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The Monte Carlo Method: an Introduction

The Monte Carlo Method: an Introduction. Detlev Reiter. Research Centre Jülich (FZJ) D -52425 Jülich http://www.fz-juelich.de e-mail: d.reiter@fz-juelich.de Tel.: 02461 / 61-5841. Vorlesung HHU Düsseldorf , WS 07/08 March 2008. There are two dominant methods of simulation

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The Monte Carlo Method: an Introduction

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  1. The Monte Carlo Method: an Introduction Detlev Reiter Research Centre Jülich (FZJ) D -52425 Jülich http://www.fz-juelich.de e-mail: d.reiter@fz-juelich.de Tel.: 02461 / 61-5841 Vorlesung HHU Düsseldorf, WS 07/08 March 2008

  2. There are two dominant methods of simulation • for complex many particle systems • Molecular Dynamics • Solve the classical equations of motion from mechanics. • Particles interact via a given interaction potential. • Deterministic behaviour (within numerical precision). • Find temporal evolution. • 2) Monte Carlo Simulation • Find mean values (expectation values) of some system components. • Random behaviour from given probability distribution laws. The Monte Carlo technique is a very far spread technique, because it is not limited to systems of particles.

  3. This lecture • Brief introduction: simulation • What is the Monte Carlo Method • Random number generation • Integration by Monte Carlo • Tomorrow: one (of many) particular application: • particle transport by Monte Carlo

  4. ASDEX-UPDRADE (IPP Garching)

  5. Monte Carlo particle trajectories, ions and neutral particles

  6. Basic principle of the Monte Carlo method The task: calculate (estimate) a number I (one number only. Not an entire functional dependence). Historic example: A dull way to calculate p Numerically: look for an appropriate convergent series and evaluate this approximately by Monte Carlo: look for a stochastic model (i.e.: (W, s, p, X): probability space with random variable X) Example: throw a needle an a sheet with equidistant parallel stripes. Distance between stripes: D, length of needle: L, L<D.

  7. First application of Monte Carlo Method The needle experiment of Comte de Buffon, 1733 (french biologist, 1707-1788) What is the probability p, that a needle (length L), which randomly falls on a sheet, crosses one of the lines (distance D)? (N trials, n „hits“)

  8. Yt =1, if crossing, Yt=0 else, then

  9. Today: • Using a computer to generate random events: • We need to be able to generate random numbers X • with any given probability function f(x), or • a given cumulative distribution F(x) . • Uniformly distributed random numbers • General random numbers: can be obtained • from a sequence of independent uniform random • numbers

  10. Random number generation f(x) 1/(b-a) a b

  11. We will see next: Any continuous distribution can be generated from uniform random numbers on [0,1] Any discrete distribution can be generated from uniform random numbers on [0,1] Hence: Any given distribution can be generated from uniform random numbers on [0,1]

  12. Strategy: try to transform F to another distribution, such that inverse of new F is explicitly known.

  13. Example: Normal (Gaussian) distribution Cumulative distr. function Inverse cumul. distr. fct. best format of storing distributions for Monte Carlo applications: „Inverse cumulative distribution function F-1(x)“, x uniform [0,1]

  14. Exercise (and most important example:) Generate random numbers from a Gaussian. Let X, Y two independent Gaussian random numbers. Transform to polar coordiantes (Jacobian!) R, Φ Sample Φ (trivial, it is uniform on 2π) Apply inversion method for R Transform sampled Φ, R back to X, Y. This is a pair of Gaussians. (Box-Muller Method)

  15. Exponential distribution by „inversion“ Note: Z and 1-Z have same distrib. (see tomorrow)

  16. Cauchy: e.g.: natural Line broadening

  17. (stepwise constant, with steps at points T)

  18. Rejection enclosing rectangle Reject z Accept z, take x=z y uniform z, uniform f(x): distribution density y=f(x) sample x from f(x) X

  19. NEXT: Any Monte Carlo estimate can be regarded as a mean value, i.e. an integral (or sum) over a given probability distribution, ususally in a high dimensional space (e.g. of random walks….) Generic Monte Carlo: Integration Hence: How does Monte Carlo integration work?

  20. Hit or Miss known area miss hit x2 uniform x1, uniform I: unknown area f(x) I = ∫ f(x) dx X

  21. Suggestion: try again with previous example from dull and crude Monte Carlo

  22. Outlook: next lecture (tomorrow)

  23. END

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