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SUMMARY:QMC-integration based on arbitrary (t\,m\,s)-Nets yields optimal c
 onvergence rates for several scales of function spaces - Michael Gnewuch (
 Universität Osnabrück)
DTSTART:20240718T130000Z
DTEND:20240718T134000Z
UID:TALK218185@talks.cam.ac.uk
DESCRIPTION:We study the integration problem over the s-dimensional unit c
 ube on four scales of Banach spaces of integrands. First we consider Haar 
 wavelet spaces\, consisting of functions whose Haar wavelet coefficients e
 xhibit a certain decay behavior measured by a parameter a >0. &nbsp\;We st
 udy the worst case error of integration over the norm unit balland provide
  upper error &nbsp\;bounds for quasi-Monte Carlo (QMC) cubature rules base
 d on arbitrary (t\,m\,s)-nets as well as matching lower error bounds for a
 rbitrary cubature rules. These results show that using arbitrary (t\,m\,s)
 -nets as sample points yields the best possible rate of convergence.Via su
 itable embeddings our upper error bounds on Haar wavelet spacestransfer (w
 ith possibly different constants) to certain spaces of fractional smoothne
 ss 0 < a <1 and to Sobolev and Besov spaces of dominating mixed smoothness
  0 < a <1.Known lower bounds for Sobolev and Besov spaces of dominating mi
 xed smoothness show that arbitrary (t\,m\,s)-nets yield optimal convergenc
 e rates also on these scales of spaces.\nThis is joint work with Josef Dic
 k\, Lev Markhasin\, and Winfried Sickel
LOCATION:Seminar Room 1\, Newton Institute
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