Quasi-Monte Carlo methods for the numerical integration of multivariate Walsh series. (English) Zbl 0855.11042
The authors consider quasi-Monte Carlo methods for the numerical integration of multivariate Walsh series by approximating the integrals via net-sequences. Best possible bounds are shown for the integration error \(R_N (f)\) when the integration points form a \((t, m, s)\)-net in \([0, 1]^s\) in base \(b\). More precisely it is proved that
\[
R_N (f)\;\ll\;b^{t(\alpha- (1/2))} {{(\log N)^{s-1}} \over {N^{\alpha- (1/2)}}},
\]
provided that \(f\) belongs to an \(E_s^\alpha\)-class with respect to the Walsh series in base \(b\). This improves an earlier paper of the authors and C. Traunfellner [Math. Comput. 63, 277-291 (1994; Zbl 0806.65013)].
Reviewer: R.F.Tichy (Graz)
MSC:
| 11K45 | Pseudo-random numbers; Monte Carlo methods |
| 65C05 | Monte Carlo methods |
| 65D30 | Numerical integration |
Keywords:
quasi-Monte Carlo methods; numerical integration of multivariate Walsh series; net-sequences; integration errorCitations:
Zbl 0806.65013References:
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