Chapter

Monte Carlo and Quasi-Monte Carlo Methods 1996

Volume 127 of the series Lecture Notes in Statistics pp 238-250

Computing Discrepancies Related to Spaces of Smooth Periodic Functions

  • Karin FrankAffiliated withRUS, Universität Stuttgart
  • , Stefan HeinrichAffiliated withFB Informatik, Universität Kaiserslautern

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Abstract

A notion of discrepancy is introduced, which represents the integration error on spaces of r-smooth periodic functions. It generalizes the diaphony and constitutes a periodic counterpart to the classical L2-discrepancy as well as r-smooth versions of it introduced recently by Paskov [Pas93]. Based on previous work [FH96], we develop an efficient algorithm for computing periodic discrepancies for quadrature formulas possessing certain tensor product structures, in particular, for Smolyak quadrature rules (also called sparse grid methods). Furthermore, fast algorithms of computing periodic discrepancies for lattice rules can easily be derived from well—known properties of lattices. On this basis we carry out numerical comparisons of discrepancies between Smolyak and lattice rules.