Monte Carlo method Monte Carlo method A Box-Muller transform is a method of generating pairs of independent standard normally distributed (zero expectation, unit variance) random numbers, given a source of uniformly distributed random numbers. There are two kinds: ...more on Wikipedia about "Box-Muller transform"
In Monte Carlo methods, one or more control variates may be employed to achieve variance reduction by exploiting the correlation between statistics. ...more on Wikipedia about "Control variate"
Direct simulation Monte Carlo (DSMC) is a computational Monte Carlo algorithm for the stochastic simulation of rarefied gas flows. ...more on Wikipedia about "Direct simulation Monte Carlo"
In chemistry, Dynamic Monte Carlo (DMC) is a method for modeling the dynamic behaviors of molecules by comparing the rates of individual steps with random numbers. Unlike the Metropolis Monte Carlo method, which has been employed to study systems at equilibrium, the DMC method is used to investigate nonequilibrium systems such as reaction, diffusion, etc. This method is mainly applied to analyze the behavior of adsorbates on surfaces. ...more on Wikipedia about "Dynamic Monte Carlo method"
In mathematics and physics, Gibbs sampling is an algorithm to generate a sequence of samples from the joint probability distribution of two or more random variables. The purpose of such a sequence is to approximate the joint distribution (as with a histogram), or to compute an integral (such as an expected value). Gibbs sampling is a special case of the Metropolis-Hastings algorithm, ...more on Wikipedia about "Gibbs sampling"
Importance sampling(IS) is a variance reduction technique that can be used in the Monte Carlo method. The idea behind IS is that certain values of the input random variables in a simulation have more impact on the parameter being estimated than others. If these "important" values are emphasized by sampling more frequently, then the estimator variance can be reduced. Hence, the basic methodology in IS is to choose a distribution which "encourages" the important values. This use of a "biased" distributions will result in a biased estimator if it is applied directly in the simulation. However, the simulation outputs are weighted to correct for the use of the biased distribution, and this ensures that the new IS estimator is unbiased. The weight is given by the likelihood ratio, that is, the Radon-Nikodym derivative of the true underlying distribution with respect to the biased simulation distribution. ...more on Wikipedia about "Importance sampling"
Markov chain Monte Carlo (MCMC) methods, sometimes called random walk Monte Carlo methods, are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its stationary distribution. The state of the chain after a large number of steps is then used as a sample from the desired distribution. The quality of the sample improves as a function of the number of steps. ...more on Wikipedia about "Markov chain Monte Carlo"
This SIGGRAPH 1997 paper by Eric Veach and Leonidas J. Guibas describes an application of a variant of the Monte Carlo method called the Metropolis algorithm to the rendering equation for generating 3d images from detailed physical descriptions of three dimensional scenes. ...more on Wikipedia about "Metropolis light transport"
In mathematics and physics, the Metropolis-Hastings algorithm is an algorithm used to generate a sequence of samples from the probability distribution of one or more variables. ...more on Wikipedia about "Metropolis-Hastings algorithm"
The MISER algorithm is a method for reducing error in the Monte Carlo simulation by focusing the search in areas of the function with higher variance using recursive stratified sampling. ...more on Wikipedia about "MISER algorithm"
Monte Carlo methods are a class of computational algorithms for simulating the behavior of various physical and mathematical systems. They are distinguished from other simulation methods (such as molecular dynamics) by being stochastic, that is nondeterministic in some manner - usually by using random numbers (or more often pseudo-random numbers) - as opposed to deterministic algorithms. A classic use is for the evaluation of definite integrals, particularly multidimensional integrals with complicated boundary conditions. ...more on Wikipedia about "Monte Carlo method"
In the field of financial mathematics, many problems, for instance the problem of finding the arbitrage-free value of a particular derivative, boil down to the computation of a particular integral. In many cases these integrals can be valued analytically, and in still more cases they can be valued using numerical integration. However when the number of dimensions (or degrees of freedom) in the problem is large, numerical integration methods become intractable. In these cases it is common to resort to the more widely applicable Monte Carlo methods to solve the problem. For large dimension integrals as can very often occur in financial problems, Monte Carlo methods converge to the solution more quickly than numerical integration methods. The advantage Monte Carlo methods offer increases as the dimensions of the problem increase. ...more on Wikipedia about "Monte Carlo methods in finance"
OpenBUGS is a computer software for the Bayesian analysis of complex statistical models using Markov chain Monte Carlo (MCMC) methods. OpenBUGS is the open source variant of WinBUGS ( Bayesian inference Using Gibbs Sampling). The latest version can run on Windows and Linux, as well as from inside the R statistical package. ...more on Wikipedia about "OpenBUGS"
Parallel tempering is a simulation method aimed at improving the dynamic properties of Monte Carlo method simulations of physical systems, and of Markov chain Monte Carlo sampling methods more generally. The method originates from Geyer and was later developed, among others, by G. Parisi et al. ...more on Wikipedia about "Parallel tempering"
Particle filters, also known as Sequential Monte Carlo methods (SMC), are sophisticated model estimation techniques based on simulation. ...more on Wikipedia about "Particle filter"
Quantum Monte Carlo is the application of the Monte Carlo method for solving the Schrödinger wave equation of an electronic system. ...more on Wikipedia about "Quantum Monte Carlo"
In numerical analysis, a quasi-Monte Carlo method is a method for the computation of an integral (or some other problem) which is based on low-discrepancy sequences. This is in contrast to a regular Monte Carlo method, which is based on sequences of pseudorandom numbers. ...more on Wikipedia about "Quasi-Monte Carlo method"
The VEGAS algorithm, due to G. P. Lepage, is a method for reducing error in the Monte Carlo simulation by using a known or approximate probability distribution function to concentrate the search in those areas of the graph that make the greatest contribution to the final integral. ...more on Wikipedia about "VEGAS algorithm"
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