Recent Advances in Randomized Quasi-Monte Carlo Methods

  • Pierre L’Ecuyer
  • Christiane Lemieux
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 46)


We survey some of the recent developments on quasi-Monte Carlo (QMC) methods, which, in their basic form, are a deterministic counterpart to the Monte Carlo (MC) method. Our main focus is the applicability of these methods to practical problems that involve the estimation of a high-dimensional integral. We review several QMC constructions and different randomizations that have been proposed to provide unbiased estimators and for error estimation. Randomizing QMC methods allows us to view them as variance reduction techniques. New and old results on this topic are used to explain how these methods can improve over the MC method in practice. We also discuss how this methodology can be coupled with clever transformations of the integrand in order to reduce the variance further. Additional topics included in this survey are the description of figures of merit used to measure the quality of the constructions underlying these methods, and other related techniques for multidimensional integration.


Monte Carlo Monte Carlo Estimator Lattice Rule Polynomial Lattice Digital Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Acworth, P., M. Broadie, and P. Glasserman. (1997). A comparison of some Monte Carlo and quasi-Monte Carlo techniques for option pricing. In Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, ed. P. Hellekalek and H. Niederreiter, Number 127 in Lecture Notes in Statistics, 1–18. Springer-Verlag.Google Scholar
  2. Åkesson, F., and J. P. Lehoczy. (2000). Path generation for quasi-Monte Carlo simulation of mortgage-backed securities. Management Science 46:1171–1187.zbMATHCrossRefGoogle Scholar
  3. Antonov, I. A., and V. M. Saleev. (1979). An economic method of computing LPT-sequences. Zh. Vychisl. Mat. Mat. Fiz. 19:243–245. In Russian.MathSciNetzbMATHGoogle Scholar
  4. Avramidis, A. N., and J. R. Wilson. (1996). Integrated variance reduction strategies for simulation. Operations Research 44:327–346.zbMATHCrossRefGoogle Scholar
  5. Bakhvalov, N. S. (1959). On approximate calculation of multiple integrals. Vestnik Moskovskogo Universiteta, Seriya Matematiki, Mehaniki, Astronomi, Fiziki, Himii 4:3–18. In Russian.Google Scholar
  6. Boyle, P., M. Broadie, and P. Glasserman. (1997). Monte Carlo methods for security pricing. Journal of Economic Dynamics & Control 21(8–9):1267–1321. Computational financial modelling.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Braaten, E., and G. Weller. (1979). An improved low-discrepancy sequence for multidimensional quasi-Monte Carlo integration. Journal of Computational Physics 33:249–258.zbMATHCrossRefGoogle Scholar
  8. Bratley, P., and B. L. Fox. (1988). Algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Transactions on Mathematical Software 14(1): 88–100.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Bratley, P., B. L. Fox, and H. Niederreiter. (1992). Implementation and tests of low-discrepancy sequences. ACM Transactions on Modeling and Computer Simulation 2:195–213.zbMATHCrossRefGoogle Scholar
  10. Bratley, P., B. L. Fox, and H. Niederreiter. (1994). Algorithm 738: Programs to generate Niederreiter’s low-discrepancy sequences. ACM Transactions on Mathematical Software 20:494–495.zbMATHCrossRefGoogle Scholar
  11. Caflisch, R. E.,W. Morokoff, and A. Owen. (1997). Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension. The Journal of Computational Finance 1(1): 27–46.CrossRefGoogle Scholar
  12. Caflisch, R. E., and B. Moskowitz. (1995). Modified Monte Carlo methods using quasi-random sequences. In Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, ed. H. Niederreiter and P. J.-S. Shiue, Number 106 in Lecture Notes in Statistics, 1–16. New York: Springer-Verlag.zbMATHCrossRefGoogle Scholar
  13. Cheng, J., and M. J. Druzdzel. (2000). Computational investigation of low-discrepancy sequences in simulation algorithms for bayesian networks. In Uncertainty in Artificial Intelligence Proceedings 2000, 72–81.Google Scholar
  14. Cochran, W. G. (1977). Sampling techniques. Second ed. New York: John Wiley and Sons.zbMATHGoogle Scholar
  15. Conway, J. H., and N. J. A. Sloane. (1999). Sphere packings, lattices and groups. 3rd ed. Grundlehren der Mathematischen Wissenschaften 290. New York: Springer-Verlag.zbMATHCrossRefGoogle Scholar
  16. Couture, R., and P. L’Ecuyer. (2000). Lattice computations for random numbers. Mathematics of Computation 69(230): 757–765.MathSciNetzbMATHCrossRefGoogle Scholar
  17. Couture, R., P. L’Ecuyer,and S. Tezuka. (1993). On the distribution of κ-dimensional vectors for simple and combined Tausworthe sequences. Mathematics of Computation 60(202): 749–761, S11–S16.MathSciNetzbMATHGoogle Scholar
  18. Coveyou, R. R., and R. D. MacPherson. (1967). Fourier analysis of uniform random number generators. Journal of the ACM 14:100–119.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Cranley, R., and T. N. L. Patterson. (1976). Randomization of number theoretic methods for multiple integration. SIAM Journal on Numerical Analysis 13(6): 904–914.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Davis, P., and P. Rabinowitz. (1984). Methods of numerical integration. second ed. New York: Academic Press.zbMATHGoogle Scholar
  21. Dieter, U. (1975). How to calculate shortest vectors in a lattice. Mathematics of Computation 29(131): 827–833.MathSciNetzbMATHCrossRefGoogle Scholar
  22. Duffie, D. (1996). Dynamic asset pricing theory. second ed. Princeton University Press.Google Scholar
  23. Efron, B., and C. Stein. (1981). The jackknife estimator of variance. Annals of Statistics 9:586–596.MathSciNetzbMATHCrossRefGoogle Scholar
  24. Entacher, K. (1997). Quasi-Monte Carlo methods for numerical integration of multivariate Haar series. BIT 37:846–861.MathSciNetzbMATHCrossRefGoogle Scholar
  25. Entacher, K., P. Hellekalek, and P. L’Ecuyer. (2000). Quasi-Monte Carlo node sets from linear congruential generators. In Monte Carlo and Quasi-Monte Carlo Methods 1998, ed. H. Niederreiter and J. Spanier, 188–198. Berlin: Springer.zbMATHCrossRefGoogle Scholar
  26. Faure, H. (1982). Discrépance des suites associées à un système de numération. Acta Arithmetica 61:337–351.zbMATHCrossRefGoogle Scholar
  27. Faure, H. (2001). Variations on (0, s)-sequences. Journal of Complexity. To appear.Google Scholar
  28. Faure, H., and S. Tezuka. (2001). A new generation of (0, s)-sequences. To appear.Google Scholar
  29. Fincke, U., and M. Pohst. (1985). Improved methods for calculating vectors of short length in a lattice, including a complexity analysis. Mathematics of Computation 44:463–471.MathSciNetzbMATHCrossRefGoogle Scholar
  30. Fishman, G. S. (1990), Jan. Multiplicative congruential random number generators with modulus 2β: An exhaustive analysis for β=32 and a partial analysis for β=48. Mathematics of Computation 54(189): 331–344.MathSciNetzbMATHGoogle Scholar
  31. Fishman, G. S., and L. S. Moore III. (1986). An exhaustive analysis of multiplicative congruential random number generators with modulus 231−1. SIAM Journal on Scientific and Statistical Computing 7(1): 24–45.MathSciNetzbMATHCrossRefGoogle Scholar
  32. Fox, B. L. (1986). Implementation and relative efficiency of quasirandom sequence generators. ACM Transactions on Mathematical Software 12:362–376.zbMATHCrossRefGoogle Scholar
  33. Fox, B. L. (1999). Strategies for quasi-Monte Carlo. Boston, MA: Kluwer Academic.CrossRefGoogle Scholar
  34. Friedel, I., and A. Keller. (2001). Fast generation of randomized low-discrepancy point sets. In Monte Carlo and Quasi-Monte Carlo Methods 2000, ed. K.-T. Fang, F. J. Hickernell, and H. Niederreiter: Springer. To appear.Google Scholar
  35. Golubov, B., A. Efimov, and V. Skvortsov. (1991). Walsh series and transforms: Theory and applications, Volume 64 of Mathematics and Applications: Soviet Series. Boston: Kluwer Academic Publishers.zbMATHCrossRefGoogle Scholar
  36. Haber, S. (1983). Parameters for integrating periodic functions of several variables. Mathematics of Computation 41:115–129.MathSciNetzbMATHCrossRefGoogle Scholar
  37. Halton, J. H. (1960). On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numerische Mathematik 2:84–90.MathSciNetzbMATHCrossRefGoogle Scholar
  38. Heinrich, S., F. J. Hickernell, and R.-X. Yue. (2001a). Integration of multivariate Haar wavelet series. Submitted.Google Scholar
  39. Heinrich, S., F. J. Hickernell, and R. X. Yue. (2001b). Optimal quadrature for Haar wavelet spaces. submitted.Google Scholar
  40. Hellekalek, P. (1998). On the assessment of random and quasirandom point sets. In Random and Quasi-Random Point Sets, ed. P. Hellekalek and G. Larcher, Volume 138 of Lecture Notes in Statistics, 49–108. New York: Springer.zbMATHCrossRefGoogle Scholar
  41. Hellekalek, P., and G. Larcher. (Eds.) (1998). Random and quasi-random point sets, Volume 138 of Lecture Notes in Statistics. New York: Springer.zbMATHGoogle Scholar
  42. Hellekalek, P., and H. Leeb. (1997). Dyadic diaphony. Acta Arithmetica 80:187–196.MathSciNetzbMATHCrossRefGoogle Scholar
  43. Hickernell, F. J. (1998a). A generalized discrepancy and quadrature error bound. Mathematics of Computation 67:299–322.MathSciNetzbMATHCrossRefGoogle Scholar
  44. Hickernell, F. J. (1998b). Lattice rules: How well do they measure up? In Random and Quasi-Random Point Sets, ed. P. Hellekalek and G. Larcher, Volume 138 of Lecture Notes in Statistics, 109–166. New York: Springer.CrossRefGoogle Scholar
  45. Hickernell, F. J. (1999). Goodness-of-fit statistics, discrepancies and robust designs. Statistical and Probability Letters 44:73–78.MathSciNetzbMATHCrossRefGoogle Scholar
  46. Hickernell, F. J. (2000). What affects accuracy of quasi-Monte Carlo quadrature? In Monte Carlo and Quasi-Monte Carlo Methods 1998, ed. H. Niederreiter and J. Spanier, 16–55. Berlin: Springer.zbMATHCrossRefGoogle Scholar
  47. Hickernell, F. J., and H. S. Hong. (1997). Computing multivariate normal probabilities using rank-1 lattice sequences. In Proceedings of the Workshop on Scientific Computing (Hong Kong), ed. G. H. Golub, S. H. Lui, F. T. Luk, and R. J. Plemmons, 209–215. Singapore: Springer-Verlag.Google Scholar
  48. Hickernell, F. J., H. S. Hong, P. L’Ecuyer, and C. Lemieux. (2001). Extensible lattice sequences for quasi-Monte Carlo quadrature. SIAM Journal on Scientific Computing 22(3): 1117–1138.MathSciNetzbMATHCrossRefGoogle Scholar
  49. Hickernell, F. J., and H. Wozniakowski. (2001). The price of pessimism for multidimensional quadrature. Journal of Complexity 17. To appear.Google Scholar
  50. Hlawka, E. (1961). Funktionen von beschränkter variation in der theorie der gleichverteilung. Ann. Mat. Pura. Appl. 54:325–333.MathSciNetzbMATHCrossRefGoogle Scholar
  51. Hlawka, E. (1962). Zur angenäherten berechnung mehrfacher integrale. Monatshefte für Mathematik 66:140–151.MathSciNetzbMATHCrossRefGoogle Scholar
  52. Hoeffding, W. (1948). A class of statistics with asymptotically normal distributions. Annals of Mathematical Statistics 19:293–325.MathSciNetzbMATHCrossRefGoogle Scholar
  53. Hong, H. S., and F. H. Hickernell. (2001). Implementing scrambled digital sequences. Submitted for publication.Google Scholar
  54. Knuth, D. E. (1998). The art of computer programming, volume 2: Seminumerical algorithms. Third ed. Reading, Mass.: Addison-Wesley.zbMATHGoogle Scholar
  55. Korobov, N. M. (1959). The approximate computation of multiple integrals. Dokl. Akad. Nauk SSSR 124:1207–1210. in Russian.MathSciNetzbMATHGoogle Scholar
  56. Korobov, N. M. (1960). Properties and calculation of optimal coefficients. Dokl. Akad. Nauk SSSR 132:1009–1012. in Russian.MathSciNetGoogle Scholar
  57. Larcher, G. (1998). Digital point sets: Analysis and applications. In Random and Quasi-Random Point Sets, ed. P. Hellekalek and G. Larcher, Volume 138 of Lecture Notes in Statistics, 167–222. New York: Springer.zbMATHCrossRefGoogle Scholar
  58. Larcher, G., A. Lauss, H. Niederreiter, and W. C. Schmid. (1996). Optimal polynomials for (t, m, s)-nets and numerical integration of multivariate Walsh series. SIAM Journal on Numerical Analysis 33(6):2239–2253.MathSciNetzbMATHCrossRefGoogle Scholar
  59. Larcher, G., H. Niederreiter, and W. C. Schmid. (1996). Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration. Monatshefte für Mathematik 121(3):231–253.MathSciNetzbMATHCrossRefGoogle Scholar
  60. Larcher, G., and G. Pirsic. (1999). Base change problems for generalized Walsh series and multivariate numerical integration. Pacific Journal of Mathematics 189:75–105.MathSciNetzbMATHCrossRefGoogle Scholar
  61. L’Ecuyer, P. (1994). Uniform random number generation. Annals of Operations Research 53:77–120.MathSciNetzbMATHCrossRefGoogle Scholar
  62. L’Ecuyer, P. (1996). Maximally equidistributed combined Tausworthe generators. Mathematics of Computation 65(213): 203–213.MathSciNetzbMATHCrossRefGoogle Scholar
  63. L’Ecuyer, P. (1999). Tables of linear congruential generators of different sizes and good lattice structure. Mathematics of Computation 68(225): 249–260.MathSciNetzbMATHCrossRefGoogle Scholar
  64. L’Ecuyer, P., and R. Couture. (1997). An implementation of the lattice and spectral tests for multiple recursive linear random number generators. INFORMS Journal on Computing 9(2): 206–217.MathSciNetzbMATHCrossRefGoogle Scholar
  65. L’Ecuyer, P., and C. Lemieux. (2000). Variance reduction via lattice rules. Management Science 46(9): 1214–1235.zbMATHCrossRefGoogle Scholar
  66. L’Ecuyer, P., and F. Panneton. (2000). A new class of linear feedback shift register generators. In Proceedings of the 2000 Winter Simulation Conference, ed. J. A. Joines, R. R. Barton, K. Kang, and P. A. Fishwick, 690–696. Pistacaway, NJ: IEEE Press.CrossRefGoogle Scholar
  67. Lemieux, C.(2000), May. L’utilisation de règles de réseau en simulation comme technique de réduction de la variance. Ph. D. thesis, Université de Montréal.Google Scholar
  68. Lemieux, C., M. Cieslak, and K. Luttmer. (2001). RandQMC user’s guide. In preparation.Google Scholar
  69. Lemieux, C., and P. L’Ecuyer. (2001). Selection criteria for lattice rules and other low-discrepancy point sets. Mathematics and Computers in Simulation 55(1–3): 139–148.MathSciNetzbMATHCrossRefGoogle Scholar
  70. Lemieux, C., and A. B. Owen. (2001). Quasi-regression and the relative importance of the ANOVA components of a function. In Monte Carlo and Quasi-Monte Carlo Methods 2000, ed. K.-T. Fang, F. J. Hickernell, and H. Niederreiter: Springer. To appear.Google Scholar
  71. Lidl, R., and H. Niederreiter. (1994). Introduction to finite fields and their applications. Revised ed. Cambridge: Cambridge University Press.zbMATHCrossRefGoogle Scholar
  72. Loh, W.-L. (1996a). A combinatorial central limit theorem for randomized orthogonal array sampling designs. Annals of Statistics 24:1209–1224.MathSciNetzbMATHCrossRefGoogle Scholar
  73. Loh, W.-L. (1996b). On Latin hypercube sampling. The Annals of Statistics 24:2058–2080.MathSciNetzbMATHCrossRefGoogle Scholar
  74. Maisonneuve, D. (1972). Recherche et utilisation des “bons treillis”, programmation et résultats numériques. In Applications of Number Theory to Numerical Analysis, ed. S. K. Zaremba, 121–201. New York: Academic Press.CrossRefGoogle Scholar
  75. Maize, E. (1981). Contributions to the theory of error reduction in quasi-Monte Carlo methods. Ph. D. thesis, Claremont Graduate School, Claremont, CA.Google Scholar
  76. Matousěk, J. (1998). On the L2-discrepancy for anchored boxes. Journal of Complexity 14:527–556.MathSciNetzbMATHCrossRefGoogle Scholar
  77. Matsumoto, M., and Y. Kurita. (1994). Twisted GFSR generators II. ACM Transactions on Modeling and Computer Simulation 4(3): 254–266.zbMATHCrossRefGoogle Scholar
  78. Matsumoto, M., and T. Nishimura. (1998). Mersenne twister: A 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Transactions on Modeling and Computer Simulation 8(1): 3–30.zbMATHCrossRefGoogle Scholar
  79. Mckay, M. D., R. J. Beckman, and W. J. Conover. (1979). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21:239–245.MathSciNetzbMATHGoogle Scholar
  80. Morohosi, H., and M. Fushimi. (2000). A practical approach to the error estimation of quasi-Monte Carlo integration. In Monte Carlo and Quasi-Monte Carlo Methods 1998, ed. H. Niederreiter and J. Spanier, 377–390. Berlin: Springer.zbMATHCrossRefGoogle Scholar
  81. Morokoff, W. J., and R. E. Caflisch. (1994). Quasi-random sequences and their discrepancies. SIAM Journal on Scientific Computing 15:1251–1279.MathSciNetzbMATHCrossRefGoogle Scholar
  82. Niederreiter, H. (1986). Multidimensional numerical integration using pseudorandom numbers. Mathematical Programming Study 27:17–38.MathSciNetzbMATHCrossRefGoogle Scholar
  83. Niederreiter, H. (1987). Point sets and sequences with small discrepancy. Monatshefte für Mathematik 104:273–337.MathSciNetzbMATHCrossRefGoogle Scholar
  84. Niederreiter, H. (1988). Low-discrepancy and low-dispersion sequences. Journal of Number Theory 30:51–70.MathSciNetzbMATHCrossRefGoogle Scholar
  85. Niederreiter, H. (1992a). Low-discrepancy point sets obtained by digital constructions over finite fields. Czechoslovak Math. Journal 42:143–166.MathSciNetzbMATHGoogle Scholar
  86. Niederreiter, H. (1992b). Random number generation and quasi-Monte Carlo methods, Volume 63 of SIAM CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia: SIAM.zbMATHCrossRefGoogle Scholar
  87. Niederreiter, H., and G. Pirsic. (2001). Duality for digital nets and its applications. Acta Arithmetica 97:173–182.MathSciNetzbMATHCrossRefGoogle Scholar
  88. Niederreiter, H., and C. Xing. (1997). The algebraic-geometry approach to low-discrepancy sequences. In Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, ed. P. Hellekalek, G. Larcher, H. Niederreiter, and P. Zinterhof, Volume 127 of Lecture Notes in Statistics, 139–160. New York: Springer-Verlag.Google Scholar
  89. Niederreiter, H., and C. Xing. (1998). Nets, (t, s)-sequences, and algebraic geometry. In Random and Quasi-Random Point Sets, ed. P. Hellekalek and G. Larcher, Volume 138 of Lecture Notes in Statistics, 267–302. New York: Springer.zbMATHCrossRefGoogle Scholar
  90. Ökten, G. (1996). A probabilistic result on the discrepancy of a hybrid-Monte Carlo sequence and applications. Monte Carlo methods and Applications 2:255–270.MathSciNetzbMATHCrossRefGoogle Scholar
  91. Owen, A. B. (1992a). A central limit theorem for Latin hypercube sampling. Journal of the Royal Statistical Society B 54(2): 541–551.MathSciNetzbMATHGoogle Scholar
  92. Owen, A. B. (1992b). Orthogonal arrays for computer experiments, integration and visualization. Statistica Sinica 2:439–452.MathSciNetzbMATHGoogle Scholar
  93. Owen, A. B. (1994). Lattice sampling revisited: Monte Carlo variance of means over randomized orthogonal arrays. Annals of Statistics 22:930–945.MathSciNetzbMATHCrossRefGoogle Scholar
  94. Owen, A. B. (1995). Randomly permuted (t, m, s)-nets and (t, s)-sequences. In Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, ed. H. Niederreiter and P. J.-S. Shiue, Number 106 in Lecture Notes in Statistics, 299–317. Springer-Verlag.Google Scholar
  95. Owen, A. B. (1997). Monte Carlo variance of scrambled equidistribution quadrature. SIAM Journal on Numerical Analysis 34(5): 1884–1910.MathSciNetzbMATHCrossRefGoogle Scholar
  96. Owen, A. B. (1998a). Latin supercube sampling for very high-dimensional simulations. ACM Transactions of Modeling and Computer Simulation 8(1): 71–102.zbMATHCrossRefGoogle Scholar
  97. Owen, A. B. (1998b). Scrambling Sobol and Niederreiter-Xing points. Journal of Complexity 14:466–489.MathSciNetzbMATHCrossRefGoogle Scholar
  98. Pagès, G. (1997). A space quantization method for numerical integration. Journal of Computational and Applied Mathematics 89:1–38.MathSciNetzbMATHCrossRefGoogle Scholar
  99. Paskov, S., and J. Traub. (1995). Faster valuation of financial derivatives. Journal of Portfolio Management 22:113–120.CrossRefGoogle Scholar
  100. Pirsic, G. (2001). A software implementation of Niederreiter-Xing sequences. In Monte Carlo and Quasi-Monte Carlo Methods 2000, ed. K.-T. Fang, F. J. Hickernell, and H. Niederreiter: Springer. To appear.Google Scholar
  101. Pirsic, G., and W. C. Schmid. (2001). Calculation of the quality parameter of digital nets and application to their construction. Journal of Complexity. To appear.Google Scholar
  102. Sloan, I. H., and S. Joe. (1994). Lattice methods for multiple integration. Oxford: Clarendon Press.zbMATHGoogle Scholar
  103. Sloan, I. H., and L. Walsh. (1990). A computer search of rank-2 lattice rules for multidimensional quadrature. Mathematics of Computation 54:281–302.MathSciNetzbMATHGoogle Scholar
  104. Sobol’, I. M. (1967). The distribution of points in a cube and the approximate evaluation of integrals. U.S.S.R. Comput. Math. and Math. Phys. 7:86–112.MathSciNetzbMATHCrossRefGoogle Scholar
  105. Sobol’, I. M. (1969). Multidimensional quadrature formulas and Haar functions. Moskow: Nauka. In Russian.zbMATHGoogle Scholar
  106. Sobol’, I. M.(1976). Uniformly distributed sequences with an additional uniform property. USSR Comput. Math. Math. Phys. Academy of Sciences 16:236–242.zbMATHCrossRefGoogle Scholar
  107. Sobol’, I. M., and Y. L. Levitan. (1976). The production of points uniformly distributed in a multidimensional. Technical Report Preprint 40, Institute of Applied Mathematics, USSR Academy of Sciences. In Russian.Google Scholar
  108. Spanier, J. (1995). Quasi-Monte Carlo methods for particle transport problems. In Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, ed. H. Niederreiter and P. J.-S. Shiue, Volume 106 of Lecture Notes in Statistics, 121–148. New York: Springer-Verlag.zbMATHCrossRefGoogle Scholar
  109. Spanier, J., and E. H. Maize. (1994). Quasi-random methods for estimating integrals using relatively small samples. SIAM Review 36:18–44.MathSciNetzbMATHCrossRefGoogle Scholar
  110. Tan, K. S., and P. P. Boyle. (2000). Applications of randomized low discrepancy sequences to the valuation of complex securities. Journal of Economic Dynamics and Control 24:1747–1782.MathSciNetzbMATHCrossRefGoogle Scholar
  111. Tausworthe, R. C. (1965). Random numbers generated by linear recurrence modulo two. Mathematics of Computation 19:201–209.MathSciNetzbMATHCrossRefGoogle Scholar
  112. Tezuka, S. (1987). Walsh-spectral test for GFSR pseudorandom numbers. Communications of the ACM 30(8): 731–735.MathSciNetzbMATHCrossRefGoogle Scholar
  113. Tezuka, S. (1995). Uniform random numbers: Theory and practice. Norwell, Mass.: Kluwer Academic Publishers.zbMATHCrossRefGoogle Scholar
  114. Tezuka, S., and P. L’Ecuyer. (1991). Efficient and portable combined Tausworthe random number generators. ACM Transactions on Modeling and Computer Simulation 1(2): 99–112.zbMATHCrossRefGoogle Scholar
  115. Tezuka, S., and T. Tokuyama. (1994). A note on polynomial arithmetic analogue of Halton sequences. ACM Transactions on Modeling and Computer Simulation 4:279–284.zbMATHCrossRefGoogle Scholar
  116. Tootill, J. P. R., W. D. Robinson, and D. J. Eagle. (1973). An asymptotically random Tausworthe sequence. Journal of the ACM 20:469–481.zbMATHCrossRefGoogle Scholar
  117. Tuffin, B. (1996). On the use of low-discrepancy sequences in Monte Carlo methods. Technical Report No. 1060, I.R.I.S.A., Rennes, France.zbMATHGoogle Scholar
  118. Tuffin, B. (1998). Variance reduction order using good lattice points in Monte Carlo methods. Computing 61:371–378.MathSciNetzbMATHCrossRefGoogle Scholar
  119. Wang, D., and A. Compagner. (1993). On the use of reducible polynomials as random number generators. Mathematics of Computation 60:363–374.MathSciNetzbMATHCrossRefGoogle Scholar
  120. Wang, X., and F. J. Hickernell. (2000). Randomized Halton sequences. Math. Comput. Modelling 32:887–899.MathSciNetzbMATHCrossRefGoogle Scholar
  121. Yakowitz, S., J. E. Krimmel, and F. Szidarovszky. (1978). Weighted Monte Carlo integration. SIAM Journal on Numerical Analysis 15:1289–1300.MathSciNetzbMATHCrossRefGoogle Scholar
  122. Yakowitz, S., P. L’Ecuyer, and F. Vázquez-Abad. (2000). Global stochastic optimization with low-discrepancy point sets. Operations Research 48(6): 939–950.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2002

Authors and Affiliations

  • Pierre L’Ecuyer
    • 1
  • Christiane Lemieux
    • 2
  1. 1.Département d’Informatique et de Recherche OpérationnelleUniversité de MontréalMontréalCanada
  2. 2.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

Personalised recommendations