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Quasi-random methods for estimating integrals using relatively small samples. (English) Zbl 0824.65009

The paper is a survey of the quasi-random methods for calculating multiple integrals and is aimed at emphasizing the techniques effective for “relatively small” sample size, rather than those characterized by a good asymptotic convergence. The authors mention many important results concerning the quasi-Monte Carlo methods, and chiefly the classical Koksma and Hlawka theorems, with other more or less known theorems concerning the boundary of the error. The well-known sequences of Halton, Sobol, Faure and Niederreiter are mentioned in the paper while the so called lattice methods are treated a little more extensively. These various topics are object of comments and comparisons.
In the last part of the paper the authors recall some recent techniques consisting in introducing in a quasi-random procedure some transformations of the integrand analogous to those usually applied in the classical Monte Carlo method for the so called variance reduction. The transformations mentioned in the paper are called “importance sampling” and “weighted importance sampling”, and are illustrated by short examples. The bibliography quotes 75 papers and books.
Reviewer: M.Cugiani (Milano)

MSC:

65D32 Numerical quadrature and cubature formulas
65C05 Monte Carlo methods
41A55 Approximate quadratures
41A63 Multidimensional problems
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