Basic knowledge and applications of computational mathematics
Lei Hongjun
【Abstract】
This article aims to delve into the basics of computational mathematics and its applications in various fields. We will first introduce the basic concepts and development history of computational mathematics, and then elaborate on its practical applications in various fields such as scientific computing, data analysis, artificial intelligence, etc., and support these ideas with specific cases and statistics. Finally, we will look forward to the development trend of computational mathematics.
Keywords: Computational Mathematics, Basic Knowledge, Applications, Scientific Computing, Data Analysis, Artificial Intelligence
1. Basic knowledge of computational mathematics
1.1 Basic Concepts of Numerical Computing
Numerical computing, as the core foundation of computational mathematics, covers a wide range of fields, from basic mathematical operations to complex algorithm design. Its basic concepts involve the discretization and approximation of continuous mathematical problems, and aim to solve mathematical problems efficiently and accurately through computers. Numerical computing plays a vital role in modern scientific research and engineering applications.
The basic concepts of numerical calculation include numerical approximation, numerical stability, and error analysis.
1. Numerical approximation
- Numerical approximation simplifies calculations by approximating complex functions with simple functions (e.g. polynomials).
- In physics, it simulates complex phenomena such as turbulence, predicting and interpreting them.
Example: Approximating a sinusoidal function using a quadratic polynomial
- 选择二次多项式 p(x) = ax^2 + bx + c` 逼近 `sin(x)`。
- Interpolation determines the coefficient: 'p(x)' is equal to 'sin(x)' at the point x = 0, π/2, π'.
- Solve the system of equations to obtain the coefficients and plot the 'sin(x)' and 'p(x)' graphs to compare the differences.
- Limitations of numerical approximation: Only exact matches on finite points, other points may have errors.
- By selecting the approximation method and adjusting the parameters, the error can be reduced and a more accurate approximation can be obtained.
2. Numerical stability.
Numerical stability is an important indicator to measure the sensitivity of an algorithm to small changes in input data. A stable algorithm can still output reliable results even when there are small errors in the input data, such as the Gaussian integral method.
In the analysis of bridge structures, we need to solve problems such as stress distribution through numerical calculations. To simplify the analysis process, we typically use a planar frame of beams and pillars as a model.
Fig.1. Bridge structure model
Although FEA (Finite Element Analysis) plays an important role in structural simulations, the input data is often biased due to errors in engineering measurements. In this case, stable algorithms, such as optimized FEA solvers, are particularly important to give reliable results even if there are errors in the input data, while unstable algorithms can lead to inaccurate analysis results.
Figure 2 Finite Element Analysis (FEA) example
Therefore, in numerical engineering analysis, the choice of stabilization algorithm is the key factor to ensure the reliability of the analysis results and reduce the error.
3. Error analysis
Error analysis is crucial in numerical calculations, providing a basis for algorithm selection and optimization. In economics and finance, it ensures the accuracy and reliability of quantitative models. In complex mathematical models, error analysis control is the key to ensuring the accuracy of calculations. In physics and engineering, error analysis is essential for precision measurements and experimental verification to help understand uncertainties. In data science and machine learning, error analysis evaluates model performance, optimizes parameters, and improves prediction accuracy. It is important to understand the source and nature of errors, reduce errors, and improve accuracy. Error analysis is an important support for the development of scientific research and technology, and attention should be paid to its research and application to promote scientific and technological progress.
Fig.3 Illustration of the concept of error analysis
(1) Truth value: Truth value is a logical concept that is usually used to describe the truth or falsehood of a proposition or statement. In logic, truth is the direct result of a proposition, and it indicates the true state of a proposition.
(2) Error: Error refers to the difference between the measured or calculated result and the true value. Error is a common concept in a variety of scientific, engineering, and statistical fields, and it is often used to describe the deviation between an observed value and a true value.
(3) Mean: It is usually used to express the average value of a set of values. In statistics, a mean is often used to describe a central trend or mean of a set of data. To put it simply, the mean is the sum of a set of numbers divided by the number of numbers in that group, and it reflects the average of that set of data.
(4) Standard deviation: To put it simply, standard deviation is used to measure the difference between each value and the mean value in the data set. The calculation formula is as follows:
Standard deviation is a statistic used to describe how discrete a set of numerical distributions is. It reflects the degree to which this set of values deviates from its average.
(5) Residuals: mainly used to describe the gap between the observed value and the predicted value. In data analysis, residuals are often used for model validation and prediction.
Specifically, when we use a model, such as a regression model, to make predictions on a set of data, there is a predicted value for each observation. The residual, on the other hand, is the difference between the observed value and the predicted value.
(6) Gross error: Gross error usually refers to the error that deviates significantly from the true value due to some non-random factors in the process of measurement or experiment. This error is often not caused by accuracy problems or operational errors in the measuring equipment, but due to some external factors that are not easy to control or abnormal conditions in the measurement environment.
(7) Singular value: It refers to a value that is significantly different from other data in the dataset and may be due to measurement errors, data entry errors, or the occasional result of a particular event. Singular values not only affect the accuracy of data analysis, but can also lead to misjudgments or misleading decisions in statistical models. Therefore, it is a very important step to identify and process singular values before proceeding with data analysis.
(8) Random error:
In data collection and analysis, data deviates from the true value due to unforeseen and uncontrollable factors.
(9) Systematic error:
Systematic errors originate from the measurement process and are directional and predictable, resulting in bias in the data. It can be caused by instrument calibration issues, method defects, environmental changes, etc. It seriously affects the accuracy of data analysis, especially in scientific research, engineering design, medical diagnosis and other fields.
1.2 Linear Algebra and Matrix Computation
Linear algebra and matrix computation mainly involve the basic concepts of linear algebra and the operation of matrices.
1. Fundamentals of Linear Algebra:
(1) Linear equations: including unvariate, binary, and multivariate linear equations, and their solutions.
(2) Vector space: such as vectors, the addition of vectors, the number multiplication, dot product, cross product and other basic concepts.
(3) The concept of matrix: such as the rows, columns, elements, transpose and so on of the matrix.
2. Matrix Calculation:
(1) Matrix operation: including matrix addition, subtraction, number multiplication, multiplication (especially the nature and calculation method of matrix multiplication).
(2) Advanced concepts such as inverse, determinant and eigenvalue of matrices: these are very important parts of matrix calculation, which are often used to solve linear equations, judge the properties and properties of matrices, etc.
Calculation of special matrices: such as symmetry matrices, Van der Monde matrices and other special types of matrices and their calculation methods.
Linear algebra plays a vital role in the fields of image processing, economics, finance, physics, and engineering due to its unique matrix computation. In image processing, linear algebra supports the precise implementation of image transformations, for example, image rotation is achieved by multiplying the 3x3 rotation matrix with the image matrix. In economics and finance, linear algebra is widely used in quantitative analysis, risk assessment, and prediction of market trends, providing a scientific basis for decision-making. In physics and engineering, linear algebra and matrix computation are indispensable tools for quantum mechanics and engineering computer-aided design. With the rapid development of big data and AI technology, the importance of linear algebra in the field of machine learning and deep learning has become increasingly prominent, especially in the forward and backpropagation algorithms of neural networks.
1.3 Calculus and Numerical Integration
In computational mathematics, calculus and numerical integration are crucial. Calculus is the basis for understanding continuous change and is used to resolve complex problems. Numerical integration can quickly obtain the integration value through approximate calculation, which provides convenience for engineering and scientific calculation.
In physics, many phenomena rely on the resolution of calculus equations. For example, in quantum mechanics, calculus equations are used to solve wave functions. These equations are often solved by numerical integration methods such as Gaussian integral and Simpson integral approximation to ensure computational efficiency and accuracy of results.
Gaussian integrals are based on Gaussian points, select specific points to calculate the integrand value, and weighted the sum to obtain the approximate integral. The advantage is that the Gaussian integration accuracy is higher when the number of points is the same. For example, in the [-1, 1] interval, the accuracy of two Gaussian points is comparable to the Newton-Cortez formula for three points.
The Simpson integral is based on parabolic interpolation, the equidistant point is selected to calculate the integrand value, the parabolic interpolation is used to approximate the integrand, and then the trapezoidal formula is used to obtain the approximate integral value. The advantage is that it is simple to implement and the results are good. However, when the integrand changes drastically, the accuracy may be affected.
1.4 Numerical Optimization and Algorithms
In computational mathematics, numerical optimization and algorithms are crucial.
From engineering to financial analysis, from bioinformatics to earth sciences, there is a need to solve optimization problems in all fields. Including gradient descent method, Newtonian method, quasi-Newtonian method and genetic algorithm, etc.
(1) Application of gradient descent method. In quantitative investment, it is used for asset portfolio optimization to minimize risks and maximize returns, which is superior to traditional methods.
(2) Characteristics of genetic algorithm. Simulate the principles of natural selection and genetics to quickly find near-optimal solutions in the field of engineering design, improving efficiency and quality.
1.5 Numerical stability and error analysis
Numerical stability and error analysis play an important role in the field of computational mathematics, which are directly related to the accuracy of calculation results and the reliability of scientific research. Among them, numerical stability focuses on the response and performance of the algorithm to small input changes, while error analysis focuses on evaluating the impact and contribution of computational error to the final result.
In the practice of numerical integration, different calculation methods often produce different results, and the role of error analysis is particularly important, which can scientifically evaluate the accuracy of the calculation results. At the same time, the numerical stability ensures that the algorithm can maintain a stable output in the process of complex calculations, which further enhances the reliability of the calculation results.
In CFD simulations in the aerospace sector, the requirements for algorithm stability and error control are particularly stringent to ensure design accuracy and avoid potential accident risks. To this end, researchers are constantly exploring new algorithms and techniques, such as adaptive step size and iterative control strategies, to improve the stability and accuracy of calculations. At the same time, new technologies such as parallel and distributed computing also provide new ideas and methods for solving the problems of numerical stability and error analysis.
In summary, numerical stability and error analysis are indispensable core elements in the field of computational mathematics. In the future, we need to further strengthen relevant research and applications to ensure the accuracy and reliability of calculation results and provide solid technical support for scientific research and engineering practice.
2. Advanced topics in computational mathematics
2.1 Numerical solutions to differential equations and difference equations
In the field of computational mathematics, the numerical solutions of differential equations and difference equations occupy a crucial position, and they play an indispensable role in the modeling of practical problems, covering a wide range of fields such as Newton's second law in physics and growth models in economics.
The Euler method is recognized for its simplicity, but there is room for improvement in terms of accuracy. In contrast, the Runge-Kuta method is widely used in aerospace trajectory simulation and other fields due to its high accuracy and stability. As an important tool to describe discrete systems, difference equations have shown their unique value in the fields of economics and biology. The iterative method and matrix method also show significant advantages in the field of stock price forecasting.
2.2 Numerical approximation and interpolation
Numerical approximation and interpolation are essential in computational mathematics to construct an approximation function from known data points to predict unknown points. These technologies are widely used in signal processing, image processing and other fields.
Numerical approximation is commonly used in polynomials, rational functions, splines, etc. For example, polynomial approximation is commonly used in signal processing to filter or smooth signals.
Interpolation is a branch of numerical approximation that constructs a continuous function by inserting new data points. Common methods include Lagrangian interpolation, Newtonian interpolation, and so on. Lagrangian interpolation has important applications in image processing.
2.3 Advanced algorithms for numerical linear algebra
Advanced algorithms for numerical linear algebra occupy an important place in computational mathematics. With the increasing demand for large-scale linear equation processing, it is difficult for traditional methods to meet the high efficiency and accuracy. Therefore, advanced algorithms such as iterative method, sparse matrix algorithm, parallel and distributed computing have been developed.
(1) Iterative method: such as Jacobi, Gauss-Seidel and SOR iteration, it has the advantage of dealing with large sparse matrices, and the solution vector is updated to a preset accuracy through iterative updates. In reservoir simulation, the iterative method improves the computational efficiency.
(2) Sparse matrix algorithm: According to the design of sparse matrix structure, the amount of computation and storage space are reduced by compressing storage and special computing skills.
(3) Parallel and distributed computing: Divide large linear equations into multiple subproblems, and solve them in parallel to improve the speed. In aerospace computational fluid dynamics, parallel computation shortens the computation time.
2.4 Advanced Methods of Numerical Integration
Numerical integration is crucial in computational mathematics. With the development of science and technology, traditional methods are limited, and the Gaussian integral method meets the complex needs with high precision and wide applicability. Adaptive integration reduces computational costs, such as quantum mechanical solving. In the era of big data, advanced numerical integration methods are combined with machine learning to process massive amounts of data to support research and industry. In conclusion, advanced numerical integration methods play a key role in computational mathematics, driving their application and development.
2.5 Parallel and Distributed Computing in Numerical Computation
3. The application of computational mathematics in scientific research
3.1 Applications of Computational Mathematics in Physics
In physics, computational mathematics is widely used in theory and experiments. In quantum mechanics, numerical computing techniques solve equations, simulate quantum systems, and design new materials. In condensed matter physics, computational mathematics simulates the structure and properties of materials and predicts superconducting materials. It is also used in fluid mechanics and thermodynamic simulations to improve research accuracy and efficiency, and to promote the integration of disciplines. Computational physics simulates the structure of the universe and supports biophysics and geophysics. Computers are the new laboratory for physicists, and computational mathematics will continue to play an important role in physics.
3.2 Computational Simulation in Engineering
Computational simulation has become a key tool in engineering for modeling and simulating complex systems and predicting and optimizing performance.
(1) Bridge design: simulation software simulates the stress distribution and deformation under the load of the bridge to ensure safety and stability.
(2) Automotive engineering: CAD/CAE technical support design, simulating performance under different road conditions, optimizing design and shortening cycle.
(3) Aerospace: Computational fluid dynamics simulates the airflow field of high-speed aircraft, predicts aerodynamic performance, and ensures flight safety.
4. Future prospects for computational mathematics
4.1 Artificial Intelligence and Computational Mathematics
AI and computational mathematics are combined to optimize algorithms to improve computing power and provide new perspectives for complex problems. Computational mathematics provides a mathematical foundation for machine learning, such as image recognition, which improves recognition rates through optimization algorithms. Quantum computing brings new challenges and opportunities to computational mathematics and advances science and technology.
4.2 Big Data and Computational Mathematics
Big data poses new challenges to computational mathematics, such as processing huge data volumes and complex structures. Researchers explore new algorithms and models to improve processing efficiency. The combination of computational mathematics and big data has achieved remarkable results in medical and other fields and will continue to play a central role.
4.3 Quantum Computing and Computational Mathematics
Quantum computing brings new opportunities for computational mathematics, and its parallelism and superposition provide a new paradigm. Algorithms such as QAOA can find an approximate optimal solution in polynomial time. Quantum computing deals with large-scale systems of linear equations, opening up new possibilities for simulation, while at the same time bringing new challenges.
4.4 Computational Mathematics is studied across disciplines
Computational mathematics connects different disciplines and advances science. It plays an important role in biology, physics and other fields, such as gene editing, climate simulation, etc. Computational mathematics is an important tool in interdisciplinary research, and its application should be actively explored.
4.5 Computational Mathematics in Sustainable Development
Computational mathematics plays an important role in sustainable development, such as climate change research, resource management, and environmental protection. Establish climate models to predict global climate trends, optimize resource allocation and utilization, and simulate and assess environmental pollution and ecological restoration.
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