Error bounds for quasi-Monte Carlo integration with nets

We analyze the error introduced by approximately calculating the s-dimensional Lebesgue measure of a Jordan-measurable subset of Is = [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 Is is defined by some function of bounded variation on Īs−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. Introduction Applications of quasi-Monte Carlo methods arise in problems of numerical analysis that can be reduced to numerical integration. For s ≥ 2 let I = [0, 1) be the half-open s-dimensional unit cube, and λs be the s-dimensional Lebesgue measure. If E is a Jordan-measurable subset of I and P is a set of |P | points x1, . . . ,x|P | evenly distributed over I, the volume of E can be approximated by A(E,P ) |P | , where A(E,P ) is the number of p’s, 1 ≤ p ≤ |P |, for which xp ∈ E. An analysis of the error (1) ∣∣∣∣A(E,P ) |P | − λs(E) ∣∣∣∣ was given in a paper of Niederreiter and Wills [5]. The error was bounded by means of D(P ) 1/s . The discrepancy D(P ) of the point set P is defined by D(P ) = sup J ∣∣∣∣A(J, P ) |P | − λs(J) ∣∣∣∣ , Received by the editor October 10, 1994 and, in revised form, February 15, 1995. 1991 Mathematics Subject Classification. Primary 65C05; Secondary 11K38.