An Historical Overview of Lattice Point Sets

Good lattice point sets are an important kind of low discrepancy points for multidimensional quadrature, simulation, experimental design, etc. The theoretical development of lattice point sets began over 40 years ago, but some important gaps in the theory remain. This article reviews the development of lattice point sets and highlights some open problems.

[1]  H. Weyl.  Über die Gleichverteilung von Zahlen mod. Eins , 1916 .

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

[3]  E. Hlawka.  Zur angenäherten Berechnung mehrfacher Integrale , 1962 .

[4]  I. F. Sharygin,et al.  A lower estimate for the error of quadrature formulae for certain classes of functions , 1963 .

[5]  S. K. Zaremba,et al.  Good lattice points, discrepancy, and numerical integration , 1966 .

[6]  W. Schmidt.  On irregularities of distribution vii , 1972 .

[7]  H. Niederreiter.  Existence of good lattice points in the sense of Hlawka , 1978 .

[8]  H. Niederreiter.  Quasi-Monte Carlo methods and pseudo-random numbers , 1978 .

[9]  William W. L. Chen.  On irregularities of distribution. , 1980 .

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

[11]  I. Sloan,et al.  Lattice methods for multiple integration: theory, error analysis and examples , 1987 .

[12]  Ian H. Sloan,et al.  Error bounds for the method of good lattice points , 1991 .

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

[14]  K. Fang,et al.  Number-theoretic methods in statistics , 1993 .

[15]  Harald Niederreiter.  Improved Error Bounds for Lattice Rules , 1993, J. Complex..

[16]  I. Sloan.  Lattice Methods for Multiple Integration , 1994 .

[17]  F. J. Hickernell.  Quadrature Error Bounds with Applications to Lattice Rules , 1997 .

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

[19]  Pierre L'Ecuyer,et al.  Tables of linear congruential generators of different sizes and good lattice structure , 1999, Math. Comput..

[20]  P. L’Ecuyer,et al.  Quasi-Monte Carlo Node Sets from Linear Congruential Generators , 2000 .

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

[22]  Fred J. Hickernell,et al.  Extensible Lattice Sequences for Quasi-Monte Carlo Quadrature , 2000, SIAM J. Sci. Comput..

[23]  Henryk Wozniakowski,et al.  Tractability of Multivariate Integration for Weighted Korobov Classes , 2001, J. Complex..

[24]  Ian H. Sloan,et al.  Component-by-component construction of good lattice rules , 2002, Math. Comput..

[25]  Frances Y. Kuo,et al.  On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces , 2002, Math. Comput..