Student - William Sealy Gost (13.6.1876 - 16.10.1937)albania
This birth-and-death process is suffering from labor pains; it will be the death of me yet. (Student Sayings)
Introduction
The Monte Carlo method provides approximate solutions to a variety of mathematical problems by performing statistical sampling experiments on a computer. The method applie
s to problems with no probabilistic content as well as to tho with inherent probabilistic structure. Among all numerical methods that rely on N-point evaluations in M-dimensional space to produce an approximate solution, the Monte Carlo method has absolute error of estimate that decreas as N superscript -1/2 whereas, in the abnce of exploitable special structure all others have errors that decrea as N superscript -1/M at best.
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dif>pittsburghHistory of Monte Carlo method
The method is called after the city in the Monaco principality, becau of a roulette, a simple random number generator. The name and the systematic development of Monte Carlo methods dates from about 1944.
There are however a number of isolated and undeveloped instances on much earlier occasions.
For example, in the cond half of the nineteenth century a number of people performed alios
五金工具英语experiments, in which they threw a needle in a haphazard manner onto a board ruled with parallel straight lines and inferred the value of PI = 3.14… from obrvations of the number of interctions between needle and lines. An account of this playful diversion (indulged in by certain Captain Fox, amongst others, whilst recovering from wounds incurred in the American Civil War) occurs in a paper Hall (A. HALL 1873. " On an experimental determination of PI"). The author of this WEB page has developed the software for simulating of this experiment. Try JAVA implementation of Buffon's needle experiment for the determination of PI.
In 1899 Lord Rayleigh showed that a one-dimensional random walk without absorbing barriers could provide an approximate solution to a parabolic differential equation.
A. N. Kolmogorov (12.4.1903-20.10.1987)snow leopard
In 1931 Kolmogorov showed the relationship between Markov stochastic process and certain integro-differential equations.
In early part of the twentieth century, British statistical schools indulged in a fair amount of unsophisticated Monte Carlo work. Most of this ems to have been of didactic character and rarely ud for rearch or discovery. Only on a few rare occasions was the emphasis on original discovery rather than comforting verification. In 1908 Student (W.S. Gost) ud experimental sampling to help him towards his discovery of the distribution of the correlation coefficient. In the same year Student also ud sampling to bolster his faith in his so-called t-distribution, which he had derived by a somewhat shaky and incomplete theoretical analysis.
The real u of Monte Carlo methods as a rearch tool stems from work on the atomic bomb during the cond world war. This work involved a direct simulation of the probabilistic problems concerned with random neutron diffusion in fissile material; but eve
n at an early stage of the investigations, von Neumann and Ulam refined this particular " Russian roulette" and "splitting" methods. However, the systematic development of the ideas had to await the work of Harris and Herman Kahn in 1948. About 1948 Fermi, Metropolis, and Ulam obtained Monte Carlo estimates for the eigenvalues of Schrodinger equation.
John von Neumann (28.12.1903-8.2.1957)
In about 1970, the newly developing theory of computational complexity began to provide a more preci and persuasive rationale for employing the Monte Carlo method. The theo
marry me什么意思中文ry identified a class of problems for which the time to evaluate the exact solution to a problem within the class grows, at least, exponentially with M. The question to be resolved was whether or not the Monte Carlo method could estimate the solution to a problem in this intractable class to within a specified statistical accuracy in time bounded above by a polynomial in M. Numerous examples now support this contention. Karp (1985) shows this property for estimating reliability in a planar multiterminal network with randomly failing edges. Dyer (1989) establish it for estimating the volume of a convex body in M-dimensional Euclidean space. Broder (1986) and Jerrum and Sinclair (1988) establish the property for estimating the permanent of a matrix or, equivalently, the number of perfect matchings in a bipartite graph.
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