Tractability versus curse of dimensionality for geometric $L_p$-discrepancies

Abstract

This paper studies tractability versus the curse of dimensionality for several geometric $L_p$-discrepancies through a unified discrepancy--integration duality framework, where worst case integration errors in suitable function spaces equal the corresponding discrepancies. A general lower bound method for non-negative linear rules in spaces under broad tensor-product assumptions establishes exponential information complexity in the dimension $d$, yielding the curse of dimensionality for the respective discrepancy. We complement this overview by new results on discrepancy--integration duality and the curse of dimensionality for the periodic $L_p$-discrepancy. The current state of research on this general problem is summarized in a clearly laid out table, which also highlights the remaining open questions.