An Algorithm to Compute Bounds for the Star Discrepancy

We propose an algorithm to compute upper and lower bounds for the star discrepancy of an arbitrary sequence of points in the s-dimensional unit cube. The method is based on a particular partition of the unit cube into subintervals and on a specialized procedure for orthogonal range counting. The cardinality of the partition depends on the dimension and on an accuracy parameter that has to be specified. We have implemented this method and here we present results of some computational experiments obtained with this implementation.

[1]  George Marsaglia,et al.  In: Applications of Number Theory to Numerical Analysis , 1972 .

[2]  David Eppstein,et al.  Computing the discrepancy , 1993, SCG '93.

[3]  S. Tezuka.  Uniform Random Numbers: Theory and Practice , 1995 .

[4]  Tony Warnock,et al.  Computational investigations of low-discrepancy point-sets. , 1972 .

[5]  E. Novak,et al.  The inverse of the star-discrepancy depends linearly on the dimension , 2001 .

[6]  H. Keng,et al.  Applications of number theory to numerical analysis , 1981 .

[7]  J. Halton.  On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals , 1960 .

[8]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[9]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[10]  H. Niederreiter.  Point sets and sequences with small discrepancy , 1987 .

[11]  H. Faure.  Discrépance de suites associées à un système de numération (en dimension s) , 1982 .

[12]  Kurt Mehlhorn,et al.  Multi-dimensional searching and computational geometry , 1984 .

[13]  I. Sobol.  On the distribution of points in a cube and the approximate evaluation of integrals , 1967 .

[14]  Henri Faure.  Multidimensional quasi-Monte-Carlo methods , 1994 .

[15]  George S. Lueker,et al.  A data structure for orthogonal range queries , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[16]  Pierre L'Ecuyer,et al.  Good Parameters and Implementations for Combined Multiple Recursive Random Number Generators , 1999, Oper. Res..

[17]  Harald Niederreiter,et al.  Discrepancy and convex programming , 1972 .

[18]  I. A. Antonov,et al.  An economic method of computing LPτ-sequences , 1979 .

[19]  Henri Faure.  Méthodes quasi-Monte-Carlo multidimensionnelles , 1994, Theor. Comput. Sci..

[20]  Paul Bratley,et al.  Algorithm 659: Implementing Sobol's quasirandom sequence generator , 1988, TOMS.

[21]  Russel E. Caflisch,et al.  Quasi-Random Sequences and Their Discrepancies , 1994, SIAM J. Sci. Comput..

[22]  Kurt Mehlhorn,et al.  Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry , 2012, EATCS Monographs on Theoretical Computer Science.

[23]  Jon Louis Bentley,et al.  Decomposable Searching Problems , 1979, Inf. Process. Lett..

[24]  K. F. Roth.  On irregularities of distribution , 1954 .

[25]  K. Fang,et al.  Application of Threshold-Accepting to the Evaluation of the Discrepancy of a Set of Points , 1997 .

[26]  Eric Thiémard.  Computing Bounds for the Star Discrepancy , 2000, Computing.