# Monte Carlo method

**Monte CarloMonte Carlo simulationMonte Carlo simulationsMonte-CarloMonte Carlo techniquessimulationMonte Carlo analysisMonte Carlo calculationsMonte Carlo methodsMonte Carlo algorithm**

Monte Carlo methods (or Monte Carlo experiments) are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results.wikipedia

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### Mean field particle methods

**mean field particlemean field interacting particle methodsmean field**

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.

### 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.

### Stanislaw Ulam

**UlamUlam, S.M.Stanisław 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.

### Pseudorandom number generator

**pseudo-random number generatorPRNGpseudorandom number generation**

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.

### 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

### 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.

### Markov chain Monte Carlo

**MCMCMarkov Chain Monte Carlo (MCMC)Markov chain Monte Carlo methods**

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.

### Simulated annealing

**simulated annealing algorithmannealing optimizationsSA**

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.

### 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.

### 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.

### Monte Carlo integration

**Monte Carlocompute an integralMonte Carlo integrator**

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.

### 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.

### 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.

### Molecular dynamics

**dynamicsmolecular dynamicMD**

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.

### Nicholas Metropolis

**MetropolisN. MetropolisMetropolis, 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.

### 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.

### Pseudo-random number sampling

**random samplesOther Types of Random Quantitiespseudo-random number**

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.

### Buffon's needle

**his needlehis needle problem**

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.

### Path tracing

**path-tracingpathtracerbidirectional path tracing**

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.

### Monte Carlo methods in finance

**Monte Carlo simulationsimulationsimulation techniques**

Monte Carlo methods in finance are often used to evaluate investments in projects at a business unit or corporate level, or to evaluate financial derivatives.

Monte Carlo methods are used in finance and mathematical finance to value and analyze (complex) instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the distribution of their value over the range of resultant outcomes.

### 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.

### Biology Monte Carlo method

Biology Monte Carlo method

Biology Monte Carlo methods (BioMOCA) have been developed at the University of Illinois at Urbana-Champaign to simulate ion transport in an electrolyte environment through ion channels or nano-pores embedded in membranes.

### Simultaneous localization and mapping

**SLAMvisual simultaneous localization and mappingMapping**

In autonomous robotics, Monte Carlo localization can determine the position of a robot. It is often applied to stochastic filters such as the Kalman filter or particle filter that forms the heart of the SLAM (simultaneous localization and mapping) algorithm.

Monte Carlo methods).

### Probability distribution

**distributionprobability distributionscontinuous probability distribution**

When the probability distribution of the variable is parametrized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler. 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. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a 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.

### Comparison of risk analysis Microsoft Excel add-ins

**risk-analysis add-in**

Comparison of risk analysis Microsoft Excel add-ins

The following is a comparison of various add-in packages available to do Monte Carlo probabilistic modeling and risk analysis.