Monte-Carlo and Quasi-Monte-Carlo Methods for Numerical Integration

We consider the problem of numerical integration in dimension s, with eventually large s; the usual rules need a very huge number of nodes with increasing dimension to obtain some accuracy, say an error bound less than 10−2; this phenomenon is called ”the curse of dimensionality”; to overcome it, two kind of methods have been developped: the so-called Monte-Carlo and Quasi-Monte-Carlo methods. Very good and up-to-date monographs on the subject exist; our purpose in the present survey is only to present the basic constructions in the two approaches, with a special insight on the second one which performs better for numerical integration and sets the trend with randomized hybridations . To avoid technicalities, we restrict ourselves to the integral domain I = [0, 1] and to the more commonly used sets of nodes which are already implemented in computers routines; we also leave out the so-called Lattice methods to keep an appropriate length to this proceeding paper. The reader interested who should like to go further in the subject should consult the monographs of H.Niederreiter [N], S.Tezuka [T], M.Drmota-R.Tichy [DT], B.L.Fox[F], J.Matoušek [M] (in chronological order) and also of J.E.Gentle [G] for MC Methods and I.H.Sloan-S.Joe [SJ] for Lattice methods; the collective Springer Lecture notes in Statistics 138 [HL] and the proceedings of the three Conferences MCQMC published by Springer (Lectures Notes in Statistics 106 [NSh], 127 [NHLZ], and the book published this year [NSp]) present the main contributions during the five last years; good references on the Number Theory background for Irregularities of Distribution are the books of J.Beck-W.W.L.Chen [BC] and L.Kuipers-H.Niederreiter [KN].