TY - JOUR
T1 - The discrepancy and gain coefficients of scrambled digital nets
AU - Yue, Rong Xian
AU - Hickernell, Fred J.
N1 - Funding Information:
Digital sequences and nets are among the most popular kinds of low discrepancy sequences and sets and are often used for quasi-Monte Carlo quadrature rules. Several years ago Owen proposed a method of scrambling digital sequences and recently Faure and Tezuka have proposed another method. This article considers the discrepancy of digital nets under these scramblings. The first main result of this article is a formula for the discrepancy of a scrambled digital (l, t, m, s)-net in base b with n=lbm points that requires only O(n) operations to evaluate. The second main result is exact formulas for the gain coefficients of a digital (t, m, s)-net in terms of its generator matrices. The gain coefficients, as defined by Owen, determine both the worst-case and random-case analyses of quadrature error. © 2002 Elsevier Science (USA) 1This work was partially supported by a Hong Kong Research Grants Council Grant HKBU/2030/99P, by Hong Kong Baptist University Grant FRG/97-98/II-99, and by Shanghai NSF Grant 00JC14057 and a Shanghai Higher Education STF Grant.
PY - 2002/3
Y1 - 2002/3
N2 - Digital sequences and nets are among the most popular kinds of low discrepancy sequences and sets and are often used for quasi-Monte Carlo quadrature rules. Several years ago Owen proposed a method of scrambling digital sequences and recently Faure and Tezuka have proposed another method. This article considers the discrepancy of digital nets under these scramblings. The first main result of this article is a formula for the discrepancy of a scrambled digital (λ, t, m, s)-net in base b with n = λbm points that requires only O(n) operations to evaluate. The second main result is exact formulas for the gain coefficients of a digital (t, m, s)-net in terms of its generator matrices. The gain coefficients, as defined by Owen, determine both the worst-case and random-case analyses of quadrature error.
AB - Digital sequences and nets are among the most popular kinds of low discrepancy sequences and sets and are often used for quasi-Monte Carlo quadrature rules. Several years ago Owen proposed a method of scrambling digital sequences and recently Faure and Tezuka have proposed another method. This article considers the discrepancy of digital nets under these scramblings. The first main result of this article is a formula for the discrepancy of a scrambled digital (λ, t, m, s)-net in base b with n = λbm points that requires only O(n) operations to evaluate. The second main result is exact formulas for the gain coefficients of a digital (t, m, s)-net in terms of its generator matrices. The gain coefficients, as defined by Owen, determine both the worst-case and random-case analyses of quadrature error.
UR - http://www.scopus.com/inward/record.url?scp=0036222370&partnerID=8YFLogxK
U2 - 10.1006/jcom.2001.0630
DO - 10.1006/jcom.2001.0630
M3 - Journal article
AN - SCOPUS:0036222370
SN - 0885-064X
VL - 18
SP - 135
EP - 151
JO - Journal of Complexity
JF - Journal of Complexity
IS - 1
ER -