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Calculation of Discrepancy Measures and Applications

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Book cover A Panorama of Discrepancy Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2107))

Abstract

In this book chapter we survey known approaches and algorithms to compute discrepancy measures of point sets. After providing an introduction which puts the calculation of discrepancy measures in a more general context, we focus on the geometric discrepancy measures for which computation algorithms have been designed. In particular, we explain methods to determine L 2-discrepancies and approaches to tackle the inherently difficult problem to calculate the star discrepancy of given sample sets. We also discuss in more detail three applications of algorithms to approximate discrepancies.

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Notes

  1. 1.

    BMO stands for “bounded mean oscillation”.

  2. 2.

    NP stands for “non-deterministic polynomial time”.

References

  1. A.V. Aho, J.E. Hopcroft, J.D. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, 1974)

    MATH  Google Scholar 

  2. E.I. Atanassov, On the discrepancy of the Halton sequences. Math. Balkanica (N. S.) 18, 15–32 (2004)

    MathSciNet  MATH  Google Scholar 

  3. J. Beck, Some upper bounds in the theory of irregulatities of distribution. Acta Arithmetica 43, 115–130 (1984)

    MathSciNet  MATH  Google Scholar 

  4. J. Beck, On the discrepancy of convex plane sets. Monatsh. Math., 91–106 (1988)

    Google Scholar 

  5. J. Beck, W.W.L. Chen, Irregularities of Distribution (Cambridge University Press, Cambridge, 1987)

    MATH  Google Scholar 

  6. J.L. Bentley, 1977, Algorithms for Klee’s rectangle problem.Unpublished notes, Dept. of Computer Science, Carnegie Mellon University

    Google Scholar 

  7. D. Bilyk, Roth’s orthogonal function method in discrepancy theory and some new connections, in A Panorama of Discrepancy Theory, ed. by W.W.L. Chen, A. Srivastav, G. Travaglini (Springer, Berlin, 2012)

    Google Scholar 

  8. D. Bilyk, M.T. Lacey, I. Parissis, A. Vagharshakyan, Exponential sqared integrability of the discrepancy function in two dimensions. Mathematika 55, 1–27 (2009)

    MathSciNet  MATH  Google Scholar 

  9. E. Braaten, G. Weller, An improved low-discrepancy sequence for multidimensional quasi-Monte Carlo integration. J. Comput. Phys. 33, 249–258 (1979)

    MATH  Google Scholar 

  10. L. Brandolini, C. Choirat, L. Colzani, G. Gigante, R. Seri, G. Travaglini, Quadrature rules and distribution of points on manifolds, 2011. To appear in: Ann. Scuola Norm. Sup. Pisa Cl. Sci.

    Google Scholar 

  11. P. Bundschuh, Y.C. Zhu, A method for exact calculation of the discrepancy of low-dimensional point sets I. Abh. Math. Sem. Univ. Hamburg 63, 115–133 (1993)

    MathSciNet  MATH  Google Scholar 

  12. T.M. Chan, A (slightly) faster algorithm for Klee’s measure problem. Comput. Geom. 43(3), 243–250 (2010)

    MathSciNet  MATH  Google Scholar 

  13. B. Chazelle, The Discrepancy Method (Cambridge University Press, Cambridge, 2000)

    MATH  Google Scholar 

  14. J. Chen, B. Chor, M. Fellows, X. Huang, D.W. Juedes, I.A. Kanj, G. Xia, Tight lower bounds for certain parameterized NP-hard problems. Inf. Comput. 201(2), 216–231 (2005)

    MathSciNet  MATH  Google Scholar 

  15. J. Chen, X. Huang, I.A. Kanj, G. Xia, Strong computational lower bounds via parameterized complexity. J. Comput. Syst. Sci. 72(8), 1346–1367 (2006)

    MathSciNet  MATH  Google Scholar 

  16. W.W.L. Chen, M.M. Skriganov, Upper bounds in irregularities of point distribution, in A Panorama of Discrepancy Theory, ed. by W.W.L. Chen, A. Srivastav, G. Travaglini (Springer, Berlin, 2012)

    Google Scholar 

  17. W.W.L. Chen, G. Travaglini, Discrepancy with respect to convex polygons. J. Complexity 23, 662–672 (2007)

    MathSciNet  MATH  Google Scholar 

  18. H. Chi, M. Mascagni, T. Warnock, On the optimal Halton sequence. Math. Comput. Simul. 70, 9–21 (2005)

    MathSciNet  MATH  Google Scholar 

  19. S.C. Chuang, Y.C. Hung, Uniform design over general input domains with applications to target region estimation in computer experiments. Comput. Stat. Data Anal. 54, 219–232 (2010). http://dx.doi.org/10.1016/j.csda.2010.01.032

  20. T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms (3. ed.)(MIT Press, Cambridge, 2009)

    Google Scholar 

  21. M. de Berg, O. Cheong, M. van Kreveld, M.H. Overmars, Computational Geometry. Algorithms and Applications (Springer, Berlin, 2008)

    Google Scholar 

  22. L. de Clerck, A method for exact calculation of the star-discrepancy of plane sets applied to the sequence of Hammersley. Monatsh. Math. 101, 261–278 (1986)

    MathSciNet  MATH  Google Scholar 

  23. F.-M. De Rainville, C. Gagné, O. Teytaud, D. Laurendeau, Evolutionary optimization of low-discrepancy sequences. ACM Trans. Model. Comput. Simul. 22 (2012). Article 9

    Google Scholar 

  24. J. Dick, F. Pillichshammer, Digital Nets and Sequences (Cambridge University Press, Cambridge, 2010)

    MATH  Google Scholar 

  25. J. Dick, F. Pillichshammer, Discrepancy theory and quasi-Monte Carlo integration, in A Panorama of Discrepancy Theory, ed. by W.W.L. Chen, A. Srivastav, G. Travaglini (Springer, Berlin, 2012)

    Google Scholar 

  26. J. Dick, I.H. Sloan, X. Wang, H. Woźniakowski, Liberating the weights. J. Complexity 20(5), 593–623 (2004). http://dx.doi.org/10.1016/j.jco.2003.06.002

  27. D.P. Dobkin, D. Eppstein, D.P. Mitchell, Computing the discrepancy with applications to supersampling patterns. ACM Trans. Graph. 15, 354–376 (1996)

    Google Scholar 

  28. B. Doerr, Generating randomized roundings with cardinality constraints and derandomizations, in Proceedings of the 23rd Annual Symposium on Theoretical Aspects of Computer Science (STACS’06), ed. by B. Durand, W. Thomas LNiCS, vol. 3884 (Springer, Berlin and Heidelberg, 2006)

    Google Scholar 

  29. B. Doerr, M. Gnewuch, Construction of low-discrepancy point sets of small size by bracketing covers and dependent randomized rounding, in Monte Carlo and Quasi-Monte Carlo Methods 2006, ed. by A. Keller, S. Heinrich, H. Niederreiter (Springer, Berlin and Heidelberg, 2008)

    Google Scholar 

  30. B. Doerr, M. Gnewuch, A. Srivastav, Bounds and constructions for the star discrepancy via δ-covers. J. Complexity 21, 691–709 (2005)

    MathSciNet  MATH  Google Scholar 

  31. B. Doerr, M. Gnewuch, M. Wahlström, Implementation of a component-by-component algorithm to generate low-discrepancy samples, in Monte Carlo and Quasi-Monte Carlo Methods 2008, ed. by P. L’Ecuyer, A.B. Owen (Springer, Berlin and Heidelberg, 2009)

    Google Scholar 

  32. B. Doerr, M. Gnewuch, M. Wahlström, Algorithmic construction of low-discrepancy point sets via dependent randomized rounding. J. Complexity 26, 490–507 (2010)

    Google Scholar 

  33. B. Doerr, M. Gnewuch, P. Kritzer, F. Pillichshammer, Component-by-component construction of low-discrepancy point sets of small size. Monte Carlo Methods Appl. 14, 129–149 (2008)

    MathSciNet  MATH  Google Scholar 

  34. B. Doerr, M. Wahlström, Randomized rounding in the presence of a cardinality constraint, in Proceedings of the Workshop on Algorithm Engineering and Experiments (ALENEX 2009), ed. by I. Finocchi, J. Hershberger (SIAM, Philadelphia, 2009)

    Google Scholar 

  35. C. Doerr (nee Winzen), F.-M. De Rainville, Constructing Low Star Discrepancy Point Sets with Genetic Algorithms, in Proc. of Genetic and Evolutionary Computation Conference (GECCO’13) (ACM, New York, 2013)

    Google Scholar 

  36. R.G. Downey, M.R. Fellows, Parameterized Complexity (Springer, Berlin, 1999)

    Google Scholar 

  37. M. Drmota, R.F. Tichy, Sequences, Discrepancies and Applications. Lecture Notes in Mathematics, vol. 1651 (Springer, Berlin and Heidelberg, 1997)

    Google Scholar 

  38. K.-T. Fang, D.K.J. Lin, P. Winker, Y. Zhang, Uniform design: theory and application. Technometrics 42, 237–248 (2000)

    MathSciNet  MATH  Google Scholar 

  39. H. Faure, Discrépance de suites associées à un systéme de numération (en dimension un). Bull. Soc. Math. France 109, 143–182 (1981)

    MathSciNet  MATH  Google Scholar 

  40. H. Faure, Discrépance de suites associées à un systéme de numération (en dimension s). Acta Arithmetica 41, 338–351 (1982)

    MathSciNet  Google Scholar 

  41. H. Faure, C. Lemieux, Generalized Halton sequences in 2008: A comparative study. ACM Trans. Model. Comput. Simul. 19, 15–131 (2009)

    Google Scholar 

  42. J. Flum, M. Grohe, Parameterized Complexity Theory (Springer, New York, 2006)

    Google Scholar 

  43. K. Frank, S. Heinrich, Computing discrepancies of Smolyak quadrature rules. J. Complexity 12, 287–314 (1996)

    MathSciNet  MATH  Google Scholar 

  44. M.L. Fredman, B.W. Weide, On the complexity of computing the measure of \(\bigcup [a_{i},b_{i}]\). Commun. ACM 21(7), 540–544 (1978)

    Google Scholar 

  45. H. Gabai, On the discrepancy of certain sequences mod 1. Indaq. Math. 25, 603–605 (1963)

    MathSciNet  Google Scholar 

  46. M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W. H. Freeman and Co., San Francisco, 1979)

    MATH  Google Scholar 

  47. T. Gerstner, M. Griebel, Numerical integration using sparse grids. Numer. Algorithms, 209–232 (1998)

    Google Scholar 

  48. P. Giannopoulus, C. Knauer, M. Wahlström, D. Werner, Hardness of discrepancy computation and epsilon-net verification in high dimensions. J. Complexity 28, 162–176 (2012)

    MathSciNet  Google Scholar 

  49. M. Gnewuch, Bounds for the average L p-extreme and the L ∞-extreme discrepancy. Electron. J. Combin., 1–11 (2005). Research Paper 54

    Google Scholar 

  50. M. Gnewuch, Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy. J. Complexity 24, 154–172 (2008)

    MathSciNet  MATH  Google Scholar 

  51. M. Gnewuch, Construction of minimal bracketing covers for rectangles. Electron. J. Combin. 15 (2008). Research Paper 95

    Google Scholar 

  52. M. Gnewuch, Entropy, Randomization, Derandomization, and Discrepancy, in Monte Carlo and Quasi-Monte Carlo Methods 2010, ed. by L. Plaskota, H. Woźniakowski (Springer, Berlin and Heidelberg, 2012)

    Google Scholar 

  53. M. Gnewuch, Weighted geometric discrepancies and numerical integration on reproducing kernel Hilbert spaces. J. Complexity 28, 2–17 (2012)

    MathSciNet  MATH  Google Scholar 

  54. M. Gnewuch, A. Roşca, On G-discrepancy and mixed Monte Carlo and quasi-Monte Carlo sequences. Acta Univ. Apul. Math. Inform. 18, 97–110 (2009)

    MATH  Google Scholar 

  55. M. Gnewuch, A. Srivastav, C. Winzen, Finding optimal volume subintervals with k points and calculating the star discrepancy are NP-hard problems. J. Complexity 25, 115–127 (2009)

    MathSciNet  MATH  Google Scholar 

  56. M. Gnewuch, M. Wahlström, C. Winzen, A new randomized algorithm to approximate the star discrepancy based on threshold accepting. SIAM J. Numer. Anal. 50, 781–807 (2012)

    MathSciNet  MATH  Google Scholar 

  57. J.H. Halton, On the efficiency of certain quasi-random sequences of points in evaluating multidimensional integrals. Numer. Math. 2, 84–90 (1960)

    MathSciNet  MATH  Google Scholar 

  58. J.H. Halton, S.K. Zaremba, The extreme and L 2-discrepancies of some plane sets. Monatshefte Math. 73, 316–328 (1969)

    MathSciNet  MATH  Google Scholar 

  59. J.M. Hammersley, Monte Carlo methods for solving multivariate problems. Ann. New York Acad. Sci. 86, 844–874 (1960)

    MathSciNet  MATH  Google Scholar 

  60. S. Heinrich, Efficient algorithms for computing the L 2 discrepancy. Math. Comp. 65, 1621–1633 (1996)

    MathSciNet  MATH  Google Scholar 

  61. S. Heinrich, E. Novak, G.W. Wasilkowski, H. Woźniakowski, The inverse of the star-discrepancy depends linearly on the dimension. Acta Arithmetica 96, 279–302 (2001)

    MathSciNet  MATH  Google Scholar 

  62. P. Hellekalek, H. Leeb, Dyadic diaphony. Acta Arithmetica 80, 187–196 (1997)

    MathSciNet  Google Scholar 

  63. R. Henrion, C. Küchler, W. Römisch, Discrepancy distances and scenario reduction in two-stage stochastic mixed-integer programming. J. Ind. Manag. Optim. 4, 363–384 (2008)

    Google Scholar 

  64. R. Henrion, C. Küchler, W. Römisch, Scenario reduction in stochastic programming with respect to discrepancy distances. Comput. Optim. Appl. 43, 67–93 (2009)

    MathSciNet  MATH  Google Scholar 

  65. F.J. Hickernell, A generalized discrepancy and quadrature error bound. Math. Comp. 67, 299–322 (1998)

    MathSciNet  MATH  Google Scholar 

  66. F.J. Hickernell, I.H. Sloan, G.W. Wasilkowski, On tractability of weighted integration over bounded and unbounded regions in \(\mathbb{R}^{s}\). Math. Comp. 73, 1885–1901 (2004)

    MathSciNet  MATH  Google Scholar 

  67. F.J. Hickernell, X. Wang, The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimensions. Math. Comp. 71, 1641–1661 (2001)

    MathSciNet  Google Scholar 

  68. A. Hinrichs, Discrepancy of Hammersley points in Besov spaces of dominated mixed smoothness. Math. Nachr. 283, 477–483 (2010)

    Google Scholar 

  69. A. Hinrichs, Discrepancy, Integration and Tractability, in Monte Carlo and Quasi-Monte Carlo Methods 2012, ed. by J. Dick, F.Y. Kuo, G. Peters, I.H. Sloan (Springer, Berlin and Heidelberg, 2013). pp. 129–172

    Google Scholar 

  70. A. Hinrichs, H. Weyhausen, Asymptotic behavior of average L p -discrepancies. J. Complexity 28, 425–439 (2012)

    MathSciNet  MATH  Google Scholar 

  71. E. Hlawka, Discrepancy and uniform distribution of sequences. Compositio Math. 16, 83–91 (1964)

    MathSciNet  MATH  Google Scholar 

  72. J.H. Holland, Adaptation In Natural And Artificial Systems (University of Michigan Press, Ann Arbor, 1975)

    Google Scholar 

  73. J. Hromković, Algorithms for Hard Problems, 2nd edn (Springer, Berlin and Heidelberg, 2003)

    Google Scholar 

  74. R. Impagliazzo, R. Paturi, The complexity of k-SAT, in Proc. 14th IEEE Conf. on Computational Complexity (1999)

    Google Scholar 

  75. R. Impagliazzo, R. Paturi, F. Zane, Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)

    MathSciNet  MATH  Google Scholar 

  76. S. Joe, An intermediate bound on the star discrepancy, in Monte Carlo and Quasi-Monte Carlo Methods 2010, ed. by L. Plaskota, H. Woźniakowski (Springer, Berlin and Heidelberg, 2012)

    Google Scholar 

  77. S. Joe, F.Y. Kuo, Constructing Sobol’ sequences with better two-dimensional projections. SIAM J. Sci. Comput. 30, 2635–2654 (2008)

    MathSciNet  MATH  Google Scholar 

  78. S. Joe, I.H. Sloan, On computing the lattice rule criterion R. Math. Comp. 59, 557–568 (1992)

    Google Scholar 

  79. W. Kahan, Further remarks on reducing truncation errors. Commun. ACM 8, 40 (1965)

    Google Scholar 

  80. S. Kirckpatrick, C. Gelatt, M. Vecchi, Optimization by simulated annealing. Science 20, 671–680 (1983)

    Google Scholar 

  81. V. Klee, Can the measure of \(\bigcup _{1}^{n}[a_{i},b_{i}]\) be computed in less than O(nlogn) steps? The American Mathematical Monthly 84(4), 284–285 (1977). http://www.jstor.org/stable/2318871

  82. D.E. Knuth, The Art of Computer Programming. Vol. 2. Seminumerical Algorithms, 3rd edn. Addison-Wesley Series in Computer Science and Information Processing (Addison-Wesley, Reading, 1997)

    Google Scholar 

  83. L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences (Wiley-Interscience [Wiley], New York, 1974). Pure and Applied Mathematics

    Google Scholar 

  84. F.Y. Kuo, Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces. J. Complexity 19, 301–320 (2003)

    MathSciNet  MATH  Google Scholar 

  85. G. Larcher, F. Pillichshammer, A note on optimal point distributions in [0, 1)s. J. Comput. Appl. Math. 206(2), 977–985 (2007). http://dx.doi.org/10.1016/j.cam.2006.09.004

  86. P. L’Ecuyer, Good parameter sets for combined multiple recursive random number generators. Oper. Res. 47, 159–164 (1999)

    MATH  Google Scholar 

  87. P. L’Ecuyer, P. Hellekalek, Random number generators: selection criteria and testing, in Random and quasi-random point sets. Lecture Notes in Statist., vol. 138 (Springer, New York, 1998). http://dx.doi.org/10.1007/978-1-4612-1702-2_5

    Google Scholar 

  88. C. Lemieux, Monte Carlo and Quasi-Monte Carlo Sampling (Springer, New York, 2009)

    MATH  Google Scholar 

  89. G. Leobacher, F. Pillichshammer, Bounds for the weighted L p discrepancy and tractability of integration. J. Complexity 529–547 (2003)

    Google Scholar 

  90. D.K.J. Lin, C. Sharpe, P. Winker, Optimized U-type designs on flexible regions. Comput. Stat. Data Anal. 54(6), 1505–1515 (2010). http://dx.doi.org/10.1016/j.csda.2010.01.032

  91. L. Markhasin, Discrepancy of generalized Hammersley type point sets in Besov spaces with dominating mixed smoothness. Uniform Distribution Theory 8, 135–164 (2013)

    MathSciNet  Google Scholar 

  92. L. Markhasin, Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension. J. Complexity 29(5), 370–388 (2013)

    MathSciNet  MATH  Google Scholar 

  93. J. Matoušek, On the L 2-discrepancy for anchored boxes. J. Complexity 14, 527–556 (1998)

    MathSciNet  MATH  Google Scholar 

  94. J. Matoušek, Geometric Discrepancy, 2nd edn. (Springer, Berlin, 2010)

    MATH  Google Scholar 

  95. K. Mehlhorn, Multi-dimensional Searching and Computational Geometry, Data Structures and Algorithms 3 (Springer, Berlin/New York, 1984)

    Google Scholar 

  96. W. Morokoff, R. Caflisch, Quasi-random sequences and their discrepancies. SIAM J. Sci. Comput. 15, 1251–1279 (1994)

    MathSciNet  MATH  Google Scholar 

  97. H. Niederreiter, Discrepancy and convex programming. Ann. Mat. Pura Appl. 93, 89–97 (1972)

    MathSciNet  MATH  Google Scholar 

  98. H. Niederreiter, Methods for estimating discrepancy, in Applications of Number Theory to Numerical Analysis, ed. by S.K. Zaremba (Academic Press, New York, 1972)

    Google Scholar 

  99. H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods. SIAM CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63 (SIAM, Philadelphia, 1992)

    Google Scholar 

  100. E. Novak, K. Ritter, High dimensional integration of smooth functions over cubes. Numer. Math. 75, 79–97 (1996)

    MathSciNet  MATH  Google Scholar 

  101. E. Novak, H. Woźniakowski, Tractability of Multivariate Problems. Vol. 1, Linear Information. EMS Tracts in Mathematics (European Mathematical Society (EMS), Zürich, 2008)

    Google Scholar 

  102. E. Novak, H. Woźniakowski, L 2 discrepancy and multivariate integration, in Analytic number theory. Essays in honour of Klaus Roth., ed. by W.W.L. Chen, W.T. Gowers, H. Halberstam, W.M. Schmidt, R.C. Vaughan (Cambridge University Press, Cambridge, 2009)

    Google Scholar 

  103. E. Novak, H. Woźniakowski, Tractability of Multivariate Problems. Vol. 2, Standard Information for Functionals. EMS Tracts in Mathematics (European Mathematical Society (EMS), Zürich, 2010)

    Google Scholar 

  104. D. Nuyens, R. Cools, Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comp. 75, 903–920 (2006)

    MathSciNet  MATH  Google Scholar 

  105. G. Ökten, Error reduction techniques in quasi-Monte Carlo integration. Math. Comput. Modelling 30, 61–69 (1999)

    MathSciNet  MATH  Google Scholar 

  106. G. Ökten, M. Shah, Y. Goncharov, Random and deterministic digit permutations of the Halton sequence, in Monte Carlo and Quasi-Monte Carlo Methods 2010, ed. by L. Plaskota, H. Woźniakowski (Springer, Berlin and Heidelberg, 2012)

    Google Scholar 

  107. M.H. Overmars, C.-K. Yap, New upper bounds in Klee’s measure problem. SIAM J. Comput. 20(6), 1034–1045 (1991)

    MathSciNet  MATH  Google Scholar 

  108. S.H. Paskov, Average case complexity of multivariate integration for smooth functions. J. Complexity 9, 291–312 (1993)

    Google Scholar 

  109. P. Peart, The dispersion of the Hammersley sequence in the unit square. Monatshefte Math. 94, 249–261 (1982)

    MathSciNet  MATH  Google Scholar 

  110. T. Pillards, R. Cools, A note on E. Thiémard’s algorithm to compute bounds for the star discrepancy. J. Complexity 21, 320–323 (2005)

    MathSciNet  MATH  Google Scholar 

  111. T. Pillards, B. Vandewoestyne, R. Cools, Minimizing the L 2 and L ∞ star discrepancies of a single point in the unit hypercube. J. Comput. Appl. Math. 197(1), 282–285 (2006). http://dx.doi.org/10.1016/j.cam.2005.11.005

  112. W. Römisch, Scenario reduction techniques in stochastic programming, in SAGA 2009, ed. by O. Watanabe, T. Zeugmann LNCS, vol. 43 (Springer, Berlin-Heidelberg, 2009)

    Google Scholar 

  113. W.M. Schmidt, Irregularities of distribution IX. Acta Arithmetica, 385–396 (1975)

    Google Scholar 

  114. M. Shah, A genetic algorithm approach to estimate lower bounds of the star discrepancy. Monte Carlo Methods Appl. 16, 379–398 (2010)

    MathSciNet  MATH  Google Scholar 

  115. I.H. Sloan, S. Joe, Lattice Methods for Multiple Integration (Clarendon Press, Oxford, 1994)

    MATH  Google Scholar 

  116. I.H. Sloan, A.V. Reztsov, Component-by-component construction of good lattice rules. Math. Comp. 71, 263–273 (2002)

    MathSciNet  MATH  Google Scholar 

  117. I.H. Sloan, H. Woźniakowski, When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complexity 14, 1–33 (1998)

    MathSciNet  MATH  Google Scholar 

  118. I.H. Sloan, F.Y. Kuo, S. Joe, On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces. Math. Comp. 71, 1609–1640 (2002)

    MathSciNet  MATH  Google Scholar 

  119. S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk. SSSR 4, 1042–1045 (1963)

    Google Scholar 

  120. I.M. Sobol’, The distribution of points in a cube and the approximate evaluation of integrals. Zh. Vychisl. Mat. i. Mat. Fiz. 7, 784–802 (1967)

    Google Scholar 

  121. I.M. Sobol’, D. Asotsky, A. Kreinin, S. Kucherenko, Construction and comparison of high-dimensional Sobol’ generators. Wilmott Mag. Nov. 2011, 64–79 (2011)

    Google Scholar 

  122. A. Srinivasan, Distributions on level-sets with applications to approximation algorithms, in Proceedings of FOCS’01 (2001)

    Google Scholar 

  123. S. Steinerberger, The asymptotic behavior of the average L p -discrepancies and a randomized discrepancy. Electron. J. Combin., 1–18 (2010). Research Paper 106

    Google Scholar 

  124. W. Stute, Convergence rates for the isotrope discrepancy. Ann. Probab., 707–723 (1977)

    Google Scholar 

  125. V.N. Temlyakov, On a way of obtaining lower estimates for the errors of quadrature formulas. Math. USSR Sbornik 71, 247–257 (1992)

    MathSciNet  MATH  Google Scholar 

  126. S. Tezuka, Uniform Random Numbers: Theory and Practice (Kluwer Academic, Boston, 1995)

    MATH  Google Scholar 

  127. E. Thiémard, 1998, Economic generation of low-discrepancy sequences with a b-ary Gray code. EPFL-DMA-ROSO, RO981201, http://rosowww.epfl.ch/papers/grayfaure/

  128. E. Thiémard, Computing bounds for the star discrepancy. Computing 65, 169–186 (2000)

    MathSciNet  MATH  Google Scholar 

  129. E. Thiémard, Sur le calcul et la majoration de la discrépance à l’origine (PhD thesis École polytechnique fédérale de Lausanne EPFL, nbr 2259, Lausanne, 2000). Available from http://infoscience.epfl.ch/record/32735

  130. E. Thiémard, An algorithm to compute bounds for the star discrepancy. J. Complexity 17, 850–880 (2001)

    MathSciNet  MATH  Google Scholar 

  131. E. Thiémard, Optimal volume subintervals with k points and star discrepancy via integer programming. Math. Meth. Oper. Res. 54, 21–45 (2001)

    MATH  Google Scholar 

  132. H. Triebel, Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration. EMS Tracts in Mathematics 11 (European Mathematical Society, Zürich, 2010)

    Google Scholar 

  133. B. Vandewoestyne, R. Cools, Good permutations for deterministic scrambled Halton sequences in terms of L 2-discrepancy. J. Comput. Appl. Math. 189, 341–361 (2006)

    MathSciNet  MATH  Google Scholar 

  134. X. Wang, F.J. Hickernell, Randomized Halton sequences. Math. Comput. Modell. 32, 887–899 (2000)

    MathSciNet  MATH  Google Scholar 

  135. T.T. Warnock, Computational investigations of low-discrepancy point sets, in Applications of number theory to numerical analysis, ed. by S.K. Zaremba (Academic Press, New York, 1972)

    Google Scholar 

  136. T.T. Warnock, 2013. Personal communication

    Google Scholar 

  137. G.W. Wasilkowski, H. Woźniakowski, Explicit cost bounds of algorithms for multivariate tensor product problems. J. Complexity 11, 1–56 (1995)

    MathSciNet  MATH  Google Scholar 

  138. B.E. White, On optimal extreme-discrepancy point sets in the square. Numer. Math. 27, 157–164 (1976/77)

    Google Scholar 

  139. P. Winker, K.T. Fang, Applications of threshold-accepting to the evaluation of the discrepancy of a set of points. SIAM J. Numer. Anal. 34, 2028–2042 (1997)

    MathSciNet  MATH  Google Scholar 

  140. C. Winzen, Approximative Berechnung der Sterndiskrepanz (Christian-Albrechts-Universität zu Kiel, Kiel, 2007)

    Google Scholar 

  141. H. Woźniakowski, Average case complexity of multivariate integration. Bull. Am. Math. Soc. (N. S.) 24, 185–191 (1991)

    MATH  Google Scholar 

  142. S.K. Zaremba, Some applications of multidimensional integration by parts. Ann. Polon. Math. 21, 85–96 (1968)

    MathSciNet  MATH  Google Scholar 

  143. S.K. Zaremba, La discrépance isotrope et l’intégration numérique. Ann. Mat. Pura Appl. 37, 125–136 (1970)

    MathSciNet  Google Scholar 

  144. C. Zenger, Sparse grids, in Parallel Algorithms for Partial Differential Equations, ed. by W. Hackbusch (Vieweg, Braunschweig, 1991)

    Google Scholar 

  145. P. Zinterhof, Über einige Abschätzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden. Sitzungsber. Österr. Akad. Wiss. Math.-Naturwiss. Kl. II 185, 121–132 (1976)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Sergei Kucherenko, Shu Tezuka, Tony Warnock, Greg Wasilkowski, Peter Winker, and an anonymous referee for their valuable comments.

Carola Doerr is supported by a Feodor Lynen postdoctoral research fellowship of the Alexander von Humboldt Foundation and by the Agence Nationale de la Recherche under the project ANR-09-JCJC-0067-01.

The work of Michael Gnewuch was supported by the German Science Foundation DFG under grant GN-91/3 and the Australian Research Council ARC.

The work of Magnus Wahlström was supported by the German Science Foundation DFG via its priority program “SPP 1307: Algorithm Engineering” under grant DO 749/4-1.

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Authors and Affiliations

  1. Université Pierre et Marie Curie - Paris 6, LIP6, équipe RO, Paris, France

    Carola Doerr

  2. Max-Planck-Institut für Informatik, Saarbrücken, Germany

    Carola Doerr & Magnus Wahlström

  3. Mathematisches Seminar, Kiel University, Kiel, Germany

    Michael Gnewuch

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  1. Carola Doerr
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  2. Michael Gnewuch
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  3. Magnus Wahlström
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Correspondence to Michael Gnewuch .

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  1. Department of Mathematics, Macquarie University, Sydney, New South Wales, Australia

    William Chen

  2. Department of Computer Science, Kiel University, Kiel, Germany

    Anand Srivastav

  3. Dipartimento di Statistica e Metodi Quantitativi, Università di Milano-Bicocca, Milano, Italy

    Giancarlo Travaglini

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Doerr, C., Gnewuch, M., Wahlström, M. (2014). Calculation of Discrepancy Measures and Applications. In: Chen, W., Srivastav, A., Travaglini, G. (eds) A Panorama of Discrepancy Theory. Lecture Notes in Mathematics, vol 2107. Springer, Cham. https://doi.org/10.1007/978-3-319-04696-9_10

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