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Published in:
2007 | OriginalPaper | Chapter
Exact Error Estimates and Optimal Randomized Algorithms for Integration
Authors : Ivan T. Dimov, Emanouil Atanassov
Published in: Numerical Methods and Applications
Publisher: Springer Berlin Heidelberg
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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.