"The underlying concept is to use randomness to solve problems that might be deterministic in principle."
Statistical method for simulating probability distributions.
Markov Chains: :.
Sampling methods: :.
Metropolis Monte Carlo: A method used to simulate the movement of a system of particles by randomly proposing new positions and accepting or rejecting the new configuration based on energy changes.
Importance Sampling Monte Carlo: A method used to increase the efficiency of Monte Carlo simulations by generating samples from a weighted distribution that more closely approximates the target distribution being simulated.
Replica Exchange Monte Carlo: A method used to simulate the behavior of systems with multiple energy minima by allowing replicas of the system to exchange configurations and explore different energy states.
Hybrid Monte Carlo: A method combining the Metropolis algorithm with molecular dynamics to sample phase space more efficiently.
Dynamic Monte Carlo: A method used to calculate time-dependent properties by simulating the evolution of systems over time.
Quantum Monte Carlo: A method used to solve the Schrödinger equation to study systems in quantum mechanics.
Path Integral Monte Carlo: A method used to study the behavior of quantum systems by transforming the problem into a classical one in imaginary time.
Sequential Monte Carlo: A method used to simulate systems with complex, non-linear dynamics and data by re-weighting samples and updating them according to new data.
Markov Chain Monte Carlo: A method used to sample from a probability distribution by constructing a Markov chain whose stationary distribution is the target distribution.
"Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution."
"They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches."
"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."
"Monte Carlo–based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative 'soft' methods."
"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."
"Evaluation of multidimensional definite integrals with complicated boundary conditions."
"When the probability distribution of the variable is parameterized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler. The central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution."
"The samples being generated by the MCMC method will be samples from the desired (target) distribution."
"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."
"These mean-field particle techniques rely on sequential interacting samples."
"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."
"The computational cost associated with a Monte Carlo simulation can be staggeringly high."
"The embarrassingly parallel nature of the algorithm allows this large cost to be reduced (perhaps to a feasible level) through parallel computing strategies in local processors, clusters, cloud computing, GPU, FPGA, etc."