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Exact Error Estimates and Optimal Randomized Algorithms for Integration

  • Ivan T. Dimov
  • Emanouil Atanassov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4310)

Abstract

Exact error estimates for evaluating multi-dimensional integrals are considered. An estimate is called exact if the rates of convergence for the low- and upper-bound estimate coincide. The algorithm with such an exact rate is called optimal. Such an algorithm has an unimprovable rate of convergence.

The problem of existing exact estimates and optimal algorithms is discussed for some functional spaces that define the regularity of the integrand. Important for practical computations data classes are considered: classes of functions with bounded derivatives and Hölder type conditions.

The aim of the paper is to analyze the performance of two optimal classes of algorithms: deterministic and randomized for computing multi-dimensional integrals. It is also shown how the smoothness of the integrand can be exploited to construct better randomized algorithms.

Keywords

Functional Space Deterministic Algorithm Monte Carlo Algorithm Integration Problem Bulgarian Academy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Ivan T. Dimov
    • 1
  • Emanouil Atanassov
    • 2
  1. 1.Institute for Parallel Processing, Bulgarian Academy of Sciences Acad. G. Bonchev Str., bl. 25 A, 1113 Sofia, Bulgaria and ACET Centre, University of Reading Whiteknights, PO Box 217, Reading, RG6 6AHUK
  2. 2.Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl. 25 A, 1113 SofiaBulgaria

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