Related papers: Tractability results for the weighted star-discrep…
We analyze the weighted star discrepancy of so-called $p$-sets which go back to definitions due to Korobov in the 1950s and Hua and Wang in the 1970s. Since then, these sets have largely been ignored since a number of other constructions…
We study tractability properties of the weighted $L_p$-discrepancy. The concept of {\it weighted} discrepancy was introduced by Sloan and Wo\'{z}\-nia\-kowski in 1998 in order to prove a weighted version of the Koksma-Hlawka inequality for…
In this paper we study tractability properties of the weighted star discrepancy with general coefficients of centered regular grids with different mesh-sizes. We give exact characterizations of the weight sequences $(\gamma_j)_{j \ge 1}$…
We study high-dimensional numerical integration in the worst-case setting. The subject of tractability is concerned with the dependence of the worst-case integration error on the dimension. Roughly speaking, an integration problem is…
We study the weighted star discrepancy of the Halton sequence. In particular, we show that the Halton sequence achieves strong polynomial tractability for the weighted star discrepancy for product weights $(\gamma_j)_{j \ge 1}$ under the…
In this paper, we present some new (in-)tractability results related to the integration problem in subspaces of the Wiener algebra over the $d$-dimensional unit cube. We show that intractability holds for multivariate integration in the…
We present two main contributions to the expected star discrepancy theory. First, we derive a sharper expected upper bound for jittered sampling, improving the leading constants and logarithmic terms compared to the state-of-the-art [Doerr,…
The technique of $\star$-translation is important in arguments calibrating the strengths of determinacy theories against large cardinals, for example in [9] and [1] in the paper's bibliography. It has also been used in analysing the…
In the theory of tractability of multivariate problems one usually studies problems with finite smoothness. Then we want to know which $s$-variate problems can be approximated to within $\varepsilon$ by using, say, polynomially many in $s$…
In this paper we consider $L_p$-approximation, $p \in \{2,\infty\}$, of periodic functions from weighted Korobov spaces. In particular, we discuss tractability properties of such problems, which means that we aim to relate the dependence of…
The $L_{\infty}$ star discrepancy is a measure for the regularity of a finite set of points taken from $[0,1)^d$. Low discrepancy point sets are highly relevant for Quasi-Monte Carlo methods in numerical integration and several other…
In this paper, we study tractability of $L_2$-approximation of one-periodic functions from weighted Korobov spaces in the worst-case setting. The considered weights are of product form. For the algorithms we allow information from the class…
In this paper, we consider the upper bound of the probabilistic star discrepancy based on Hilbert space filling curve sampling. This problem originates from the multivariate integral approximation, but the main result removes the strict…
We consider approximation problems for a special space of d variate functions. We show that the problems have small number of active variables, as it has been postulated in the past using concentration of measure arguments. We also show…
We study numerical integration for a weighted Korobov space of analytic periodic functions for which the Fourier coefficients decay exponentially fast. In particular, we are interested in how the error depends on the dimension $d$. Many…
It is well-known that the problem of sampling recovery in the $L_2$-norm on unweighted Korobov spaces (Sobolev spaces with mixed smoothness) as well as classical smoothness classes such as H\"older classes suffers from the curse of…
We prove trace theorems for weighted mixed norm Sobolev spaces in the upper-half space where the weight is a power function of the vertical variable. The results show the differentiability order of the trace functions depends only on the…
The star discrepancy $D_N^*(\mathcal{P})$ is a quantitative measure for the irregularity of distribution of a finite point set $\mathcal{P}$ in the multi-dimensional unit cube which is intimately related to the integration error of…
Astronomers generally assume planet-forming disks are aligned with the rotation of their host star. However, recent observations have shown evidence of warping in protoplanetary disks. One can measure the statistical alignment between the…
We show that the minimal discrepancy of a point set in the $d$-dimensional unit cube with respect to Orlicz norms can exhibit both polynomial and weak tractability. In particular, we show that the $\psi_\alpha$-norms of exponential Orlicz…