Extensible Lattice Sequences for Quasi-Monte Carlo Quadrature

Integration lattices are one of the main types of low discrepancy sets used in quasi-Monte Carlo methods. However, they have the disadvantage of being of fixed size. This article describes the construction of an infinite sequence of points, the first bm of which forms a lattice for any nonnegative integer m. Thus, if the quadrature error using an initial lattice is too large, the lattice can be extended without discarding the original points. Generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice. The spectral test used for finding pseudorandom number generators is one important example of such a discrepancy. The performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems.

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

[2]  A. Genz.  Numerical Computation of Multivariate Normal Probabilities , 1992 .

[3]  J. McNamee,et al.  Construction of fully symmetric numerical integration formulas of fully symmetric numerical integration formulas , 1967 .

[4]  U. Dieter,et al.  How to calculate shortest vectors in a lattice , 1975 .

[5]  A. Owen,et al.  Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension , 1997 .

[6]  Jerome Spanier,et al.  Quasi-Monte Carlo Methods for Particle Transport Problems , 1995 .

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

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

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

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

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

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

[13]  H. Wozniakowski.  Average case complexity of multivariate integration , 1991 .

[14]  William J. Morokoff.  Generating Quasi-Random Paths for Stochastic Processes , 1998, SIAM Rev..

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

[16]  S. Haber.  Parameters for integrating periodic functions of several variables , 1983 .

[17]  Terje O. Espelid,et al.  An adaptive algorithm for the approximate calculation of multiple integrals , 1991, TOMS.

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

[19]  U. Fincke,et al.  Improved methods for calculating vectors of short length in a lattice , 1985 .

[20]  S. Joe.  Letter section: Randomization of lattice rules for numerical multiple integration , 1990 .

[21]  Fred J. Hickernell,et al.  The asymptotic efficiency of randomized nets for quadrature , 1999, Math. Comput..

[22]  Anargyros Papageorgiou,et al.  Faster Evaluation of Multidimensional Integrals , 2000, ArXiv.

[23]  G. Marsaglia.  Random numbers fall mainly in the planes. , 1968, Proceedings of the National Academy of Sciences of the United States of America.

[24]  Frank Stenger,et al.  Con-struction of fully symmetric numerical integration formulas , 1967 .

[25]  Fred J. Hickernell,et al.  A Comparison of Random and Quasirandom Points for Multidimensional Quadrature , 1995 .

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

[27]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[28]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[29]  Christiane Lemieux,et al.  Efficiency improvement by lattice rules for pricing Asian options , 1998, 1998 Winter Simulation Conference. Proceedings (Cat. No.98CH36274).

[30]  Peter Hellekalek,et al.  On the assessment of random and quasi-random point sets , 1998 .

[31]  Paul A. Ruud,et al.  Simulation of multivariate normal rectangle probabilities and their derivatives theoretical and computational results , 1996 .

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

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

[34]  S. C. Zaremba.  Some applications of multidimensional integration by parts , 1968 .

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

[36]  P. Gruber,et al.  Funktionen von beschränkter Variation in der Theorie der Gleichverteilung , 1990 .

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

[38]  R. Cranley,et al.  Randomization of Number Theoretic Methods for Multiple Integration , 1976 .

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

[40]  R. Caflisch,et al.  Quasi-Monte Carlo integration , 1995 .

[41]  B. Keister.  Multidimensional quadrature algorithms , 1996 .

[42]  Fred J. Hickernell,et al.  The mean square discrepancy of randomized nets , 1996, TOMC.

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

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

[45]  Jerome Spanier,et al.  Quasi-Random Methods for Estimating Integrals Using Relatively Small Samples , 1994, SIAM Rev..

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

[47]  E. Hlawka.  Funktionen von beschränkter Variatiou in der Theorie der Gleichverteilung , 1961 .

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

[49]  P. L’Ecuyer,et al.  A Comparison of Monte Carlo, Lattice Rules and Other Low-Discrepancy Point Sets , 2000 .

[50]  T. Patterson,et al.  The optimum addition of points to quadrature formulae. , 1968 .

[51]  Joseph F. Traub,et al.  Faster Valuation of Financial Derivatives , 1995 .

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

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

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

[55]  J HickernellF,et al.  Computing Multivariate Normal Probabilities Using Rank-1 Lattice Sequences , 1997 .

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

[57]  Alan C. Genz.  A Lagrange Extrapolation Algorithm for Sequences of Approximations to Multiple Integrals , 1982 .

[58]  D. Hunter.  Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension , 2000 .

[59]  I. Olkin,et al.  Generating Correlation Matrices , 1984 .

[60]  Henryk Wozniakowski,et al.  An intractability result for multiple integration , 1997, Math. Comput..