the spaces we consider decay very fast.

Helene Laimer

The standard method for constructing generating vectors for good lattice point sets is the component-by-component construction. Numerical experiments have shown that the generating vectors found by these constructions sometimes tend to have recurring components, which can lead to the problem of having projections with all lattice points lying on the main diagonal. In this paper we combine methods of Dick and Kritzer to avoid this problem with a reduced fast component-by-component construction. That is, we give a variation of the standard component-by-component construction which avoids repeated components and simultaneously results in a considerable speed-up in comparison to the standard construction.