Returns favorable parameters for Faure sequences.
[nsim,skip,leap] = lowdisc_fauresuggest ( dim , base ) [nsim,skip,leap] = lowdisc_fauresuggest ( dim , base , nsimmin )
a floating point integer, the spatial dimension.
a floating point integer, the base used in Faure's sequence.
a floating point integer, the minimum required number of simulations. Default nsimmin = 1.
a floating point integer, the number of simulations to perform, with nsim >= nsimmin.
a floating point integer, the number of initial elements to skip in the sequence.
a floating point integer, the number of elements to ignore each time an element is generated.
This routine provides favorable parameters to be used with a Faure sequence.
We use suggestions from : "Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators", B. L. Fox, ACM Transactions on Mathematical Software, Volume 12, Number 4, pages 362-376, 1986.
The citation is the following. "For QS greater than a half-dozen or so, one sees that the Faure sequence starts out as a series of points lying on slightly warped, nearly parallel hyperplanes with several points unduly close to (0,0,...,0) or (1,1,...,1). Our first two preliminary versions of the code initialized NEXTN to 0 (following [5]) and to 1, both leading to terrible results at s=20 and s=25. Deleting the first QS^4-2 terms, for example, eliminates these effects because of the scrambling then generated by the weigted sums in Faure's scheme."
With respect to the number of simulations, the citation p. 647 is the following. "[Sobol' says (personal communication, 1985) that] taking N equal to an integral power of qs favors the generator Faure, apparently because then again D*(N)=O((log(N)^(s-1)) [...]." And p. 374 : "[...] otherwise, use Faure and check for stability at N equal to successive powers of qs, starting at N = qs^3."
We return skip = b^4 - 2. We compute k = max( 3 , ceil(log(nsimmin)/log(b)) ) so that the number of simulations is greater than nsimmin. We compute nsim = b^k. We return leap = 0.
We might also be interested by the experiments in : "Computational investigations of low-discrepancy sequences", Kocis, L. and Whiten, W. J. 1997. ACM Trans. Math. Softw. 23, 2 (Jun. 1997), 266-294. Especially p. 277. The citation p. 277 is the following. "Properties of the Faure sequence leaped were studied experimentally using the Fox implementation for s = 40 with base b = 41 and leaps 2 to 31. Observation of the projections of the points of the Faure sequence leaped in two dimensional planes selected at random demonstrated a considerable improvement on the original Faure sequence. Any value of the leap (except L = b = 41) improved the Faure sequence. The best results were achieved for leaps L = 5, 6, 12 to 17 and then around 30."