Monte Carlo method | Wikipedia audio article
Channel: wikipedia tts
Date: 06/10/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Monte_Carlo_method


00:00:08 1 Overview
00:00:14 2 History
00:00:25 3 Definitions
00:00:31 3.1 Monte Carlo and random numbers
00:00:37 3.2 Monte Carlo simulation versus "what if" scenarios
00:00:42 4 Applications
00:00:48 4.1 Physical sciences
00:00:54 4.2 Engineering
00:00:59 4.3 Climate change and radiative forcing
00:01:05 4.4 Computational biology
00:01:11 4.5 Computer graphics
00:01:17 4.6 Applied statistics
00:01:22 4.7 Artificial intelligence for games
00:01:28 4.8 Design and visuals
00:01:34 4.9 Search and rescue
00:01:39 4.10 Finance and business
00:01:45 4.11 Law
00:01:51 5 Use in mathematics
00:01:57 5.1 Integration
00:02:02 5.2 Simulation and optimization
00:02:08 5.3 Inverse problems



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SUMMARY
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Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. 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 mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.
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 mathematics, evaluation of multidimensional definite integrals  with complicated boundary conditions.  In application to systems engineering problems (space, oil exploration, aircraft design, etc.), Monte Carlo–based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative "soft" methods.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.k.a. the sample mean) of independent samples of the variable. When the probability distribution of the variable is parametrized, 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. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution. By the ergodic theorem, the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler.
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). In other instances we are given a flow of probability distributions with an increasing level of sampling complexity (path spaces models with an increasing time horizon, Boltzmann–Gibbs measures associated with decreasing temperature parameters, and many others). These models can also be seen as the evolution of the law of the random states of a nonlinear Markov chain. A natural way to simulate these sophisticated nonlinear Markov process ...