The discrepancy and gain coefficients of scrambled digital nets
Journal of Complexity
Integration of Multivariate Haar Wavelet Series
WAA '01 Proceedings of the Second International Conference on Wavelet Analysis and Its Applications
The distribution of the discrepancy of scrambled digital (t, m, s)-nets
Mathematics and Computers in Simulation - Special issue: 3rd IMACS seminar on Monte Carlo methods - MCM 2001
Algorithm 823: Implementing scrambled digital sequences
ACM Transactions on Mathematical Software (TOMS)
Journal of Complexity
Variance with alternative scramblings of digital nets
ACM Transactions on Modeling and Computer Simulation (TOMACS)
I-binomial scrambling of digital nets and sequences
Journal of Complexity
Optimal quadrature for Haar wavelet spaces
Mathematics of Computation
Constructions of (t ,m,s)-nets and (t,s)-sequences
Finite Fields and Their Applications
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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 $(\lam,t,m,s)$-nets and (t,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.