Information-based complexity
Quadrature formulae for functions of several variables satisfying a general Lipschitz condition
USSR Computational Mathematics and Mathematical Physics
Complexity theory of real functions
Complexity theory of real functions
Monte Carlo algorithms: performance analysis for some computer architectures
Journal of Computational and Applied Mathematics
A Method for Increasing the Efficiency of Monte Carlo Integration
Journal of the ACM (JACM)
Monte Carlo algorithms for evaluating Sobol' sensitivity indices
Mathematics and Computers in Simulation
The b-adic diaphony as a tool to study pseudo-randomness of nets
NMA'10 Proceedings of the 7th international conference on Numerical methods and applications
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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.