I.The Basic Information of the Course
Course Number:202012420010
The EnglishName of theCourse：Introduction to Scientific Computation
The ChineseName of theCourse：科学计算引论
In-classHours andAllocation：Total class hours: 48, 40 in class, 8 in class discussion and on computer
Credit(s):3
Semester:1
AppliedDiscipline(Professional Degree Category)：Mathematics
CourseObjectOriented：Doctor of Science
EvaluationMode:Open-book examination + Process Evaluation
TeachingMethod:Blended Teaching
CourseOpening Department:School of Mathematical Science
Notes:Filling explanation to applied discipline: the general courses like foreign languages, ideological and political theory course may fill in “AllDisciplines”,Mathematics general course may fill inScience andEngineeringDiscipline, and the other courses fill in “Name ofApplicableDiscipline” according to actual teaching objects, and the number of applicable discipline may be more than one.
II.Prerequisite Course
Numerical Analysis, Real Variable Function, Functional Analysis, Ordinary Differential Equation, Partial Differential Equation
III. The Objectives and Requirements of the Course
The Objectives of the Course:
Scientific computing is the numerical computation using computers to solve mathematical problems in science and engineering. It refers to the whole process of using computers to reproduce, predict and discover the laws and evolutionary characteristics of objective world movements. The specific content of this course includes the overall content including the theory of computational mathematics and application software, information science, operations research and control. The course is aimed at doctoral students of the School of Mathematical Sciences. The purpose of the course is to teach students the classic numerical methods, and at the same time, in the era of big data and the era of intelligent science, emphasize the scientific nature of computational theory while focusing on systemic and interdisciplinary In-depth integration, focusing on teaching some new methods developed in recent years and numerical processing methods for cross-application problems in some disciplines, while emphasizing the implementation of effective algorithms, explaining the methods of scientific computing in the new situation in a simple way, improving students’ respective scientific research and expansion capabilities in new research directions.
The Requirements of the Course:
Through the course study, you can understand the latest scientific research results related to the course at home and abroad, be familiar with the relevant theories and applications of numerical analysis, scientific computation, and matrix computation. Details as follows:
(1) Be familiar with and master the basic concepts and theories of numerical approximation, numerical algebra and numerical solution of ordinary and differential equations in the theory of "numerical analysis"; focus on some classical numerical computation methods, such as interpolation, function approximation, numerical integration, solution of linear equations and solution of nonlinear equations.
(2) Master several standard methods of matrix computation, such as matrix sequence, matrix series, matrix computes, matrix factorization and its application, such as LU factorization, QR factorization, singular value factorization, etc.; understand the estimation and characterization of matrix eigenvalues, such as the boundary estimation of eigenvalues and the included areas, etc.; understand Krylov subspace method, and be familiar with the steepest descent method and Conjugate Gradient method, etc.
(3) Understand the finite difference method, including understanding the general concept of partial differential equations, the difference format of the parabolic equation, the difference format of the hyperbolic equation, the difference format of the convection diffusion equation, the difference format of the elliptic equation; understand and be familiar with the finite element method, including understanding from function expansion to variational principle, one-dimensional finite element method, two-dimensional finite element method, Galerkin method and its extension, error estimation, etc.
(4) Master modern compression technology, including fast Fourier transform, triangle interpolation, discrete cosine transform.
(5) The final score is based on a combination of open-book examination (60%) and course design report (40%), and is evaluated on a 100-point scale.
IV. The Content of the Course
Part 1: Related theories of numerical analysis (10 class hours)
This part mainly includes the review and extension of the content of the previous "numerical computation" course, focusing on the best continuous approximation and best square approximation in the approximation algorithm, spline interpolation and B-spline interpolation, adaptive integration method; classroom study the computation of eigenvalue and eigenvector, wavelet transform, etc.
Part 2: Matrix calculation related theory (20 class hours)
The main content includes the basic problems and sources of matrix computation, ill-posed problems and numerical stability; several standard methods of basic tools for matrix computation, including multiple factorization forms of matrix and their applications, such as Schur factorization and singular value factorization, QR, QL factorization and standard orthonormalization process, tridiagonalization of matrix, Krylov subspace method, singular value factorization for least squares solution; class discussion of generalized inverse matrix; disc theorem of matrix eigenvalue problem and so on.
Part 3:The numerical solution method of the differential equation mathematical model (10 class hours)
The main contents of this part include the finite difference method and the finite element method, such as the general concept of partial differential equations, the difference format of the parabolic equation, the difference format of the hyperbolic equation, the difference format of the convection diffusion equation, the difference format of the elliptic equation; To the variational principle, one-dimensional finite element method, two-dimensional finite element method, Galerkin method and its extension, error estimation.
Part 4:Introduction to Modern Numerical Computing Technology (8 class hours)
Compression is the core of numerical analysis, although it is often hidden in interpolation, least squares, and Fourier analysis. The main contents of this part include fast Fourier transform, triangle interpolation, discrete cosine transform and their applications. Classroom discusses several methods in the global optimization problem, including simulated annealing algorithm, genetic algorithm and so on.
Ideological and political case teaching in the course:The teaching process emphasizes equal emphasis on ideological and moral education and the cultivation of knowledge. For example, the speed of the conjugate gradient method depends on the spectral distribution of the coefficient matrix. When the eigenvalues are concentrated, the condition number of the coefficient matrix is very small. The conjugate gradient method converges quickly. "The iterative method of the system of super linear convergence equations is based on an initial estimate of the solution, by continuously updating the matrix, and gradually improving the solution, knowing that the required accuracy is reached. In theory, after infinite iterations, you can get the true solution, but in fact you only need to achieve the required accuracy. So, the essence of iteration is to replace the old solution with the newly generated solution. The solution we get is constantly evolving and gradually approaching the true value. During the course of this course, the following three cases are mainly described:
Case 1: The contribution of ancient Chinese mathematicians to computational mathematics (Zu Chongzhi, Liu Hui);
Zu Chongzhi's outstanding achievement in mathematics is about the calculation of pi. Before the Qin and Han dynasties, people used "the diameter of a week" as the pi, which is the "old rate". Later, it was found that the error of the ancient rate is too large, and the pi should be "one round diameter and more than Wednesday", but there are different opinions on how much is left. Until the Three Kingdoms period, Liu Hui proposed a scientific method for calculating the pi-"cutting circle", which approximates the circumference by using the circumference of the regular polygon inscribed in the circle. Liu Hui calculated that 96 polygons were inscribed in the circle, and π=3.14 was obtained, and pointed out that the more the number of sides inscribed in the regular polygon, the more accurate the value of π obtained. Based on the achievements of his predecessors, Zu Chongzhi worked hard and repeated calculations to find π between 7.1415926 and 3.1415927. And the approximate value in the form of π fraction is obtained, which is taken as the approximate rate and as the density rate, where the six decimal places is 3.141929, which is the fraction of the numerator denominator within 1000 that is closest to the value of π. What method Zu Chongzhi used to obtain this result cannot be investigated at present. If we envisage him to follow Liu Hui's "circumcision technique", he must calculate the 16.384 polygon inscribed in the circle. How much time and labor will be required! It can be seen that his tenacious perseverance and intelligence in academics are admirable. Zu Chong's calculation of the density rate has been obtained by foreign mathematicians for more than a thousand years. To commemorate Zu Chongzhi's outstanding contributions, some foreign mathematics historians suggested that π = be called "Zu rate".
Case 2: What is computational mathematics
Modern science and technology are developing very quickly. They have a common feature, that is, they all have a lot of data problems.
For example, to launch a satellite to detect the mysteries of the universe, from the beginning of the satellite century to the launch and recovery, scientists, engineering technicians, and workers will have to comprehensively design and produce the overall satellite and components, and design the selected rocket. And production, there are a lot of data to be calculated accurately. When launching and recovering, there are precise computations regarding launch angle, orbit, remote control, recovery fall angle, etc.As another example, high-energy physics experiments are conducted in high-energy accelerators to study the properties of elementary particles with high energy, their interactions and conversion laws. There are also a lot of data computation problems.
The computation problem can be counted as a common problem in all fields of modern society. Industry, agriculture, transportation, medical care, culture and education, etc., which line and industry have a lot of data to be calculated, through data analysis, in order to grasp the development of things about the law.
A discipline that studies the solution of computational problems and related mathematical theoretical problems is called computational mathematics.Computational mathematics belongs to the category of applied mathematics. It mainly studies how to solve related mathematical and logical problems effectively by computers.
The main content of computational mathematics:
Computational mathematics is also called numerical calculation method or numerical analysis. The main content includes the numerical solution of algebraic equations, linear algebraic equations, differential equations, numerical approximation of functions, the solution of matrix eigenvalues, optimization calculation problems, probability and statistical calculation problems, etc., as well as the existence and uniqueness of solution, convergence and error analysis and other theoretical issues.
We know that there are no root-finding formulas for algebraic equations of degree five and above. Therefore, the solution to a higher-order algebraic equation of degree five or more is generally only an approximate solution to it. The method of seeking an approximate solution is numerical analysis method. For general transcendental equations, such as logarithmic equations, triangular equations, etc., only numerical analysis can be used. How to find out the computation method which is more concise, less error and less time is the main subject of numerical analysis.
In the method of solving equations, one of the commonly used methods is the iterative method, also known as the successive approximation method. The computation of the iterative method is relatively simple and easy to carry out. Iterative methods can also be used to solve linear equations. To find the approximate solution of the system of equations, an appropriate iterative formula should also be selected, so that the convergence speed is fast and the approximation error is small.
In the solution of linear algebraic equations, the Seidel iteration method, the conjugate slope method, the ultra-relaxation iteration method, etc. are commonly used. In addition, some relatively old common elimination methods, such as the Gaussian method and the chasing method, can also be widely used under the condition of using computers.
In the computation method, numerical approximation is also a commonly used basic method. Numerical approximation is also called approximate replacement, that is, a simple function is used to replace a more complex function, or a function that cannot be represented by an analytical expression. The basic method of numerical approximation is interpolation. The trigonometric function table in elementary mathematics and the correction value in the logarithm table are made according to the interpolation method.
When encountering differentiation and integration, how to use a simple function to approximate replace the given function in order to easily find and integrate is also a main content of the calculation method. The numerical solution of differential equations is also an approximate solution. The numerical solutions of ordinary differential equations are Euler method, prediction correction method and so on. For the initial value problem or boundary value problem of partial differential equations, the finite difference method and the finite element method are commonly used at present.
The basic idea of the finite difference method is to replace the differential equations and definite solution conditions of continuous variables with discrete difference equations containing only a limited number of unknowns. Find the solution of the difference equation as the approximate solution of the partial differential equation.
The finite element method was developed in modern times, and it is based on the variational principle and the difference between the divisions. It has been widely used in solving elliptic equation boundary value problems. Many people are studying the finite element method to solve hyperbolic and parabolic equations.
Computational mathematics is very rich in content, and it is playing an increasingly important role in science and technology.
Case 3: The concept of differential equations
Equations are relatively familiar to those who have studied mathematics in middle school; in elementary mathematics, there are various equations, such as linear equations, quadratic equations, higher-order equations, exponential equations, logarithmic equations, triangular equations and Equations, etc. These equations are to find out the relationship between the known and unknown numbers in the research problem, list one or more equations containing an unknown number or several unknown numbers, and then find the solution of the equation. However, in actual work, there are often problems with completely different characteristics from the above equations. For example: the movement of a substance under certain conditions needs to seek its movement and changes; an object falls freely under the action of gravity, and it must seek the law of the falling distance changing with time; the rocket is flying in space under the propulsion of the engine, To seek the orbit of its flight, etc. The motion of matter and its changing laws are described mathematically by functional relationships. Therefore, this type of problem is to seek one or several unknown functions that satisfy certain conditions. That is to say, any problem of this kind does not simply seek one or several fixed values, but requires one or several unknown functions.
The basic idea of solving this type of problem is very similar to the basic idea of solving equations in elementary mathematics. It is also necessary to find out the relationship between the known function and the unknown function in the research problem. Find the expression of the unknown function in the equation. However, in terms of the form of the equation, the specific method of solving, and the nature of the solution, there are many differences from the solution of the equation in elementary mathematics. In mathematics, solving such equations requires knowledge of differentials and derivatives. Therefore, any equation that expresses the derivative of an unknown function and the relationship between independent variables is called a differential equation. Differential equations are generated at the same time as calculus. When Scottish mathematician Nipper founded the logarithm, he discussed the approximate solutions of differential equations. While establishing calculus, Newton used simple series to solve simple differential equations. Later, Swiss mathematicians Jacob Bernoulli, Euler, French mathematicians Cello, Dalambert, Lagrange and others continued to study and enrich the theory of differential equations.
The formation and development of ordinary differential equations are closely related to the development of mechanics, astronomy, physics, and other scientific technologies. New developments in other branches of mathematics, such as complex variable functions, combinatorial topology, etc., have had a profound impact on the development of ordinary differential equations. The current development of computers has provided a very powerful tool for the application and theoretical research of ordinary differential equations.
When Newton studied celestial mechanics and mechanical mechanics, he used the tool of differential equations to theoretically obtain the laws of planetary motion. Later, French astronomer Le Ville and British astronomer Adams used differential equations to calculate the position of Neptune, which had not been discovered at that time. All these have made mathematicians more convinced of the great power of differential equations in understanding and transforming nature.
When the theory of differential equations is gradually improved, it can be used to accurately express the basic laws followed by changes in things. As long as the corresponding differential equations are listed, there is a way to understand the equations. Differential equations have become the most vital branch of mathematics.
(1) Contents of ordinary differential equations:
If an unknown function appearing in a differential equation contains only one independent variable, this equation is called an ordinary differential equation, or it can be simply called a differential equation. Generally speaking, the solution of n-order differential equations contains many arbitrary constants. That is to say, the number of arbitrary constants in the solution of the differential equation is the same as the number of solutions of the equation. This solution is called the general solution of the differential equation. The general solution constitutes a family of functions.
If a solution that satisfies certain specified conditions is required according to the actual problem, then the problem of seeking such a solution is called a definite solution problem, and a solution that satisfies the definite solution conditions for an ordinary differential equation is called a special solution. For higher-order differential equations, a new unknown function can be introduced, which can be transformed into multiple first-order differential equations.
(2) Characteristics of ordinary differential equations:
There are many concepts, solutions, and other theories of ordinary differential equations, for example, the types and solutions of equations and systems of equations, the existence and uniqueness of solutions, singular solutions, qualitative theory, etc. The following is a brief description of the relevant points of the equation solution to understand the characteristics of ordinary differential equations.
Seeking general solutions has historically been the main goal of differential equations. Once the expressions of general solutions are found, it is easy to get the special solutions needed for the problems. The expression of the general solution can also be used to understand the dependence on certain parameters, which is convenient for the appropriate value of the parameter, so that its corresponding solution has the required performance, and is also helpful for other research on the solution. Later developments have shown that there are not many situations where general solutions can be found, and what is needed in practical applications is to find special solutions that satisfy certain specified conditions. Of course, the general solution is helpful to study the properties of the solution, but people have shifted the research focus to the problem of definite solutions.
(3) Discussion on the solvability of ordinary differential equations:
Does an ordinary differential equation have special solutions? If so, how many? This is a basic problem in the theory of differential equations, and mathematicians summarize it into a basic theorem, called the theorem of existence and uniqueness. Because if there is no solution and we are going to solve it, it is meaningless; if there is a solution and it is not the only one, it is not easy to determine. Therefore, existence and uniqueness theorems are very important for solving differential equations.
Most ordinary differential equations cannot find very accurate solutions, but only approximate solutions. Of course, the accuracy of this approximate solution is relatively high. It should also be pointed out that the differential equations used to describe the physical process and the initial conditions determined by experiments are also approximate, and the effects and changes between such approximations must also be solved in theory.
Nowadays, ordinary differential equations have important applications in many disciplines, such as automatic control, design of various electronic devices, calculation of trajectory, research on the stability of aircraft and missile flight, research on the stability of chemical reaction processes, etc. These problems can be reduced to finding solutions to ordinary differential equations, or to problems that study the nature of solutions. It should be said that the application of ordinary differential equation theory has made great achievements, but its existing theory is still far from meeting the needs, and it needs to be further developed to make the theory of this discipline more perfect.
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1. E. Cresseg. Higher Engineering Mathematics (Sixth Edition in 1988). World Book Publishing Company, 1992.
2. Zhang Pingwen, Li Tiejun. Numerical Analysis. Peking University Press, 2007.
3. Xiong Chunguang, Li Yuan. Scientific and Engineering Computing (Second Edition). Hua University Press, 2015.
4. Shen Yan, Yang Lihong and others. Higher numerical computation. Tsinghua University Press, 2016.
5. Liu Jijun. Modern numerical computation methods. Science Press, 2016.
6. Gene H. Golub, Charles F. Van Loan. Matrix computations. People's Posts and Telecommunications Press, 2014.
7. Li Qingyang, Yi Dayi and others. Numerical algorithm analysis and efficient algorithm design. Huazhong University of Science and Technology Press, 2018.
8. Wu Qun, Zhou Lingjun, Yin Junfeng. Matrix analysis. Tongji University Press, 2017.
9. David Kincaid, Ward Cheney. Numerical Analysis: Mathematics of Scientific Compution (Third Edition). China Machine Press,2003.
10. Walter Gautschi. Numerical analysis. World Book Publishing Company, 2015.
B. Learning Resources
1. http://math.hrbeu.edu.cn
2. Address of the MOOC learning platform:
https://mooc1.chaoxing.com/course/201826509.html(Numerical analysis MOOC).
3. Chinese University MOOC platform: calculation method. Dalian University of Technology.
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