TY - JOUR
T1 - Optimal quadrature for haar wavelet spaces
AU - Heinrich, Stefan
AU - Hickernell, Fred J.
AU - Yue, Rong Xian
N1 - Funding information:
This 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, by Shanghai NSF Grant 00JC14057, and by Shanghai Higher Education STF grant 01D01-1.
Publisher copyright:
© 2003 American Mathematical Society
PY - 2004/1
Y1 - 2004/1
N2 - This article considers the error of the scrambled equidistribution quadrature rules in the worst-case, random-case, and average-case settings. The underlying space of integrands is a Hilbert space of multidimensional Haar wavelet series, ℋwav. The asymptotic orders of the errors are derived for the case of the scrambled (λ, t, m, s)-nets and (t, s)-sequences. These rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands ℋ wav.
AB - This article considers the error of the scrambled equidistribution quadrature rules in the worst-case, random-case, and average-case settings. The underlying space of integrands is a Hilbert space of multidimensional Haar wavelet series, ℋwav. The asymptotic orders of the errors are derived for the case of the scrambled (λ, t, m, s)-nets and (t, s)-sequences. These rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands ℋ wav.
KW - High dimensional integration
KW - Lower bounds
KW - Monte Carlo methods
KW - Quasi-Monte Carlo methods
UR - https://www.jstor.org/stable/4099869
UR - http://www.scopus.com/inward/record.url?scp=0942278836&partnerID=8YFLogxK
U2 - 10.1090/S0025-5718-03-01531-X
DO - 10.1090/S0025-5718-03-01531-X
M3 - Journal article
AN - SCOPUS:0942278836
SN - 0025-5718
VL - 73
SP - 259
EP - 277
JO - Mathematics of Computation
JF - Mathematics of Computation
IS - 245
ER -