Aicke Hinrichs, David Krieg, Erich Novak et al.
We study approximation and integration problems and compare the quality of optimal information with the quality of random information. For some problems random information is almost optimal and for some other problems random information is much worse than optimal information. We prove new results and give a short survey of known results.
Aicke Hinrichs, Joscha Prochno, Mario Ullrich
We prove the curse of dimensionality in the worst case setting for multivariate numerical integration for various classes of smooth functions. We prove the results when the domains are isotropic convex bodies with small diameter satisfying a universal $ψ_2$-estimate. In particular, we obtain the result for the important class of volume-normalized $\ell_p^d$-balls in the complete regime $2\leq p \leq \infty$. This extends a result in a work of A. Hinrichs, E. Novak, M. Ullrich and H. Woźniakowski [J. Complexity, 30(2), 117-143, 2014] to the whole range $2\leq p \leq \infty$, and additionally provides a unified approach. The key ingredient in the proof is a deep result from the theory of Asymptotic Geometric Analysis, the thin-shell volume concentration estimate due to O. Guédon and E. Milman. The connection of Asymptotic Geometric Analysis and Information-based Complexity revealed in this work seems promising and is of independent interest.
Aicke Hinrichs, Joscha Prochno, Mario Ullrich et al.
In this manuscript we introduce and study an extended version of the minimal dispersion of point sets, which has recently attracted considerable attention. Given a set $\mathscr P_n=\{x_1,\dots,x_n\}\subset [0,1]^d$ and $k\in\{0,1,\dots,n\}$, we define the $k$-dispersion to be the volume of the largest box amidst a point set containing at most $k$ points. The minimal $k$-dispersion is then given by the infimum over all possible point sets of cardinality $n$. We provide both upper and lower bounds for the minimal $k$-dispersion that coincide with the known bounds for the classical minimal dispersion for a surprisingly large range of $k$'s.