Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Quasi-random sequences and their discrepancies
SIAM Journal on Scientific Computing
Programs to generate Niederreiter's low-discrepancy sequences
ACM Transactions on Mathematical Software (TOMS)
Numerical Recipes in FORTRAN: The Art of Scientific Computing
Numerical Recipes in FORTRAN: The Art of Scientific Computing
Algorithm 823: Implementing scrambled digital sequences
ACM Transactions on Mathematical Software (TOMS)
Robust monte carlo methods for light transport simulation
Robust monte carlo methods for light transport simulation
Quasi-random integration in high dimensions
Mathematics and Computers in Simulation
Applied Numerical Mathematics
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We have computed true star-discrepancies D*(N) for Halton and Niederreiter point sequences in dimensions s = 2, 3, 4, and 6 with N up to 250000, 10000, 2000 and 300, respectively. For comparison, we also calculated some L2 discrepancies T*. The mean behaviour of D*(N) can well be approximated by power laws with an exponent between about -0.7 and -0.9, which slowly decreases with N. This behaviour is far from the generally assumed asymptotic one ∼ C(s)(log N)s/N. The factor between the true and the asymptotic behaviour increases strongly with s and reaches many orders of magnitude for large s. The ratios of D*(N) for different low discrepancy sequences are not proportional to the presumed asymptotic pre-factors C(s). Especially for the range of bases investigated, Niederreiter sequences have about the same D* as Halton ones in a range of N, which we conjecture to be at least of the order of 1010.