Monte Carlo methodWord文档下载推荐.docx

1、Data analysis VisualizationPotentialsshowLennard-Jones potential Yukawa potential Morse potentialFluid dynamicsshowFinite element Riemann solverSmoothed particle hydrodynamicsMonte Carlo methodsshowIntegration Gibbs sampling Metropolis algorithmParticleshowN-body Particle-in-cellMolecular dynamicsSc

2、ientistsshowGodunov Ulam von Neumann v t eMonte Carlo methods （or Monte Carlo experiments） are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems. These me

3、thods are most suited to calculation by a computer and tend to be used when it is infeasible to compute an exact result with a deterministic algorithm.1 This method is also used to complement theoretical derivations.Monte Carlo methods are especially useful for simulating systems with many coupled d

4、egrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures （see cellular Potts model）. They are used to model phenomena with significant uncertainty in inputs, such as the calculation of risk in business. They are widely used in mathematics, for example

5、 to evaluate multidimensional definite integrals with complicated boundary conditions. When Monte Carlo simulations have been applied in space exploration and oil exploration, their predictions of failures, cost overruns and schedule overruns are routinely better than human intuition or alternative

6、soft methods.2The Monte Carlo method was coined in the 1940s by John von Neumann, Stanislaw Ulam and Nicholas Metropolis, while they were working on nuclear weapon projects （Manhattan Project） in the Los Alamos National Laboratory. It was named after the Monte Carlo Casino, a famous casino where Ula

7、ms uncle often gambled away his money.3Contentshide 1 Introduction 2 History 3 Definitions o 3.1 Monte Carlo and random numberso 3.2 Monte Carlo simulation versus what if scenarios 4 Applications o 4.1 Physical scienceso 4.2 Engineeringo 4.3 Computational biologyo 4.4 Computer Graphicso 4.5 Applied

8、statisticso 4.6 Gameso 4.7 Design and visualso 4.8 Finance and businesso 4.9 Telecommunications 5 Use in mathematics o 5.1 Integrationo 5.2 Simulation - Optimizationo 5.3 Inverse problemso 5.4 Computational mathematics 6 See also 7 Notes 8 References 9 External linksedit IntroductionMonte Carlo meth

9、od applied to approximating the value of . After placing 30000 random points, the estimate for is within 0.07% of the actual value.Monte Carlo methods vary, but tend to follow a particular pattern:1. Define a domain of possible inputs.2. Generate inputs randomly from a probability distribution over

10、the domain.3. Perform a deterministic computation on the inputs.4. Aggregate the results.For example, consider a circle inscribed in a unit square. Given that the circle and the square have a ratio of areas that is /4, the value of can be approximated using a Monte Carlo method:41. Draw a square on

11、the ground, then inscribe a circle within it.2. Uniformly scatter some objects of uniform size （grains of rice or sand） over the square.3. Count the number of objects inside the circle and the total number of objects.4. The ratio of the two counts is an estimate of the ratio of the two areas, which

12、is /4. Multiply the result by 4 to estimate .In this procedure the domain of inputs is the square that circumscribes our circle. We generate random inputs by scattering grains over the square then perform a computation on each input （test whether it falls within the circle）. Finally, we aggregate th

13、e results to obtain our final result, the approximation of .If grains are purposefully dropped into only the center of the circle, they are not uniformly distributed, so our approximation is poor. Second, there should be a large number of inputs. The approximation is generally poor if only a few gra

14、ins are randomly dropped into the whole square. On average, the approximation improves as more grains are dropped.edit HistoryBefore the Monte Carlo method was developed, simulations tested a previously understood deterministic problem and statistical sampling was used to estimate uncertainties in t

15、he simulations. Monte Carlo simulations invert this approach, solving deterministic problems using a probabilistic analog （see Simulated annealing）.An early variant of the Monte Carlo method can be seen in the Buffons needle experiment, in which can be estimated by dropping needles on a floor made o

16、f parallel strips of wood. In the 1930s, Enrico Fermi first experimented with the Monte Carlo method while studying neutron diffusion, but did not publish anything on it.3In 1946, physicists at Los Alamos Scientific Laboratory were investigating radiation shielding and the distance that neutrons wou

17、ld likely travel through various materials. Despite having most of the necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus, and how much energy the neutron was likely to give off following a collision, the Los Alamos physicists

18、 were unable to solve the problem using conventional, deterministic mathematical methods. Stanisaw Ulam had the idea of using random experiments. He recounts his inspiration as follows:The first thoughts and attempts I made to practice the Monte Carlo Method were suggested by a question which occurr

19、ed to me in 1946 as I was convalescing from an illness and playing solitaires. The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whethe

20、r a more practical method than abstract thinking might not be to lay it out say one hundred times and simply observe and count the number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron d

21、iffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations. Later in 1946, I described the idea to John von Neumann, and we began to plan actua

22、l calculations.Stanisaw Ulam5Being secret, the work of von Neumann and Ulam required a code name. Von Neumann chose the name Monte Carlo. The name refers to the Monte Carlo Casino in Monaco where Ulams uncle would borrow money to gamble.167 Using lists of truly random random numbers was extremely sl

23、ow, but von Neumann developed a way to calculate pseudorandom numbers, using the middle-square method. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, and also noted that when it went awry it did s

24、o obviously, unlike methods that could be subtly incorrect.Monte Carlo methods were central to the simulations required for the Manhattan Project, though severely limited by the computational tools at the time. In the 1950s they were used at Los Alamos for early work relating to the development of t

25、he hydrogen bomb, and became popularized in the fields of physics, physical chemistry, and operations research. The Rand Corporation and the U.S. Air Force were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they bega

26、n to find a wide application in many different fields.Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of pseudorandom number generators, which were far quicker to use than the tables of random numbers that had been previously use

27、d for statistical sampling.edit DefinitionsThere is no consensus on how Monte Carlo should be defined. For example, Ripley8 defines most probabilistic modeling as stochastic simulation, with Monte Carlo being reserved for Monte Carlo integration and Monte Carlo statistical tests. Sawilowsky9 disting

28、uishes between a simulation, a Monte Carlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality, a Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem, and a Monte Carlo simulation uses repeated sampling to determi

29、ne the properties of some phenomenon （or behavior）. Examples: Simulation: Drawing one pseudo-random uniform variable from the interval 0,1 can be used to simulate the tossing of a coin: If the value is less than or equal to 0.50 designate the outcome as heads, but if the value is greater than 0.50 d

30、esignate the outcome as tails. This is a simulation, but not a Monte Carlo simulation. Monte Carlo method: The area of an irregular figure inscribed in a unit square can be determined by throwing darts at the square and computing the ratio of hits within the irregular figure to the total number of d

31、arts thrown. This is a Monte Carlo method of determining area, but not a simulation. Monte Carlo simulation: Drawing a large number of pseudo-random uniform variables from the interval 0,1, and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a Monte Carlo simulation of the behavior of repeatedly tossing a coin.Kalos and Whitlock4 point out that such distinctions are not always easy to maintain. For example, the emission of radiation fro