Monte Carlo method

Monte Carlo simulation or Monte Carlo study, and MC simulation is a stochastic method, with the most frequently performed random experiments form the basis. It is thereby tried using probability theory to numerically solve analytically or only consuming solvable problems. As a basis, especially the law of large numbers can be seen. The random experiments can either - about the dice, - be real or performed by generating suitable random numbers. Computer-generated operations can simulate the process at sufficiently frequent random events.

Among the pioneers of the Monte Carlo method in the 1940s include Stanislaw Ulam, Nicholas Metropolis and John von Neumann.

• 3.1 Metropolis Monte Carlo
• 3.2 Sequential Monte Carlo method (SMC )
• 3.3 Quantum Monte Carlo methods ( QMC )

Overview

The Monte Carlo simulation can statements with the following problem groups

• Alternative to analytical problem solving purely mathematical origin, the approximation of pi using Buffon's needle problem of or by the random " sprinkling " of a square on the unit circle with random dots ( here is the percentage of points that lie in the unit circle, about π / 4).
• Generalizing the calculation of the integral of a function on the interval [ 0, 1 ] ( surface area ) and then higher-dimensional integral ( volume).

• The determination of the non-central distribution of the correlation coefficients. With the help of random numbers, the realization of any number of correlation coefficients is simulated. A summary of the coefficients in a frequency table gives an empirical distribution function.
• The properties of estimators in the presence of outliers in data. With the help of the simulation can be shown that the arithmetic mean is no longer a best estimator of the expected value.
• The estimation of distribution parameters.

• Production processes in a manufacturing company to uncover bottlenecks and opportunities in production
• Weather and climate of the Earth.
• Reconstruction method in nuclear medicine.

With the Monte Carlo method can simulate problems with statistical behavior. This method has therefore been found particularly important in physics applications, and two books of the author Kurt Binder are among the most cited publications in this science division.

• It can be the path of a single raindrop simulate colliding with randomly distributed other drops. After the simulation of multiple concrete drops statements about the average droplet size are possible or even temperature and droplet density, where snow or hail arise.
• For the Galtonbrett can calculate the distribution of the balls on the subjects by means of the Gaussian distribution in case. Obstacles for the probability that a ball falls to the right, each exactly 50 % is If this is not given, the overall experiment can be modeled in a Monte Carlo simulation. For this purpose, an appropriate probability, it is assumed for each obstacle and to simulate a large number of ball throwing according to this probability.
• If no analytical formula is known for the valuation of a financial product, can be obtained by Monte Carlo simulation find suitable distribution assumptions of the relevant random variables and put a price in a simple way complex financial contracts ( "exotic" options like ).

History and Origin of name

Enrico Fermi had in the 1930s, the first ideas to Monte Carlo simulations. Were carried out in 1946 by Stanislaw Ulam this and he therefore contacted John von Neumann. This happened while working on the then secret project at the Los Alamos Scientific Laboratory, for which a code name had to be forgiven. Von Neumann chose the name " Monte Carlo", in reference to the casino in Monte Carlo, which lies in the same district of the city-state of Monaco, at the Ulam uncle would borrow money to play with.

Mathematics

Mathematically, the system is a probability weighted path in phase space (general state space ). Monte Carlo simulations are particularly suitable to statistical mean values ​​of size,

Or high-dimensional integrals ( Monte - Carlo integration ) as

To calculate. should be (such as a Boltzmann weight) in this context, a normalized statistical weight. is the value of the quantity in the state. The summation or integration here runs through a space, so the phase space of the particles in the system.

Often the room is so large that the summation can not be completely performed. Instead, now produces a Markov chain of states in whose frequency is distributed as the predetermined weight. Areas of the environment with high weight should therefore be frequently represented in the Markov chain as areas with low weight. ( This is known as importance sampling. ) If this is successful, then the expectation values ​​can be easily calculated as the arithmetic mean of the size of these states of the Markov chain, ie, as

This relationship is based on the law of large numbers. Depending on the physical system, it may be difficult to generate this Markov chain. In particular, one must ensure that the Markov chain is actually covered the entire room and not only scans a part of the room. They say that the algorithm has to be ergodic.

Methods

Metropolis Monte Carlo

The published by Metropolis method to study statistical- mechanical systems using computer simulation is derived from the Monte Carlo integration.

Sequential Monte Carlo method (SMC )

Sequential Monte Carlo methods are suitable for Bayesian state estimation of dynamic systems. The objective is to estimate the system condition as a function of time based on a series of observations of the system and a priori knowledge of the system dynamics. To the complex probability density of the state is approximated by a discrete quantity of particles. Sequential Monte Carlo methods are also called particle filters.

Quantum Monte Carlo methods ( QMC )

Quantum Monte Carlo methods are used for the calculation of physical observables in quantum field theoretical models. Examples are models of theoretical solid state physics such as the Hubbard model or the tJ model.

Common programming packages with Monte Carlo methods

• PYTHIA is a simulation program for particle physics. Here as many different branches of a tree decomposition can be realized by randomly realized a branch at each fork.
• SHERPA is a simulation program for the high-energy particle physics. Built at the TU Dresden, it is now developed by an international working group distributed to Frank Krauss.
• SPICE is a simulation program for analog, digital and mixed electronic circuits. With the Monte Carlo simulation, it is possible to calculate the impact of the variance of the components values ​​within the specified tolerance.