Monte Carlo Method

(redirected from Monte Carlo methods)
Also found in: Dictionary, Medical, Financial.

Monte Carlo method

[′män·tē ′kär·lō ‚meth·əd]
(statistics)
A technique which obtains a probabilistic approximation to the solution of a problem by using statistical sampling techniques.

Monte Carlo Method

 

a numerical method for solving mathematical problems using the modeling of random processes and events. The term “Monte Carlo method” was coined in 1949, although some calculations using the modeling of random events had been previously performed by statisticians. (The method is named after the city of Monte Carlo, which is well known for its Casino.) The Monte Carlo method gained wide-spread use only after the development of high-speed computers. Programs for carrying out calculations on computers using the Monte Carlo method are comparatively simple and, as a rule, do not require large-capacity internal memory.


Monte Carlo Method

 

a method in computational and applied mathematics based on the simulation of random variables and the construction of statistical estimates of desired values.

The Monte Carlo method is generally considered to have been formulated in 1944 by the American scientists J. von Neumann and S. Ulam. In connection with work on the development of atomic reactors, von Neumann and Ulam made extensive use of the apparatus of probability theory in the computer solution of applied problems. At first the Monte Carlo method was used primarily to solve complex problems of radiation transport theory and neutron physics for which traditional numerical methods were not suited. The method was subsequently applied to a large number of different problems in statistical mechanics. The method is now used in such areas as game theory, queuing theory, mathematical economics, and the theory of message transmission in the presence of interference.

To solve a deterministic problem by the Monte Carlo method, a probabilistic model is constructed, and the desired quantity, for example, a multidimensional integral, is represented as the mathematical expectation of a function of a stochastic process. The process is then simulated on a computer. Probabilistic models are known, for example, for the computation of integrals, the solving of integral equations of the second kind, the solving of systems of linear algebraic equations, the solving of boundary problems for elliptic equations, and the estimation of the eigenvalues of linear operators. An estimate with a low degree of error can be obtained by proper selection of the probabilistic model.

The simulation of random variables with given distributions plays a special role in various applications of the Monte Carlo method. Such simulation is generally done by transforming one or more independent values of a random number a that is uniformly distributed in the interval (0, 1). The sequences of sample values of a are usually obtained with a computer through the use of theoretical numerical algorithms, among which the residue method has become the most common. Such numbers are said to be pseudorandom; they are checked through statistical tests and the solution of standard problems.

If, in an estimate based on the Monte Carlo method, the random variables involved are determined by the actual process being simulated, then we speak of direct modeling. Such an estimate is inefficient if rare events are involved because the actual process contains little information about such events. This inefficiency is usually manifested in too large a probabilistic error (variance) of the random estimates of the desired quantities. Many techniques have been developed for reducing the variance of these estimates within the framework of the Monte Carlo method. Almost all the techniques are based on modifications of the simulation through the use of information on the function of the random variables whose expectation is being computed.

The Monte Carlo method has had, and continues to have, a considerable influence on the development of other methods of computational mathematics—for example, methods of numerical integration. The Monte Carlo method has been successfully used as a supplementary method in combination with other methods to solve many problems. (See also.)

REFERENCES

Metod Monte-Karlo v problème perenosa izluchenii. Moscow, 1967.
Metodstatisticheskikh ispytanii (Metod Monte-Karlo). Moscow, 1962.
Reshenie priamykh i nekotorykh obratnykh zadach atmosfernoi optiki metodom Monte-Karlo. Novosibirsk, 1968.
Ermakov, S. M. Metod Monte-Karlo i smezhnye voprosy. Moscow, 1971.
Mikhailov, G. A. Nekotorye voprosy teorii metodov Monte-Karlo. Novosibirsk, 1974.

G. I. MARCHUK

Monte Carlo method

A technique that provides approximate solutions to problems expressed mathematically. Using random numbers and trial and error, it repeatedly calculates the equations to arrive at a solution. Many of the Monte Carlo methods and practices used to be referred to as rather generic "statistical sampling." Monte Carlo, of course, is a historical reference to the famous casino in Monaco.
References in periodicals archive ?
Thus, it is preferable to estimate them by means of Monte Carlo methods.
The simulation by means of the Monte Carlo method also included an index of instability of the EU funds accessed in an aggregation algorithm of TFA (Total Funds Accessed), focused on two variables, the allocated funds (F[A.
The reason for this issue is that the failure probability is too small in this case, so that the random numbers are so low that the Monte Carlo method fails to predict the solution at an acceptable level of accuracy.
Among statistical methods, "the error in the approximation is smaller for Monte Carlo methods than for other comparable methods.
Making Monte Carlo experiments an integral part of the introductory course can help support this objective in several ways: first, developing an understanding of Monte Carlo methods leads to a deeper understanding of fundamental principles of econometrics and statistics; second, Monte Carlo experiments can help illustrate key econometric concepts and results; third, conducting Monte Carlo experiments gives students hands-on experience that is both instructive and enjoyable; and finally, experience with simulation gives beginning students an understanding of an important method of econometric research.
KEYWORDS: continuous random variables, computer simulations, Monte Carlo methods, MATLAB.
The proposal is divided into three research projects whose goals are: (1) Obtain upper and lower bounds for the density of the solution to the two types of SPDEs (i) and (ii), by means of the Malliavin Calculus; (2) Use these bounds in order to prove the local asymptotic normality (LAN) for the models (i) and (ii), and then apply Hajek-Lecam~s theorem to obtain asymptotically efficient estimators for the parameter of the equations; (3) Study Monte Carlo methods and exact simulation of the SDE model with jumps (i), and apply these computational methods to the following financial problems: jump volatility models and numerical computations of greeks.
His topics are random number generation, simulating statistical models, Monte Carlo methods, Markov Chain Monte Carlo methods, beyond Monte Carlo, and continuous-time models.
Models and analyses are developed using Bayesian approaches and tools including Markov chain and sequential Monte Carlo methods.
For a tensile component of a landing gear was simulate an accelerated life testing with Monte Carlo methods.
And applications of operation research, linear programing, game theory, probability theory, queuing theory, Monte Carlo methods, decision trees and CPM are going to be around for a long time.

Full browser ?