The Discrepancy and Gain Coefficients of Scrambled Digital Nets

Digital sequences and nets are among the most popular kinds of low discrepancy sequences and sets and are often used for quasi-Monte Carlo quadrature rules. Several years ago Owen proposed a method of scrambling digital sequences and recently Faure and Tezuka have proposed another method. This article considers the discrepancy of digital nets under these scramblings. The first main result of this article is a formula for the discrepancy of a scrambled digital (?, t, m, s)-net in base b with n=?bm points that requires only O(n) operations to evaluate. The second main result is exact formulas for the gain coefficients of a digital (t, m, s)-net in terms of its generator matrices. The gain coefficients, as defined by Owen, determine both the worst-case and random-case analyses of quadrature error.

[1]  H. Niederreiter,et al.  Nets, ( t, s )-Sequences, and Algebraic Geometry , 1998 .

[2]  J. M. Sek,et al.  On the L2-discrepancy for anchored boxes , 1998 .

[3]  Jirí Matousek,et al.  On the L2-Discrepancy for Anchored Boxes , 1998, J. Complex..

[4]  Art B. Owen,et al.  Monte Carlo, Quasi-Monte Carlo, and Randomized Quasi-Monte Carlo , 2000 .

[5]  H. Niederreiter.  Low-discrepancy and low-dispersion sequences , 1988 .

[6]  Fred J. Hickernell,et al.  The Price of Pessimism for Multidimensional Quadrature , 2001, J. Complex..

[7]  David Yao,et al.  Quasi-Monte Carlo methods and their randomizations , 2002 .

[8]  Shu Tezuka,et al.  Another Random Scrambling of Digital ( t , s )-Sequences , 2002 .

[9]  Fred J. Hickernell,et al.  A generalized discrepancy and quadrature error bound , 1998, Math. Comput..

[10]  A. Owen.  Monte Carlo Variance of Scrambled Net Quadrature , 1997 .

[11]  Fred J. Hickernell,et al.  Monte Carlo and Quasi-Monte Carlo Methods 2000 , 2002 .

[12]  G. Larcher.  Digital Point Sets: Analysis and Application , 1998 .

[13]  J HickernellF,et al.  Implementing Scrambled Digital Nets , 2003 .

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

[15]  A. Owen.  Randomly Permuted (t,m,s)-Nets and (t, s)-Sequences , 1995 .

[16]  Harald Niederreiter,et al.  Monte-Carlo and Quasi-Monte Carlo Methods 1998 , 2000 .

[17]  P. Hellekalek,et al.  Random and Quasi-Random Point Sets , 1998 .

[18]  Harald Niederreiter.  Constructions of (t, m, s)-Nets , 2000 .

[19]  Art B. Owen,et al.  Scrambling Sobol' and Niederreiter-Xing Points , 1998, J. Complex..

[20]  Fred J. Hickernell,et al.  The Mean Square Discrepancy of Scrambled (t, s)-Sequences , 2000, SIAM J. Numer. Anal..

[21]  Rong-Xian Yue.  VARIANCE OF QUADRATURE OVER SCRAMBLED UNIONS OF NETS , 1999 .

[22]  Harald Niederreiter,et al.  The Microstructure of (t, m, s)-Nets , 2001, J. Complex..

[23]  A. Owen.  Scrambled net variance for integrals of smooth functions , 1997 .

[24]  F. J. Hickernell.  Lattice rules: how well do they measure up? in random and quasi-random point sets , 1998 .

[25]  Harald Niederreiter,et al.  Quasirandom points and global function fields , 1996 .

[26]  F. J. Hickernell.  What Affects the Accuracy of Quasi-Monte Carlo Quadrature? , 2000 .

[27]  Rong-Xian Yue,et al.  On the variance of quadrature over scrambled nets and sequences , 1999 .

[28]  R. Caflisch.  Monte Carlo and quasi-Monte Carlo methods , 1998, Acta Numerica.

[29]  Gerhard Larcher.  On the Distribution of Digital Sequences , 1998 .