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Optimal Summation and Integration by Deterministic, Randomized, and Quantum Algorithms

  • We survey old and new results about optimal algorithms for summation of finite sequences and for integration of functions from Hölder or Sobolev spaces. First we discuss optimal deterministic and randornized algorithms. Then we add a new aspect, which has not been covered before on conferences about (quasi-) Monte Carlo methods: quantum computation. We give a short introduction into this setting and present recent results of the authors on optimal quantum algorithms for summation and integration. We discuss comparisons between the three settings. The most interesting case for Monte Carlo and quantum integration is that of moderate smoothness \(k\) and large dimension \(d\) which, in fact, occurs in a number of important applied problems. In that case the deterministic exponent is negligible, so the \(n^{-1/2}\) Monte Carlo and the \(n^{-1}\) quantum speedup essentially constitute the entire convergence rate.
Metadaten
Author:Stefan Heinrich, Erich Novak
URN (permanent link):urn:nbn:de:hbz:386-kluedo-50305
Serie (Series number):Interner Bericht des Fachbereich Informatik (313)
Document Type:Report
Language of publication:English
Publication Date:2017/11/06
Year of Publication:2001
Publishing Institute:Technische Universität Kaiserslautern
Date of the Publication (Server):2017/11/06
Number of page:15
Faculties / Organisational entities:Fachbereich Informatik
DDC-Cassification:0 Allgemeines, Informatik, Informationswissenschaft / 004 Informatik
Licence (German):Creative Commons 4.0 - Namensnennung, nicht kommerziell, keine Bearbeitung (CC BY-NC-ND 4.0)