Entropy, Randomization, Derandomization, and Discrepancy

Springer Proceedings in Mathematics and Statistics

Volumn 23, Issue , 2012, Pages 43-78

  Gnewuch, Michael ^a  

^a UNIVERSITY OF KIEL   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

ENTROPY; GEOMETRY; QUANTUM CHEMISTRY; RANDOM PROCESSES; STARS;

DERANDOMIZATION; FUNCTION CLASS; HIGH-DIMENSIONAL; HIGH-DIMENSIONAL INTEGRALS; METRIC ENTROPY; NUMERICAL INTEGRATIONS; STAR-DISCREPANCY; WORST-CASE ERRORS;

MONTE CARLO METHODS;

EID: 84893581716     PISSN: 21941009     EISSN: 21941017     Source Type: Conference Proceeding    
DOI: 10.1007/978-3-642-27440-4_3     Document Type: Conference Paper

References (94)

1. Aistleitner, C.: Covering numbers, dyadic chaining and discrepancy. J. Complexity 27, 531-540 (2011).
2. Atanassov, E.: On the discrepancy of the Halton sequences. Math. Balkanica (N.S.) 18, 15-32 (2004).
3. Beck, J.: Some upper bounds in the theory of irregularities of distribution. Acta Arith. 43, 115-130 (1984).
4. Beck, J.: A two-dimensional van Aardenne-Ehrenfest theorem in irregularities of distribution. Composito Math. 72, 269-339 (1989).
5. Beck, J., Chen, W.W.L.: Irregularities of Distribution. Cambridge University Press, Cambridge (1987).
6. Bilyk, D., Lacey, M.T.: On the small ball inequality in three dimensions. Duke Math. J. 143, 81-115 (2008).
7. Bilyk, D., Lacey, M.T., Vagharshakyan, A.: On the small ball inequality in all dimensions. J. Funct. Anal. 254, 2470-2502 (2008).
8. Bundschuh, P., Zhu, Y.C.: A method for exact calculation of the discrepancy of lowdimensional point sets I. Abh. Math. Sem. Univ. Hamburg 63, 115-133 (1993).
9. Chari, S., Rohatgi, P., Srinivasan, A.: Improved algorithms via approximation of probability distributions. J. Comput. System Sci. 61, 81-107 (2000).
10. de Clerck, L.: A method for exact calculation of the star-discrepancy of plane sets applied to the sequence of Hammersley. Monatsh. Math. 101, 261-278 (1986).
11. van der Corput, J.G.: Verteilungsfunktionen I. Nederl. Akad. Wetensch. Proc. Ser. B 38, 813-821 (1935).
12. van der Corput, J.G.: Verteilungsfunktionen II. Nederl. Akad. Wetensch. Proc. Ser. B 38, 1058-1066 (1935).
13. Davenport, H.: Note on irregularities of distribution. Matematika 3, 131-135 (1956).
14. Dick, J.: A note on the existence of sequences with small star discrepancy. J. Complexity 23, 649-652 (2007).
15. Dick, J.: Koksma-Hlawka type inequalities of fractional order. Ann. Mat. Pura Appl. 187, 385-403 (2008).
16. Dick, J., Pillichshammer, F.: Digital Nets and Sequences. Cambridge University Press, Cambridge (2010).
17. Dobkin, D.P., Eppstein, D., Mitchell, D.P.: Computing the discrepancy with applications to supersampling patterns. ACM Trans. Graph. 15, 354-376 (1996).
18. Doerr, B.: Generating randomized roundings with cardinality constraints and derandomizations. In: B. Durand, W. Thomas (eds.) STACS 2006, 23rd Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, vol. 3884, pp. 571-583. Springer (2006).
19. Doerr, B., Gnewuch, M., Kritzer, P., Pillichshammer, F.: Component-by-component construction of low-discrepancy point sets of small size. Monte Carlo Methods Appl. 14, 129-149 (2008).
20. Doerr, B., Gnewuch, M., Srivastav, A.: Bounds and construction for the star discrepancy via i{dotless}-covers (2004). Berichtsreihe des Mathematischen Seminars der Christian-Albrechts-Universität zu Kiel, Report 04-13 (Preliminary preprint version of [21]).
21. Doerr, B., Gnewuch, M., Srivastav, A.: Bounds and constructions for the star discrepancy via i{dotless}-covers. J. Complexity 21, 691-709 (2005).
22. Doerr, B., Gnewuch, M., Wahlström, M.: Implementation of a component-by-component algorithm to generate low-discrepancy samples. In: P. L'Ecuyer, A.B. Owen (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2008, pp. 323-338. Springer, Berlin Heidelberg (2009).
23. Doerr, B., Gnewuch, M., Wahlström, M.: Algorithmic construction of low-discrepancy point sets via dependent randomized rounding. J. Complexity 26, 490-507 (2010).
24. Doerr, B., Wahlström, M.: Randomized rounding in the presence of a cardinality constraint. In: I. Finocchi, J. Hershberger (eds.) Proceedings of the 10thWorkshop on Algorithm Engineering and Experiments (ALENEX 2009), pp. 162-174. SIAM, New York, USA (2009).
25. Drmota, M., Tichy, R.F.: Sequences, Discrepancies, and Applications, Lecture Notes in Math., vol. 1651. Springer, Berlin Heidelberg New York (1997).
26. Dueck, G., Scheuer, T.: Threshold accepting: A general purpose algorithm appearing superior to simulated annealing. J. Comp. Phys. 90 (1990).
27. Even, G., Goldreich, O., Luby, M., Nisan, N., Veličković, B.: Approximations of general independent distributions. In: Proceedings of the 24th ACM Symposium on Theory of Computing (STOC), pp. 10-16 (1992).
28. Faure, H.: Discrépance de suites associées à un système de numération (en dimension s). Acta Arith. 41, 337-351 (1982).
29. Frank, K., Heinrich, S.: Computing discrepancies of Smolyak quadrature rules. J. Complexity 12, 287-314 (1996).
30. Frolov, K.K.: An upper estimate for the discrepancy in the Lp-metric, 2-p-1. Soviet. Math. Dokl. 21, 840-842 (1980).
31. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NPCompleteness. W. H. Freeman and Co., San Francisco (1979).
32. Giannopoulos, P., Knauer, C., Wahlström, M., Werner, D.: Hardness of discrepancy computation and "-net verification in high dimension. J. Complexity 28, 162-176 (2012).
33. Gnewuch, M.: Weighted geometric discrepancies and numerical integration on reproducing kernel Hilbert spaces. J. Complexity 28, 2-17 (2012).
34. Gnewuch, M.: Bounds for the average Lp-extreme and the L1-extreme discrepancy. Electron. J. Combin. 12, Research Paper 54 (2005).
35. Gnewuch, M.: Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy. J. Complexity 24, 154-172 (2008).
36. Gnewuch, M.: Construction of minimal bracketing covers for rectangles. Electron. J. Combin. 15, Research Paper 95 (2008).
37. Gnewuch, M.: On probabilistic results for the discrepancy of a hybrid-Monte Carlo sequence. J. Complexity 23, 312-317 (2009).
38. Gnewuch, M., Srivastav, A., Winzen, C.: Finding optimal volume subintervals with k points and calculating the star discrepancy are NP-hard problems. J. Complexity 25, 115-127 (2009).
39. Gnewuch, M., Wahlström, M., Winzen, C.: A new randomized algorithm to approximate the star discrepancy based on threshold-accepting. SIAM J. Numer. Anal. 50, 781-807 (2012).
40. Halton, J.H.: On the efficiency of certain quasi-random sequences of points in evaluating multidimensional integrals. Numer. Math. 2, 84-90 (1960).
41. Hardy, G.H., Littlewood, J.E.: Some problems of diophantine approximation: the lattice points of a right-angled triangle. Part I. Proc. London Math. Soc. 20, 212-249 (1922).
42. Haussler, D.: Sphere packing numbers for subsets of the Boolean n-cube with bounded Vapnik-Chervonenkis dimension. J. Combin. Theory A 69, 279-302 (1995).
43. Heinrich, S.: Efficient algorithms for computing the L2 discrepancy. Math. Comp. 65, 1621-1633 (1996).
44. Heinrich, S.: Some open problems concerning the star-discrepancy. J. Complexity 19, 416-419 (2003).
45. Heinrich, S., Novak, E., Wasilkowski, G.W., Wózniakowski, H.: The inverse of the stardiscrepancy depends linearly on the dimension. Acta Arith. 96, 279-302 (2001).
46. Hickernell, F.J.: A generalized discrepancy and quadrature error bound. Math. Comp. 67, 299-322 (1998).
47. Hickernell, F.J., Sloan, I.H., Wasilkowski, G.W.: On tractability of weighted integration over bounded and unbounded regions in Rs. Math. Comp. 73, 1885-1901 (2004).
48. Hickernell, F.J., Wang, X.: The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimensions. Math. Comp. 71, 1641-1661 (2001).
49. Hinrichs, A.: Covering numbers, Vapnik-Červonenkis classes and bounds for the stardiscrepancy. J. Complexity 20, 477-483 (2004).
50. Hinrichs, A., Pillichshammer, F., Schmid, W.C.: Tractability properties of the weighted star discrepancy. J. Complexity 23, 134-143 (2008).
51. Hlawka, E.: Funktionen von beschränkter Variation in der Theorie der Gleichverteilung. Ann. Mat. Pura Appl. 54, 325-333 (1961).
52. Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, 13-30 (1963).
53. Hromković, J.: Algorithms for Hard Problems. Springer, Berlin Heidelberg (2003). 2nd edition.
54. Johnson, D.S.: The NP-completeness column: an ongoing guide. J. Algorithms 8, 438-448 (1987).
55. Kirckpatrick, S., Gelatt, C., Vecchi, M.: Optimization by simulated annealing. Science 20, 671-680 (1983).
56. Koksma, J.F.: Een algemeene stelling uit de theorie der gelijkmatige verdeeling modulo 1. Mathematica B (Zutphen) 11, 7-11 (1942/3).
57. Lerch, M.: Question 1547. L'Intermédiaire des Mathématiciens 11, 145-146 (1904).
58. Matoušek, J.: Geometric Discrepancy. Springer-Verlag, Berlin Heidelberg (1999). Revised second printing 2010.
59. Mhaskar, H.N.: On the tractability of multivariate integration and approximation by neural networks. J. Complexity 20, 561-590 (2004).
60. Niederreiter, H.: Discrepancy and convex programming. Ann. Mat. Pura Appl. 93, 89-97 (1972).
61. Niederreiter, H.: Methods for estimating discrepancy. In: S.K. Zaremba (ed.) Applications of number theory to numerical analysis, pp. 203-236. Academic Press, New York (1972).
62. Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (1992).
63. Niederreiter, H.: On the discrepancy of some hybrid sequences. Acta Arith. 138, 373-398 (2009).
64. Niederreiter, H.: Further discrepancy bounds and an Erdös-Turán-Koksma inequality for hybrid sequences. Monatsh. Math. 161, 193-222 (2010).
65. Niederreiter, H., Xing, C.: Low-discrepancy sequences and global function fields with many rational places. Finite Fields and their Applications 2, 241-273 (1996).
66. Novak, E., Wózniakowski, H.: Tractability of Multivariate Problems, Volume I: Linear Information. European Mathematical Society, Freiburg (2008).
67. Novak, E., Wózniakowski, H.: L2 discrepancy and multivariate integration. In: W.W.L. Chen, W.T. Gowers, H. Halberstam, W.M. Schmidt, R.C. Vaughan (eds.) Essays in honour of Klaus Roth, pp. 359-388. Cambridge Univ. Press, Cambridge (2009).
68. Novak, E., Wózniakowski, H.: Tractability of Multivariate Problems, Volume II: Standard Information for Functionals. European Mathematical Society, Freiburg (2010).
69. Ökten, G.: A probabilistic result on the discrepancy of a hybrid Monte Carlo sequence and applications. Monte Carlo Methods Appl. 2, 255-270 (1996).
70. Ökten, G., Shah, M., Goncharov, Y.: Random and deterministic digit permutations of the Halton sequence. In: L. Plaskota, H. Wózniakowski (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2010, pp. 609-622. Springer-Verlag, Berlin (2012).
71. Ökten, G., Tuffin, B., Burago, V.: A central limit theorem and improved error bounds for a hybrid-Monte Carlo sequence with applications in computational finance. J. Complexity 22, 435-458 (2006).
72. Ostrowski, A.: Bemerkungen zur Theorie der Diophantischen Approximationen I. Abh. Math. Semin. Univ. Hamburg 1, 77-98 (1922).
73. Ostrowski, A.: Bemerkungen zur Theorie der Diophantischen Approximationen II. Abh. Math. Semin. Univ. Hamburg 1, 250-251 (1922).
74. Panconesi, A., Srinivasan, A.: Randomized distributed edge coloring via an extension of the Chernoff-Hoeffding bounds. SIAM J. Comput. 26, 350-368 (1997).
75. Raghavan, P.: Probabilistic construction of deterministic algorithms: approximating packing integer programs. J. Comput. Syst. Sci. 37, 130-143 (1988).
76. Roth, K.F.: On irregularities of distribution. Mathematika 1, 73-79 (1954).
77. Roth, K.F.: On irregularities of distribution III. Acta Arith. 35, 373-384 (1979).
78. Roth, K.F.: On irregularities of distribution IV. Acta Arith. 37, 67-75 (1980).
79. Schmidt, W.M.: On irregularities of distribution VII. Acta Arith. 21, 45-50 (1972).
80. Shah, M.: A genetic algorithm approach to estimate lower bounds of the star discrepancy. Monte Carlo Methods Appl. 16, 379-398 (2010).
81. Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Oxford University Press, New York (1994).
82. Sloan, I.H., Wózniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complexity 14, 1-33 (1998).
83. Spanier, J.: Quasi-Monte Carlo methods for particle transport problems. In: H. Niederreiter, P.J.S. Shiue (eds.) Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Lecture Notes in Statistic, vol. 106, pp. 121-148. Springer, Berlin (1995).
84. Srinivasan, A.: Distributions on level-sets with applications to approximation algorithms. In: Proceedings of FOCS'01, pp. 588-597 (2001).
85. Steinerberger, S.: The asymptotic behavior of the average Lp-discrepancies and a randomized discrepancy. Electron. J. Combin. 17(1), Research paper 106 (2010).
86. Talagrand, M.: Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22, 28-76 (1994).
87. Thiémard, E.: Computing bounds for the star discrepancy. Computing 65, 169-186 (2000).
88. Thiémard, E.: An algorithm to compute bounds for the star discrepancy. J. Complexity 17, 850-880 (2001).
89. Thiémard, E.: Optimal volume subintervals with k points and star discrepancy via integer programming. Math. Meth. Oper. Res. 54, 21-45 (2001).
90. van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, New York (1996).
91. Warnock, T.T.: Computational investigations of low-discrepancy point sets. In: S.K. Zaremba (ed.) Applications of number theory to numerical analysis, pp. 319-343. Academic Press, New York (1972).
92. Winker, P., Fang, K.T.: Application of threshold-accepting to the evaluation of the discrepancy of a set of points. SIAM J. Numer. Anal. 34, 2028-2042 (1997).
93. Wózniakowski, H.: Average case complexity of multivariate integration. Bull. Amer. Math. Soc. (N. S.) 24, 185-191 (1991).
94. Zaremba, S.K.: Some applications of multidimensional integration by parts. Ann. Polon. Math. 21, 85-96 (1968).