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Randomness

randomchancerandomly
Their essential idea is using randomness to solve problems that might be deterministic in principle.
Monte Carlo methods, which rely on random input (such as from random number generators or pseudorandom number generators), are important techniques in science, as, for instance, in computational science.

Mean field particle methods

mean fieldmean field interacting particle methodsmean field particle
In contrast with traditional Monte Carlo and MCMC methodologies these mean field particle techniques rely on sequential interacting samples.
Mean field particle methods are a broad class of interacting type Monte Carlo algorithms for simulating from a sequence of probability distributions satisfying a nonlinear evolution equation These flows of probability measures can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depends on the distributions of the current random states.

Pseudorandom number generator

pseudo-random number generatorPRNGpseudorandom
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.
PRNGs are central in applications such as simulations (e.g. for the Monte Carlo method), electronic games (e.g. for procedural generation), and cryptography.

Stanislaw Ulam

UlamStanisław UlamStanislaw Marcin Ulam
The modern version of the Markov Chain Monte Carlo method was invented in the late 1940s by Stanislaw Ulam, while he was working on nuclear weapons projects at the Los Alamos National Laboratory.
He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, discovered the concept of cellular automaton, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion.

Markov chain Monte Carlo

MCMCMarkov Chain Monte Carlo (MCMC)Bayesian MCMC
When the probability distribution of the variable is parametrized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler.
Random walk Monte Carlo methods are a kind of random simulation or Monte Carlo method.

Simulation

simulatorsimulatesimulations
Monte Carlo methods were central to the simulations required for the Manhattan Project, though severely limited by the computational tools at the time.
Stochastic Simulation is a simulation where some variable or process is regulated by stochastic factors and estimated based on Monte Carlo techniques using pseudo-random numbers, so replicated runs from same boundary conditions are expected to produce different results within a specific confidence band

Simulated annealing

simulated annealing algorithmannealedannealing
Monte Carlo simulations invert this approach, solving deterministic problems using a probabilistic analog (see Simulated annealing).
The method is an adaptation of the Metropolis–Hastings algorithm, a Monte Carlo method to generate sample states of a thermodynamic system, published by N. Metropolis et al. in 1953.

Quantum Monte Carlo

Monte-Carloquantum Monte Carlo methodQuantum Monte Carlo simulations
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.
The diverse flavor of quantum Monte Carlo approaches all share the common use of the Monte Carlo method to handle the multi-dimensional integrals that arise in the different formulations of the many-body problem.

Buffon's needle problem

his needlehis needle problemneedle
An early variant of the Monte Carlo method can be seen in the Buffon's needle experiment, in which can be estimated by dropping needles on a floor made of parallel and equidistant strips.
The solution, in the case where the needle length is not greater than the width of the strips, can be used to design a Monte Carlo method for approximating the number π, although that was not the original motivation for de Buffon's question.

Nicholas Metropolis

MetropolisMetropolis, NicholasMetropolis, Nicholas
A colleague of von Neumann and Ulam, Nicholas Metropolis, suggested using the name Monte Carlo, which refers to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money from relatives to gamble.
At Los Alamos, in the 1950s, a group of researchers led by Metropolis, including John von Neumann and Stanislaw Ulam, developed the Monte Carlo method.

Probability distribution

distributioncontinuous probability distributiondiscrete probability distribution
When the probability distribution of the variable is parametrized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution. 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.
A frequent problem in statistical simulations (the Monte Carlo method) is the generation of pseudo-random numbers that are distributed in a given way.

Pi

ππ\pi
Given that the ratio of their areas is undefined⁄4, the value of [[pi|]] can be approximated using a Monte Carlo method:
Monte Carlo methods, which evaluate the results of multiple random trials, can be used to create approximations of.

Monte Carlo integration

Monte Carlocompute an integralintegrating
For example, Ripley defines most probabilistic modeling as stochastic simulation, with Monte Carlo being reserved for Monte Carlo integration and Monte Carlo statistical tests.
It is a particular Monte Carlo method that numerically computes a definite integral.

Numerical integration

quadratureintegrationnumerical quadrature
Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.
Monte Carlo methods and quasi-Monte Carlo methods are easy to apply to multi-dimensional integrals.

Monte Carlo molecular modeling

In statistical physics Monte Carlo molecular modeling is an alternative to computational molecular dynamics, and Monte Carlo methods are used to compute statistical field theories of simple particle and polymer systems.
Monte Carlo molecular modeling is the application of Monte Carlo methods to molecular problems.

Random number generation

random number generatorrandom numberrandom numbers
Monte Carlo simulation methods do not always require truly random numbers to be useful (although, for some applications such as primality testing, unpredictability is vital).
Random number generators are very useful in developing Monte Carlo-method simulations, as debugging is facilitated by the ability to run the same sequence of random numbers again by starting from the same random seed.

Quasi-Monte Carlo method

quasi-Monte CarloMonte Carlo methods
Methods based on their use are called quasi-Monte Carlo methods.
This is in contrast to the regular Monte Carlo method or Monte Carlo integration, which are based on sequences of pseudorandom numbers.

Pseudo-random number sampling

random samplesmethodsOther Types of Random Quantities
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.
Historically, basic methods of pseudo-random number sampling were developed for Monte-Carlo simulations in the Manhattan project; they were first published by John von Neumann in the early 1950s.

Ensemble forecasting

ensembleensemble forecastsensemble forecast
Monte Carlo methods are also used in the ensemble models that form the basis of modern weather forecasting.
Ensemble forecasting is a form of Monte Carlo analysis.

Molecular dynamics

dynamicsMDmolecular dynamic
In statistical physics Monte Carlo molecular modeling is an alternative to computational molecular dynamics, and Monte Carlo methods are used to compute statistical field theories of simple particle and polymer systems.
Following the earlier successes of Monte Carlo simulations, the method was first developed by Fermi, Pasta, Ulam and Tsingou in the mid 50s.

ENIAC

Electronic computerElectronic Numerical Integrator And ComputerENIAC (Electronic Numerical Integrator and Computer)
Immediately after Ulam's breakthrough, John von Neumann understood its importance and programmed the ENIAC computer to carry out Monte Carlo calculations.
Related to ENIAC's role in the hydrogen bomb was its role in the Monte Carlo method becoming popular.

Expected value

expectationexpectedmean
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.k.a. the sample mean) of independent samples of the variable.
This property is often exploited in a wide variety of applications, including general problems of statistical estimation and machine learning, to estimate (probabilistic) quantities of interest via Monte Carlo methods, since most quantities of interest can be written in terms of expectation, e.g., where is the indicator function of the set \mathcal{A}.

Direct simulation Monte Carlo

In fluid dynamics, in particular rarefied gas dynamics, where the Boltzmann equation is solved for finite Knudsen number fluid flows using the direct simulation Monte Carlo method in combination with highly efficient computational algorithms.
Direct Simulation Monte Carlo (DSMC) method uses probabilistic (Monte Carlo) simulation to solve the Boltzmann equation for finite Knudsen number fluid flows.

Robert D. Richtmyer

R. D. RichtmyerRichtmyer, Robert D.Robert Davis Richtmyer
The origins of Quantum Monte Carlo methods are often attributed to Enrico Fermi and Robert Richtmyer who developed in 1948 a mean field particle interpretation of neutron-chain reactions, but the first heuristic-like and genetic type particle algorithm (a.k.a. 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 1984 In molecular chemistry, the use of genetic heuristic-like particle methodologies (a.k.a. pruning and enrichment strategies) can be traced back to 1955 with the seminal work of Marshall. N. Rosenbluth and Arianna. W. Rosenbluth.
Richtmyer used the massive IBM SSEC calculator for some of the first large-scale uses of what would be called the Monte Carlo method.

Path tracing

path-tracingbidirectional path tracingbounces of light
Path tracing, occasionally referred to as Monte Carlo ray tracing, renders a 3D scene by randomly tracing samples of possible light paths.
Path tracing is a computer graphics Monte Carlo method of rendering images of three-dimensional scenes such that the global illumination is faithful to reality.