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In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom , such as fluids, disordered materials, strongly coupled solids, and cellular structures see cellular Potts model , interacting particle systems , McKean-Vlasov processes , kinetic models of gases.
Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk in business and, in maths, evaluation of multidimensional definite integrals with complicated boundary conditions.
In application to systems engineering problems space, oil exploration , aircraft design, etc. In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation.
By the law of large numbers , integrals described by the expected value of some random variable can be approximated by taking the empirical mean a.
That is, in the limit, the samples being generated by the MCMC method will be samples from the desired target distribution. In other problems, the objective is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation.
These flows of probability distributions can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depend on the distributions of the current random states see McKean-Vlasov processes , nonlinear filtering equation.
These models can also be seen as the evolution of the law of the random states of a nonlinear Markov chain. In contrast with traditional Monte Carlo and MCMC methodologies these mean field particle techniques rely on sequential interacting samples.
The terminology mean field reflects the fact that each of the samples a. When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so that the statistical interaction between particles vanishes.
For example, consider a quadrant circular sector inscribed in a unit square. In this procedure the domain of inputs is the square that circumscribes the quadrant.
We generate random inputs by scattering grains over the square then perform a computation on each input test whether it falls within the quadrant.
Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of pseudorandom number generators , which were far quicker to use than the tables of random numbers that had been previously used for statistical sampling.
Before the Monte Carlo method was developed, simulations tested a previously understood deterministic problem, and statistical sampling was used to estimate uncertainties in the simulations.
Monte Carlo simulations invert this approach, solving deterministic problems using a probabilistic analog see Simulated annealing.
In the s, Enrico Fermi first experimented with the Monte Carlo method while studying neutron diffusion, but did not publish anything on it. The modern version of the Markov Chain Monte Carlo method was invented in the late s by Stanislaw Ulam , while he was working on nuclear weapons projects at the Los Alamos National Laboratory.
In , physicists at Los Alamos Scientific Laboratory were investigating radiation shielding and the distance that neutrons would likely travel through various materials.
Despite having most of the necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus, and how much energy the neutron was likely to give off following a collision, the Los Alamos physicists were unable to solve the problem using conventional, deterministic mathematical methods.
Ulam had the idea of using random experiments. He recounts his inspiration as follows:. Being secret, the work of von Neumann and Ulam required a code name.
Though this method has been criticized as crude, von Neumann was aware of this: Monte Carlo methods were central to the simulations required for the Manhattan Project , though severely limited by the computational tools at the time.
In the s they were used at Los Alamos for early work relating to the development of the hydrogen bomb , and became popularized in the fields of physics , physical chemistry , and operations research.
The Rand Corporation and the U. Air Force were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields.
The theory of more sophisticated mean field type particle Monte Carlo methods had certainly started by the mids, with the work of Henry P.
Harris and Herman Kahn, published in , using mean field genetic -type Monte Carlo methods for estimating particle transmission energies.
Metaheuristic in evolutionary computing. The origins of these mean field computational techniques can be traced to and with the work of Alan Turing on genetic type mutation-selection learning machines  and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey.
Quantum Monte Carlo , and more specifically Diffusion Monte Carlo methods can also be interpreted as a mean field particle Monte Carlo approximation of Feynman - Kac path integrals.
Resampled or Reconfiguration Monte Carlo methods for estimating ground state energies of quantum systems in reduced matrix models is due to Jack H.
Hetherington in  In molecular chemistry, the use of genetic heuristic-like particle methodologies a. The use of Sequential Monte Carlo in advanced signal processing and Bayesian inference is more recent.
It was in , that Gordon et al. Particle filters were also developed in signal processing in the early by P. From to , all the publications on Sequential Monte Carlo methodologies including the pruning and resample Monte Carlo methods introduced in computational physics and molecular chemistry, present natural and heuristic-like algorithms applied to different situations without a single proof of their consistency, nor a discussion on the bias of the estimates and on genealogical and ancestral tree based algorithms.
The mathematical foundations and the first rigorous analysis of these particle algorithms are due to Pierre Del Moral   in There is no consensus on how Monte Carlo should be defined.
For example, Ripley  defines most probabilistic modeling as stochastic simulation , with Monte Carlo being reserved for Monte Carlo integration and Monte Carlo statistical tests.
Sawilowsky  distinguishes between a simulation , a Monte Carlo method, and a Monte Carlo simulation: Kalos and Whitlock  point out that such distinctions are not always easy to maintain.
For example, the emission of radiation from atoms is a natural stochastic process. It can be simulated directly, or its average behavior can be described by stochastic equations that can themselves be solved using Monte Carlo methods.
The main idea behind this method is that the results are computed based on repeated random sampling and statistical analysis.
The Monte Carlo simulation is in fact random experimentations, in the case that, the results of these experiments are not well known.
Monte Carlo simulations are typically characterized by a large number of unknown parameters, many of which are difficult to obtain experimentally.
The only quality usually necessary to make good simulations is for the pseudo-random sequence to appear "random enough" in a certain sense.
What this means depends on the application, but typically they should pass a series of statistical tests. Testing that the numbers are uniformly distributed or follow another desired distribution when a large enough number of elements of the sequence are considered is one of the simplest, and most common ones.
Sawilowsky lists the characteristics of a high quality Monte Carlo simulation: Pseudo-random number sampling algorithms are used to transform uniformly distributed pseudo-random numbers into numbers that are distributed according to a given probability distribution.
Low-discrepancy sequences are often used instead of random sampling from a space as they ensure even coverage and normally have a faster order of convergence than Monte Carlo simulations using random or pseudorandom sequences.
Methods based on their use are called quasi-Monte Carlo methods. RdRand is the closest pseudorandom number generator to a true random number generator.
No statistically-significant difference was found between models generated with typical pseudorandom number generators and RdRand for trials consisting of the generation of 10 7 random numbers.
There are ways of using probabilities that are definitely not Monte Carlo simulations — for example, deterministic modeling using single-point estimates.
Scenarios such as best, worst, or most likely case for each input variable are chosen and the results recorded.
By contrast, Monte Carlo simulations sample from a probability distribution for each variable to produce hundreds or thousands of possible outcomes.
The results are analyzed to get probabilities of different outcomes occurring. The samples in such regions are called "rare events".
Monte Carlo methods are especially useful for simulating phenomena with significant uncertainty in inputs and systems with a large number of coupled degrees of freedom.
Values in the middle near the mean are most likely to occur. Examples of variables described by normal distributions include inflation rates and energy prices.
Values are positively skewed, not symmetric like a normal distribution. Examples of variables described by lognormal distributions include real estate property values, stock prices, and oil reserves.
All values have an equal chance of occurring, and the user simply defines the minimum and maximum. Examples of variables that could be uniformly distributed include manufacturing costs or future sales revenues for a new product.
The user defines the minimum, most likely, and maximum values. Values around the most likely are more likely to occur.
Variables that could be described by a triangular distribution include past sales history per unit of time and inventory levels.
The user defines the minimum, most likely, and maximum values, just like the triangular distribution. However values between the most likely and extremes are more likely to occur than the triangular; that is, the extremes are not as emphasized.
An example of the use of a PERT distribution is to describe the duration of a task in a project management model. The user defines specific values that may occur and the likelihood of each.
An example might be the results of a lawsuit: During a Monte Carlo simulation, values are sampled at random from the input probability distributions.
Each set of samples is called an iteration, and the resulting outcome from that sample is recorded. Monte Carlo simulation does this hundreds or thousands of times, and the result is a probability distribution of possible outcomes.
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