Monte Carlo method

Reprinted from: Wikipedia Monte Carlo method

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Monte Carlo method[Edit ] Wikipedia, the free encyclopedia

Monte Carlo Methods ( English:Monte Carlo method), also known as the Statistical simulation method , were the mid 1940s due to the development of science and technology and the invention of electronic computers, A kind of very important numerical calculation method, which is guided by probability statistic theory, is proposed. means using random numbers (or more common pseudo-random numbers) to solve many computational problems.

In the 1940s, von Neumann, Stanislav Uram and Nicolas Metropolis invented the Monte Carlo method at the Los Alamos National Laboratory for nuclear weapons programs. Because Ulam's uncle often loses money in the Monte Carlo Casino, and Munters ' Carlo method is based on the probability of the method.

It corresponds to a deterministic algorithm.

The Monte Carlo method is widely used in financial engineering, macroeconomics, Biomedicine, computational physics (such as particle transport calculation, quantum thermodynamic calculation, aerodynamics calculation) and other fields. ^[1]

Directory[Hide]

• 1 The basic idea of Monte Carlo method
• 2 The working process of the Monte Carlo method
• procedure for molecular simulation of 3 Monte Carlo method
• Application of 4 Monte Carlo method in mathematics
  • 4.1 points
  • 4.2 Pi
• 5 References
• 6 See also

basic ideas of Monte Carlo method [Edit ]

Generally, Monte Carlo method can be roughly divided into two categories: the problem of the solution itself has inherent randomness, with the help of computer computing ability can directly simulate this stochastic process. For example, in the study of nuclear physics, the transmission process of neutron in reactors is analyzed. The action of neutron and nucleus is restricted by the law of quantum mechanics, people can only know the probability of their interaction, but they can not obtain the position of neutron and nucleus, and the moving rate and direction of new neutron produced by fission. Scientists based on the probability of random sampling to obtain the fission position, velocity and direction, so simulating the behavior of a large number of neutrons, after statistics can obtain the range of neutron transmission, as the basis for reactor design.

Another type is the number of features that can be converted to a random distribution, such as the probability of a random event appearing, or the expected value of a random variable. By means of random sampling, the probability of the occurrence of random events is estimated, or the numerical characteristics of random variables are estimated by sampling numerical features and used as the solution of the problem. This method is used to solve complex multi-dimensional integration problems.

Suppose we want to calculate the area of an irregular graph, then the degree of irregularity of the graph is proportional to the complexity of the analytic calculations (for example, integrals). The Monte Carlo method is based on the idea that you have a bag of beans, spread the beans evenly over the graph, and then count the number of beans in the graph, the number of which is the area of the graph. The smaller the bean, the more you scatter, and the more accurate the result is. The computer program can generate a large number of evenly distributed coordinate points, and then count the numbers in the graph, through their proportion of total points and coordinates to generate a range of area can be calculated plot area.

working process of Monte Carlo method [Edit ] The π value is estimated using the Monte Carlo method. After placing 30,000 random points, Pi estimates differ by 0.07% from the true value.

In solving practical problems, the Monte Carlo method is mainly used in two parts:

1. When simulating a process with Monte Carlo method, random variables of various probability distributions are needed.
2. A statistical method is used to estimate the numerical characteristics of the model, so as to get the solution of the real problem.

procedures for molecular simulation of Monte Carlo methods [Edit ]

The simulation of molecular simulations using the Monte Carlo method is performed in the following steps:

1. Use the random number generator to produce a random molecular configuration.
2. The particle coordinates of this molecular configuration are changed irregularly, resulting in a new molecular configuration.
3. Calculates the energy of a new molecular configuration.
4. The change in the energy of the molecular configuration before the change is compared with the new molecular configuration to determine whether the configuration is acceptable.
   • If the new molecular configuration energy is lower than the energy of the original molecule configuration, the new configuration is accepted, and the next iteration is repeated using this configuration.
   • If the new molecular configuration energy is higher than the energy of the original molecule configuration, the Boltzmann factor is computed and a random number is generated.
     • If the random number is greater than the calculated Boltzmann factor, the configuration is discarded and recalculated.
     • If the random number is less than the calculated Boltzmann factor, the configuration is accepted and the next iteration is repeated using the configuration.
5. This is done iteratively until the final search for a molecular configuration bundle below the given energy condition is performed.

Application of Monte Carlo method in mathematics [edit ]

Usually the Monte Carlo method solves various mathematical problems by constructing random numbers that match certain rules. The Monte Carlo method is an effective method for solving the problem that the analytic solution is difficult to get or the analytic solution is not solved because of the complexity of computation. The most common application of Monte Carlo method in mathematics is Monte Carlo integration. Here are two simple applications of the Monte Carlo method:

points [edit ]

The non-weighted Monte Carlo integral, also known as deterministic sampling, is a random uniform sampling of the integrand variable interval, then averaging the function values of the sampled points, so that the approximate value of the function integral can be obtained. The correctness of this method is based on the central limit theorem of probability theory. When the sampling number is m, the statistical error of the approximate solution obtained by using this method is only related to M (and positive correlation) and does not change with the change of the integral dimension. Therefore, when the integral dimension is high, the Monte Carlo method is better than other numerical methods.

pi [edit ]

The Monte Carlo method can be used to approximate the calculation of pi: Let the computer randomly generate two numbers from 0 to 1, to see whether the points in which the two real numbers are horizontal are within the unit circle. Generate a series of random points, statistical units within the circle of points and total points, (the circle area and square area of the ratio of Pi:4,pi to pi), when random points obtained more time, the results closer to PI (however, accuracy is still controversial: even if you take 10 of 9 random points, The results were also matched with Pi only in the first 4 digits. Using Monte Carlo method to approximate the congenital defects of Pi is: first, the computer generated by the random number is limited by the storage format, is discrete, and can not produce continuous arbitrary real numbers, the above-mentioned practice will be partitioned into a mesh grid, in the space is not continuous, the calculated area of course with the circle more or less there is a gap.

references [edit ]

1. ^ Kroese, D. P.; Brereton, T.; Taimre, T.; Botev, Z. I. Why the Monte Carlo method was so important today. WIREs comput Stat.,6: 386–392. doi:10.1002/wics.1314.

(RPM) Monte Carlo method