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The Monte–Carlo Method

  • Simon ŠircaEmail author
Chapter
Part of the Graduate Texts in Physics book series (GTP)

Abstract

The Monte–Carlo method is introduced as a generic tool for the solution of mathematical physics models by means of computer simulation. These problems range from simple one-dimensional integration to sophisticated multi-dimensional models involving elaborate geometries and complex system states. A historical introduction and an exposition of the basic idea are followed by a basic treatment of numerical integration and discussing methods of variance reduction like importance sampling and use of quasi-random sequences. Markov-chain Monte Carlo is presented as a powerful method to generate random numbers according to arbitrary, even extremely complicated distributions. A specific implementation in the form of the Metropolis–Hastings algorithm is offered.

References

  1. 1.
    N. Metropolis, S. Ulam, The Monte Carlo method. J. Am. Stat. Assoc. 44, 335 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J.E. Gentle, Random Number Generation and Monte Carlo Methods, 2nd edn. (Springer, Berlin, 2003)Google Scholar
  3. 3.
    M.H. Kalos, P.A. Whitlock, Monte Carlo Methods, 2nd edn. (Wiley, Weinheim, 2008)Google Scholar
  4. 4.
    I.M. Sobol’, On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput. Maths. Math. Phys. 7, 86 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    I.A. Antonov, V.M. Saleev, An economic method of computing \(LP_\tau \) sequences. USSR Comput. Math. Math. Phys. 19, 252 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes: The Art of Scientific Computing, 3 edn. (Cambridge University Press, Cambridge, 2007)Google Scholar
  7. 7.
    S.H. Paskov, J.F. Traub, Faster valuation of financial derivatives. J. Portf. Manag. 22, 113 (1995)CrossRefGoogle Scholar
  8. 8.
    J.F. Traub, S.H. Paskov, I.F. Vanderhoof, A. Papageorgiou, Portfolio Structuring Using Low-discrepancy Deterministic Sequences, U.S. Patent 6,058,377, http://www.google.com/patents/US6058377
  9. 9.
    S. Chib, in Handbook of Computational Statistics, ed. by J.E. Gentle et al. Markov Chain Monte Carlo Technology, (Springer, Berlin, 2012), p. 73Google Scholar
  10. 10.
    N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087 (1953)ADSCrossRefGoogle Scholar
  11. 11.
    S. Chib, E. Greenberg, Understanding the Metropolis-Hastings algorithm. Am. Stat. 49, 327 (1995)Google Scholar
  12. 12.
    I. Beichl, F. Sullivan, The Metropolis algorithm. Comput. Sci. Eng. 2(1), 65 (2000)Google Scholar
  13. 13.
    C.P. Robert, G. Casella, Introducing Monte Carlo Methods with R (Springer, Berlin, 2010)CrossRefzbMATHGoogle Scholar
  14. 14.
    W.R. Gilks, S. Richardson, D. Spiegelhalter (eds.), Markov-Chain Monte Carlo in Practice (Chapman & Hall / CRC, New York, 1996)zbMATHGoogle Scholar
  15. 15.
    S. Brooks, A. Gelman, G. Jones, X.-Li. Meng (eds.), Handbook of Markov-Chain Monte Carlo (Chapman & Hall / CRC, New York, 2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

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