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Mathematics of Computation

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Error Bounds for quasi-Monte Carlo
integration with nets

Author: Christian Lécot
Journal: Math. Comp. 65 (1996), 179-187
MSC (1991): Primary 65C05; Secondary 11K38
MathSciNet review: 1325870
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Abstract: We analyze the error introduced by approximately calculating the $s$-dimensional Lebesgue measure of a Jordan-measurable subset of $I^s=[0,1)^s$. We give an upper bound for the error of a method using a $(t,m,s)$-net, which is a set with a very regular distribution behavior. When the subset of $I^s$ is defined by some function of bounded variation on ${\bar I}^{s-1}$, the error is estimated by means of the variation of the function and the discrepancy of the point set which is used. A sharper error bound is established when a $(t,m,s)$-net is used. Finally a lower bound of the error is given, for a method using a $(0,m,s)$-net. The special case of the 2-dimensional Hammersley point set is discussed.

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Additional Information

Christian Lécot
Affiliation: address Laboratoire de Mathématiques, Université de Savoie, 73376 Le Bourget du Lac, France

Keywords: Quasi-Monte Carlo method, $(t, m, s)$-nets, discrepancy
Received by editor(s): October 10, 1994
Received by editor(s) in revised form: February 15, 1995
Article copyright: © Copyright 1996 American Mathematical Society