10 typical mathematical modeling algorithms

Top 10 Mathematical ModelingAlgorithmMantan

 

Author: JulyJanuary 29, 2011

Reference:
I. The top ten greatest algorithms in the 20th century.
Ii. Classic Algorithm Research Series in this blog
Iii. Wikipedia

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Bloggers:
1The top 10 mathematical modeling algorithms are written based on a list on the Internet. This article gives a brief introduction to the top 10 algorithms.
This is just a list. There are still many algorithms in mathematical modeling that are not included in one. You are welcome to provide more good algorithms.
2In addition to listing common applications,
At the same time, it will also be detailed in combination with the mathematical modeling competition.
After all, these ten algorithms are widely used and important in the mathematical modeling competition.
In addition, any question marked as "a country in a certain year" is the original question of a country's mathematical modeling competition that year.
3These ten algorithms are not described too much in some classic algorithm design books.
For detailed and in-depth research, please refer to the excellent papers on the top ten algorithms in China or internationally.
Thank you.

 

 

1. Monte Carlo algorithm
In 1946, John von norann, Stan Ulam, and Nick metropolis, three scientists at the Las Vegas National Laboratory
Jointly invented the Monte Carlo method.

This algorithm was named one of the top ten greatest algorithms in the 20th century. For more information, see my blog:
Http://blog.csdn.net/v_JULY_v/archive/2011/01/10/6127953.aspx

 

The Monte Carlo method (Monte Carlo method), also known as the random sampling or statistical simulation method, is a method guided by the theory of probability statistics.

Is a very important numerical calculation method. This method uses random numbers (or more common pseudo-random numbers) to solve many computing problems.

Method.

 

Because the traditional empirical method cannot approach the real physical process, it is difficult to obtain satisfactory results.

The actual physical process is simulated on the ground. Therefore, the solution is very consistent with the actual situation and the results can be very satisfactory.

The basic principles and ideas of the Monte Carlo method are as follows:
When the problem is solved by the probability of a random event or the expected value of a random variable, an "experimental" method is used.

Estimate the probability of a random event based on the frequency of occurrence of the event, or obtain some digital features of the random variable and perform

Is the solution to the problem.

 

There is an example to give you a more intuitive understanding of the Monte Carlo method:
Suppose we want to calculate the area of an irregular graph, then the complexity of the irregular degree and Analytical computation (such as points) of the graph

Degree is proportional. How is the Monte Carlo method calculated? Suppose you have a bag of beans and scatter the beans evenly onto the graph.

The number of beans in the graph is the area of the graph. The smaller your beans, the more you scatter.

The more accurate the result is.
Here we want to assume that all beans are on a plane and there is no overlap between them.

The Monte Carlo method uses Mathematical Methods to simulate the geometric quantity and geometric features of the Motion of things.

To be tested. It is based on a probability model, according to the process described in this model, through the simulation of the results of the experiment, as a problem

Approximate Solution.

 

The Monte Carlo method is very different from the general calculation method. The general calculation method is very difficult to solve the problem of multi-dimensional or complex factors.

The Monte Carlo method is relatively simple to solve this problem. Its features are as follows:
I. Tracking particles directly, clear physical thinking and easy to understand.
II. the random sampling method is used to simulate the particle transport process, reflecting the statistical fluctuations.
III. It is not restricted by the complexity of multi-dimensional and multi-factor systems. It is a good solution to the particle transport problem of complex systems.
And so on.

This algorithm will be described in detail in this blog later.

 

2. Data processing algorithms such as data fitting, parameter estimation, and Interpolation
We usually encounter a large amount of data to be processed, and the key to data processing lies in these algorithms. MATLAB is usually used as a tool.

Data Fitting has been applied in the mathematical modeling competition. Many problems related to graphic processing are related to fitting. One example is 98 years.

Modeling: Question A of the United States, three-dimensional Interpolation of biological tissue slices, open roads to the mountains in question a of 94 years, interpolation calculation of mountain heights, and

The "sars" problem may also need to be tested through data fitting algorithms to observe the trend of data processing.

 

There are many ready-made functions in MATLAB that can be called and familiar with MATLAB. These methods can be used with ease.

 

Iii. planning issues such as linear planning, integer planning, multi-dimensional planning, and secondary planning
Many problems in the mathematical modeling competition are related to mathematical planning. It can be said that many models can be attributed to a set of inequalities as constraints.

When several function expressions are used as objective functions, solving such problems is critical. For example, Question 98, question B, using many inequalities

The problem can be clearly described. Therefore, it is easier to use Lindo, lingo and other software to solve the problem after listing the plan.

You need to be familiar with the two software.

 

Iv. Graph Theory Algorithms
There are many algorithms for such problems,
Including Dijkstra, Floyd, Prim, Bellman-Ford, maximum stream, and binary matching.

 

For more information about this graph algorithm, see Introduction to algorithms-Introduction to algorithms. For more information about graph algorithms, see Chapter 22nd-Chapter 26th.
In addition, this blog provides a brief description of the Dijkstra algorithm,
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Classical Algorithm Research Series: II. Dijkstra Algorithm
Http://blog.csdn.net/v_JULY_v/archive/2010/12/24/6096981.aspx

For more information, see the blog post updated later.

 

5. computer algorithms such as dynamic planning, backtracing search, divide and conquer, and branch and demarcation
In the mathematical modeling competition, for example, in question 92, question B was defined by branch and division, and question B in question 97 was a typical dynamic planning problem,
In addition, question B in Question 98 represents the splitting algorithm.

This problem and ACMProgramSimilar issues in design competitions,
It is recommended that you take a look at the introduction to algorithms and books related to computer algorithms, such as computer algorithm design and analysis (e-Industry Press.

 

Vi. Three classic algorithms of Optimization Theory: Simulated Annealing, neural networks, and genetic algorithms
Over the past decade, the optimization theory has developed rapidly. The three algorithms, namely simulated annealing, neural networks, and genetic algorithms, have developed rapidly.

In the mathematical modeling competition: for example, the simulated annealing algorithm of question a in 97 years, the neural network Classification Algorithm of question B in year 00, and the difficulty of question B in 01 years can also be

Using Neural Networks and the 89-year question of the U.S. competition is also related to the BP algorithm. At that time, the BP algorithm was just proposed in 86 years and it was tested in 89 years,

The competition question may be an abstract embodiment of today's cutting-edge technologies.
In, question B gamma knife is also a subject of current research. At present, the best algorithm is genetic algorithm.

 

In addition, I am very interested in artificial intelligence. genetic algorithms have been described in this blog. For more information, see.
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Classical Algorithm Research Series: 7. Exploring genetic algorithms and analyzing the essence of GA
Http://blog.csdn.net/v_JULY_v/archive/2011/01/12/6132775.aspx

 

The other two algorithms, the simulated annealing method, and the neural network, will be detailed in the blog post updates later in this blog.

 

VII. grid algorithms and exhaustion
The grid algorithm is the same as the exhaustive method, except that the grid method is a continuous problem.
For example, if optimization is required for n variables, the available space of these variables will be collected,
For example, take m + 1 point in the range [A; B], that is, a; A + (B? A) = m; A + 2 degrees (B? A) = m ;...; B

In this way, the loop requires (m + 1) N operations, so the calculation workload is large.

In the mathematical modeling competition: for example, 97-year question a and 99-year question B can all be searched using the grid method. This method is better at a relatively high computing speed.

You can also use advanced languages to do it in a fast computer. It is best not to use MATLAB as a grid. Otherwise, it will take a long time.

Everyone is familiar with the method of exhaustion.

 

VIII. Continuous Discretization Methods
Most programming solutions to physical problems are related to this method. Physical Problems indicate that we live in a continuous world.

Medium, the computer can only process the discrete amount, so the continuous amount needs to be discrete.

This method is widely used and related to many algorithms above.
In fact, grid algorithms, Monte Carlo algorithms, and simulated annealing use this idea.

 

9. Numerical Analysis Algorithm
Numerical analysis is a branch of mathematics. It mainly studies continuous mathematics (different from discrete mathematics ).

Algorithm.

If advanced languages are used for programming in the competition, some algorithms commonly used in numerical analysis include equations, matrix operations,

Function credits and other algorithms require additional library functions for calling.

These algorithms are specially designed for advanced languages. If you use MATLAB and Mathematica, you do not have to prepare them,
Because there are many mathematical software for functions in numerical analysis.

 

10. Image Processing Algorithms
In the mathematical modeling competition: for example, in question 01, you need to read BMP images, question a of Question 98 in the United States, and you need to know 3d interpolation.

Computing: Question B has higher requirements in. It requires not only programming computing but also processing, but also a lot of pictures to be presented in the digital model paper,

Therefore, image processing is the key. To solve such problems well, it is important to learn MATLAB well, especially in image processing.

 

The program source code for the top 10 mathematical modeling algorithms is packaged. Download the code here:
Http://download.csdn.net/source/3007336

 

I am particularly interested in algorithms and are getting more and more interested,
In the future, more good and classic practical algorithms will be elaborated and thoroughly studied in this blog.
.

 

 

Author's statement:
I am July and have allArticleContent and materials are copyrighted,
For more information, see the author's July and source. Thank you. May January 29, 2011.