TY - JOUR
T1 - The mean square discrepancy of scrambled (t, s)-sequences
AU - Hickernell, Fred J.
AU - Yue, Rong-Xian
N1 - This research was supported by a Hong Kong Research Grants Council grant RGC/97-98/47 and Hong Kong Baptist University grants FRG/96-97/II-67 and FRG/97-98/II-99.
PY - 2000/1
Y1 - 2000/1
N2 - The discrepancy arises in the worst-case error analysis for quasi-Monte Carlo quadrature rules. Low discrepancy sets yield good quadrature rules. This article considers the mean square discrepancies for scrambled (λ, t, m, s)-nets and (i, s)-sequences in base b. It is found that the mean square discrepancy for scrambled nets and sequences is never more than a constant multiple of that under simple Monte Carlo sampling. If the reproducing kernel defining the discrepancy satisfies a Lipschitz condition with respect to one of its variables separately, then the asymptotic order of the root mean square discrepancy is O(n-1[log n](s-1)/2) for scrambled nets. If the reproducing kernel satisfies a Lipschitz condition with respect to both of its variables, then the asymptotic order of the root mean square discrepancy is O(n-3/2[log n](s-1)/2) for scrambled nets. For an arbitrary number of points taken from a (t, s)-sequence, the root mean square discrepancy appears to be no better than O(n-1[log n](s-1)/2), regardless of the smoothness of the reproducing kernel.
AB - The discrepancy arises in the worst-case error analysis for quasi-Monte Carlo quadrature rules. Low discrepancy sets yield good quadrature rules. This article considers the mean square discrepancies for scrambled (λ, t, m, s)-nets and (i, s)-sequences in base b. It is found that the mean square discrepancy for scrambled nets and sequences is never more than a constant multiple of that under simple Monte Carlo sampling. If the reproducing kernel defining the discrepancy satisfies a Lipschitz condition with respect to one of its variables separately, then the asymptotic order of the root mean square discrepancy is O(n-1[log n](s-1)/2) for scrambled nets. If the reproducing kernel satisfies a Lipschitz condition with respect to both of its variables, then the asymptotic order of the root mean square discrepancy is O(n-3/2[log n](s-1)/2) for scrambled nets. For an arbitrary number of points taken from a (t, s)-sequence, the root mean square discrepancy appears to be no better than O(n-1[log n](s-1)/2), regardless of the smoothness of the reproducing kernel.
KW - (t, m, s)-nets
KW - Multidimensional integration
KW - Quadrature
KW - Reproducing kernel Hilbert spaces
UR - http://www.scopus.com/inward/record.url?scp=0034432296&partnerID=8YFLogxK
U2 - 10.1137/S0036142999358019
DO - 10.1137/S0036142999358019
M3 - Journal article
AN - SCOPUS:0034432296
SN - 0036-1429
VL - 38
SP - 1089
EP - 1112
JO - SIAM Journal on Numerical Analysis
JF - SIAM Journal on Numerical Analysis
IS - 4
ER -