Exact error estimates and optimal randomized algorithms for integration

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 Holder 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. © Springer-Verlag Berlin Heidelberg 2007.