Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Density Estimation by Monte Carlo and Quasi-Monte Carlo Read chapter Read first chapter

2022 | OriginalPaper | Chapter

A Tool for Custom Construction of QMC and RQMC Point Sets

Authors : Pierre L’Ecuyer, Pierre Marion, Maxime Godin, Florian Puchhammer

Published in: Monte Carlo and Quasi-Monte Carlo Methods

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We present LatNet Builder, a software tool to find good parameters for lattice rules, polynomial lattice rules, and digital nets in base 2, for quasi-Monte Carlo (QMC) and randomized quasi-Monte Carlo (RQMC) sampling over the s-dimensional unit hypercube. The selection criteria are figures of merit that give different weights to different subsets of coordinates. They are upper bounds on the worst-case error (for QMC) or variance (for RQMC) for integrands rescaled to have a norm of at most one in certain Hilbert spaces of functions. We summarize what are the various Hilbert spaces, discrepancies, types of weights, figures of merit, types of constructions, and search methods supported by LatNet Builder. We briefly discuss its organization and we provide simple illustrations of what it can do.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Quasi-Monte Carlo Software
next chapter On Dropping the First Sobol’ Point
Literature
1.
Baldeaux, J., Dick, J., Greslehner, J., Pillichshammer, F.: Construction algorithms for higher order polynomial lattice rules. J. Complex. 27, 281–299 (2011)
2.
Baldeaux, J., Dick, J., Leobacher, G., Nuyens, D., Pillichshammer, F.: Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules. Numer. Algorithms 59(3), 403–431 (2012)
3.
Dick, J.: Explicit constructions of Quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions. SIAM J. Numer. Anal. 45(5), 2141–2176 (2007)
4.
Dick, J.: Walsh spaces containing smooth functions and Quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46(3), 1519–1553 (2008)
5.
Dick, J., Kuo, F.Y., Pillichshammer, F., Sloan, I.H.: Construction algorithms for polynomial lattice rules for multivariate integration. Math. Comput. 74(248), 1895–1921 (2005)
6.
Dick, J., Pillichshammer, F.: Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces. J. Complex. 21(2), 149–195 (2005)
7.
Dick, J., Pillichshammer, F.: Strong tractability of multivariate integration of arbitrary high order using digitally shifted polynomial lattice rules. J. Complex. 23, 436–453 (2007)
8.
Dick, J., Pillichshammer, F.: Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge, U.K. (2010)
9.
Dick, J., Sloan, I.H., Wang, X., Woźniakowski, H.: Good lattice rules in weighted Korobov spaces with general weights. Numerische Mathematik 103, 63–97 (2006)
10.
Efron, B., Stein, C.: The jackknife estimator of variance. Ann. Stat. 9, 586–596 (1981)
11.
Genz, A.: Testing multidimensional integration routines. In: Proceedings of the International Conference on Tools, Methods and Languages for Scientific and Engineering Computation, pp. 81–94. Elsevier North-Holland (1984)
12.
Gilbert, A., Kuo, F., Sloan, I.: Hiding the weights-CBC black box algorithms with a guaranteed error bound. Math. Comput. Simul. 143, 202–214 (2018)
13.
Goda, T.: Good interlaced polynomial lattice rules for numerical integration in weighted Walsh spaces. J. Comput. Appl. Math. 285, 279–294 (2015)
14.
Goda, T., Dick, J.: Construction of interlaced scrambled polynomial lattice rules of arbitrary high order. Found. Comput. Math. 15, 1245–1278 (2019)
15.
Goda, T., Suzuki, K.: Recent Advances in Higher Order Quasi-Monte Carlo Methods, pp. 69–102. De Gruyter (2019)
16.
Goda, T., Suzuki, K., Yoshiki, T.: An explicit construction of optimal order Quasi-Monte Carlo rules for smooth integrands. SIAM J. Numer. Anal. 54, 2664–2683 (2016)
17.
Goda, T., Suzuki, K., Yoshiki, T.: Optimal order Quasi-Monte Carlo integration in weighted Sobolev spaces of arbitrary smoothness. SIAM J. Numer. Anal. 37, 505–518 (2017)
18.
Goda, T., Suzuki, K., Yoshiki, T.: Optimal order quadrature error bounds for infinite-dimensional higher-order digital sequences. Found. Comput. Math. 18, 433–458 (2018)
19.
Hickernell, F.J.: A generalized discrepancy and quadrature error bound. Math. Comput. 67(221), 299–322 (1998)
20.
Hickernell, F.J.: Lattice rules: How well do they measure up? In: Hellekalek, P., Larcher, G. (eds.) Random and Quasi-Random Point Sets. Lecture Notes in Statistics, vol. 138, pp. 109–166. Springer, New York (1998)
21.
Hickernell, F.J.: Obtaining \({O(N^{-2+\epsilon })}\) convergence for lattice quadrature rules. In: Fang, K.T., Hickernell, F.J., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2000, pp. 274–289. Springer, Berlin (2002)
22.
Hickernell, F.J., Hong, H.S., L’Ecuyer, P., Lemieux, C.: Extensible lattice sequences for Quasi-Monte Carlo quadrature. SIAM J. Sci. Comput. 22(3), 1117–1138 (2001)
23.
Hickernell, F.J., Yue, R.X.: The mean square discrepancy of scrambled \((t, s)\)-sequences. SIAM J. Numer. Anal. textbf38(4), 1089–1112 (2001)
24.
Hong, H.S., Hickernell, F.H.: Algorithm 823: implementing scrambled digital sequences. ACM Trans. Math. Softw. 29, 95–109 (2003)
25.
Joe, S., Kuo, F.Y.: Constructing Sobol sequences with better two-dimensional projections. SIAM J. Sci. Comput. 30(5), 2635–2654 (2008)
26.
L’Ecuyer, P.: Polynomial integration lattices. In: Niederreiter, H. (ed.) Monte Carlo and Quasi-Monte Carlo Methods 2002, pp. 73–98. Springer, Berlin (2004)
27.
L’Ecuyer, P.: Quasi-Monte Carlo methods with applications in finance. Financ. Stoch. 13(3), 307–349 (2009)
28.
29.
L’Ecuyer, P., Buist, E.: Simulation in Java with SSJ. In: Proceedings of the 2005 Winter Simulation Conference, pp. 611–620. IEEE Press, Piscataway, NJ (2005)
30.
L’Ecuyer, P., Lemieux, C.: Variance reduction via lattice rules. Manag. Sci. 46(9), 1214–1235 (2000)
31.
L’Ecuyer, P., Lemieux, C.: Recent advances in randomized Quasi-Monte Carlo methods. In: Dror, M., L’Ecuyer, P., Szidarovszky, F. (eds.) Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, pp. 419–474. Kluwer Academic, Boston (2002)
32.
L’Ecuyer, P., Munger, D.: Constructing adapted lattice rules using problem-dependent criteria. In: Proceedings of the 2012 Winter Simulation Conference, pp. 373–384. IEEE Press (2012)
33.
L’Ecuyer, P., Munger, D.: On figures of merit for randomly-shifted lattice rules. In: Woźniakowski, H., Plaskota, L. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2010, pp. 133–159. Springer, Berlin (2012)
34.
L’Ecuyer, P., Munger, D.: Algorithm 958: Lattice builder: a general software tool for constructing rank-1 lattice rules. ACM Trans. Math. Softw. 42(2), Article 15 (2016)
35.
36.
Lemieux, C., L’Ecuyer, P.: Randomized polynomial lattice rules for multivariate integration and simulation. SIAM J. Sci. Comput. 24(5), 1768–1789 (2003)
37.
Marion, P., Godin, M., L’Ecuyer, P.: An algorithm to compute the \(t\)-value of a digital net and of its projections. J. Comput. Appl. Math. 371(June), 112669 (2020)
38.
Matousěk, J.: On the \(L_2\)-discrepancy for anchored boxes. J. Complex. 14, 527–556 (1998)
39.
Niederreiter, H.: Low-discrepancy point sets obtained by digital constructions over finite fields. Czechoslovak Math. J. 42, 143–166 (1992)
40.
Niederreiter, H.: Random number generation and Quasi-Monte Carlo methods. In: SIAM CBMS-NSF Regional Conference Series in Mathematics, vol. 63. SIAM (1992)
41.
42.
Nuyens, D.: The construction of good lattice rules and polynomial lattice rules. In: Kritzer, P., Niederreiter, H., Pillichshammer, F., Winterhof, A. (eds.) Uniform Distribution and Quasi-Monte Carlo Methods: Discrepancy, Integration and Applications, pp. 223–255. De Gruyter (2014)
43.
44.
Nuyens, D., Cools, R.: Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comput. 75, 903–920 (2006)
45.
Owen, A.B.: Monte Carlo variance of scrambled equidistribution quadrature. SIAM J. Numer. Anal. 34(5), 1884–1910 (1997)
46.
Owen, A.B.: Scrambled net variance for integrals of smooth functions. Ann. Stat. 25(4), 1541–1562 (1997)
47.
Owen, A.B.: Latin supercube sampling for very high-dimensional simulations. ACM Trans. Model. Comput. Simul. 8(1), 71–102 (1998)
48.
Owen, A.B.: Variance with alternative scramblings of digital nets. ACM Trans. Model. Comput. Simul. 13(4), 363–378 (2003)
49.
Sinescu, V., L’Ecuyer, P.: Variance bounds and existence results for randomly shifted lattice rules. J. Comput. Appl. Math. 236, 3296–3307 (2012)
50.
Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Clarendon Press, Oxford (1994)
51.
Sloan, I.H., Rezstov, A.: Component-by-component construction of good lattice rules. Math. Comput. 71, 262–273 (2002)
52.
Sobol’, I.M.: The distribution of points in a cube and the approximate evaluation of integrals. U.S.S.R. Comput. Math. and Math. Phys. 7(4), 86–112 (1967)
53.
Sobol, I.M., Asotsky, D.I.: One more experiment on estimating high-dimensional integrals by quasi-Monte Carlo methods. Math. Comput. Simul. 62(3–6), 255–263 (2003)
54.
Metadata
Title
A Tool for Custom Construction of QMC and RQMC Point Sets
Authors
Pierre L’Ecuyer
Pierre Marion
Maxime Godin
Florian Puchhammer
Copyright Year
2022
Book
Monte Carlo and Quasi-Monte Carlo Methods

Print ISBN: 978-3-030-98318-5

Electronic ISBN: 978-3-030-98319-2

Copyright Year: 2022

DOI
https://doi.org/10.1007/978-3-030-98319-2_3

Premium Partner

    Image Credits