(ext.) Online Excerpt: Computational mathematics research direction and online information

Computational mathematics is used to calculate physics and engineering. The main research interests include:

Numerical functional analysis, continuous computational complexity theory, numerical partial and finite element, nonlinear numerical algebra and complex dynamical systems;

Numerical solution of nonlinear equations, numerical approximation theory, computer simulation and information processing, mathematical modeling and calculation of engineering problems, etc.

At present, the best direction of development has been combined with the CAGD direction of applied Mathematics. Now the hottest direction should be the numerical solution of differential equations, numerical algebra and manifold learning, numerical calculation of elite schools: Xi ' an Jiaotong University, Peking University, Dalian City of

From the literal view of computational mathematics, it should be closely related to computer, and also emphasize the importance of practice for computational mathematics.

Perhaps Professor Parlett's passage would best illustrate the question:

How could someone as brilliant as von Neumann think hard about a subject as mundane as triangular factoriz-ation of an Invertible matrix and not
perceive so, with suitable pivoting, the results is impressively good? Partial answers can be suggested-lack of hands-on experience, concentration on the inverse rather than on the solution of Ax = b-but I do not find them adequate. Why do Wilkinson keep the QR algorithm as a backup to a laguerre-based method for the Unsymmetric Eigenproblem for at Lea St years after the appearance of the QR? Why do more than years pass before the properties of the Lanczos algorithm were understood? I believe that the explanation must involve the impediments to comprehension of the effects of finite-precision arithmetic .

(Quoted from Www.siam.org/siamnews/11-03/matrix.pdf)

Since it is a math major, you cannot have an understanding of the experts in your field. In the early years, Chinese had a place in the field of computational mathematics because Feng Kang academician was independent of the West, founded the finite element method, and then proposed the symplectic algorithm. Here are just a few of the younger Chinese computational mathematics experts, because they represent the focus of current computational mathematics and reflect the contribution of the Chinese to the development of computational mathematics.

Hou (Caltech)
Research interests: Computational fluid dynamics, multiscale computation and simulation, multiphase flow
http://www.acm.caltech.edu/~hou/

E Wimaan (Princeton University)
Changjiang Scholar, Peking University, Research direction: Multi-scale computation and simulation
Http://ccse.pku.edu.cn/staff/weinane.htm

Baogang (Michigan State University)
Changjiang scholar, Jilin University, Research direction: Calculation of optics and electromagnetic fields
http://www.mth.msu.edu/~bao/

Jinshi (Wisconsin University)
Changjiang scholar, Tsinghua University, Research direction: Hyperbolic conservation law, computational fluid dynamics, kinetic theory, etc.
http://www.math.wisc.edu/~jin/

Bernoulli (Hong Kong Baptist University)
Chinese Academy of Sciences, research direction: Mobile Grid method, etc.
http://www.math.hkbu.edu.hk/~ttang/

Shu Qiwang (Brown University)
Zhong Ke changjiang scholar, research direction: Computational fluid dynamics, spectral method
Http://www.dam.brown.edu/people/shu/home.html

CHENHANF (Chinese University of Hong Kong)
Research direction: Numerical linear algebra
http://www.math.cuhk.edu.hk/~rchan/

Hu Jinsu (Pennsylvania State University)
Yangtze River Scholar, Peking University, Research direction: Finite element and multi-grid method
http://www.math.psu.edu/xu/

Shing
The research direction of the Chinese Academy of Sciences is nonlinear optimization
http://lsec.cc.ac.cn/~yyx/

Zhangpingwen (Peking University)
Beijing University Changjiang Scholar, the research direction for the complex fluid simulation, multi-scale calculation and
Simulation, mobile grid method, etc.
Http://www.math.pku.edu.cn/pzhang/index.html

Zhiming (Chinese Academy of Sciences)
Research direction: Scientific calculation and numerical analysis, mainly for finite element method
Http://lsec.cc.ac.cn/~zmchen/index-c.html

Other Huang Weizhang, Wu Zongmin, Xu Kun, Cheng and other people are also very prominent.

As a student of computational mathematics, it may be helpful to read the major magazines in this major regularly.

Theory: the best basic is
Mathematics of Computation
Numerische Mathematik
SIAM Journal on numerical analysis
SIAM Journal on Matrix analysis & Applications
SIAM Journal on scientific Computing

Better to have:
BIT
IMA Journal of numerical analysis
Advances in Computational mathematics
Inverse problems

There are also application-nature magazines:
Journal of Computational Physics
International Journal for numerical Methods in Engineering
Computer Methods in Applied mechanics and Engineering
International Journal for numerical Methods in fluids
Computers and Fluids
Computational mechanics

There are also a number of other subjects with computational words in the journal:

Journal of computational chemistry, computational Material Sciences

can also be browsed.

But as a starter, the review is especially helpful for the newcomers to quickly grasp and grasp an area, and thus deserve special attention.

The best part of this is the Acta numerica continuous publication published by Cambridge University Press. Acta Numerica publishes a book every year and the authors are the top figures in the field. For example, in recent years the level set method is very popular, in 05 there was a level set method, one of the founders of the Stanley Osher wrote the levels set methods in the Image science. Other topics include: Entropy stability (Tadmor E), Radial basis function (Buhmann MD), etc. This publication can be found online from a number of websites. The other one is Siam Review. Every period of SIAM Review there are several articles about computational mathematics, often from practical problems to extend the calculation, or to introduce the latest progress in each area. SIAM News also has interesting essays on calculations, so you can browse through them.

As a student of the mathematics department, it is undoubtedly necessary to read a lot of maths books. The Book of computational mathematics can be called voluminous.

The numerical solution of differential equation is the core topic in computational mathematics. The traditional methods include finite difference method, finite element method, boundary element method and spectral method.

The idea of finite difference method is the simplest and easier to understand. Li Ronghua's "numerical solution of Differential equations" introduces the most basic things: convergence, compatibility and stability.

Richtmeyer & Morton's difference Methods for initial-value problems is a classic of the difference approach.

R. Leveque also has a recent "finite difference method for differential equations" which is also interesting, introducing a new modern concept of the difference approach. Leveque's book can be downloaded on his homepage (http://www.amath.washington.edu/~rjl/), his other book numerical Methods for Conservation laws is a very good book in the numerical method of conservation law.

The finite element method is naturally recommended to use Ciarlet's "the finite Element method for Elliptic problems", which is the teaching material of the Department of Professional Science.

In addition Brenner & Scott's mathematical theory of the finite Element Method is also said to be good.

Spectral method is often the most effective method for the problem of the rule area. Professor Guo Benyu of ECNU has done a good job in this area and his "spectral Methods and their applications" has been widely praised. Professor Shen Jie of Purdue University also has a very good job, and one of his handouts can be downloaded from his homepage (http://www.math.purdue.edu/~shen/), along with the relevant MATLAB and Fortran Trefethen Spectral Methods in Matlab, and others are Canuto and others spectral Methods in Fluid Dynamics. In addition to the above methods, there are more popular in recent years meshless methods, which can refer to Zhangxiong and Liu Yan "meshless Method" (Tsinghua University Press, 2003, 50￥). Program. The best introductory book on spectral methods is

The primary tool for computational mathematics is functional analysis. The general recommended Yoshida's "Functional Analysis" (with Chinese translation: Yoshida tillage, "functional analytics") or Rudin's "functional analyses". Both of these books are very difficult, but they are also very classic books, maybe when the dictionary is more appropriate. However, the important theorem in functional analysis is not particularly useful in calculations, so we want to identify those things that may be useful, Sawyer's "Introduction to numerical functional analysis" may be a more appropriate primer. This book introduces some of the concepts of functional analysis, such as the derivation of holder inequalities, as well as the application of functional analysis in computational mathematics, such as the interpretation of Kantorovich iterative convergence criteria. Zhang Gongqing's "functional analysis" emphasizes the application of functional analysis, and there are some examples of numerical calculations, such as the Lax equivalence theorem, which is worth reading.

Computational mathematics also has many other important branches, such as matrix computation, inverse problem, computational fluid dynamics, optimization, approximation theory, and so on. As I have little to say about this, there is nothing to talk about here. Other Xu Neng that hope to calculate mathematics in these directions add up. Finally, a subscription to the mailing list is also good, so you can quickly get information about computational math conferences, new publications, and more. The recommended use of CAM in Chinese, can be registered at the following URL
Http://www.math.hkbu.edu.hk/cam-net/indexcn.html
Recommended Clever Moler na Digest in English, can be registered at the following URL
Http://www.netlib.org/na-net

Next, we introduce some books on matrix computing. Zhejiang Zhang Zhenyue teacher in this area has a very good job, the Chinese Academy of Sciences of the White, Beijing University of Xu Xiaofang, Fudan Wei Yimin and Cao Zhihao, the University of Macau Jin Xiaoqing are in this direction, and Fudan went out of the Ba Shaojun. Certainly there are many scholars in this area has a very prominent work, but I basically do not have a little involved, here also can not be listed.

The foreign Daniel has the Golub, many in this direction everybody is his student. Kahan, James Demmel, Peter Stewart, L N Trefethen,higham, this list can be very long, these people are the matrix computing aspects of everyone.

The most classical book in matrix Computing should be J H Wilkinson's "The Algebraic Eigenvalue Problem" (with Chinese translation, Shi Zhongci translation, "Algebraic eigenvalue Problem", Science Press, School library, the English version of the department). Although the book is old, it is said to be very enlightening to read. Now the classic is Golub and van Loan "Matrix Computation" (with Chinese translation, Shing translation, "Matrix Computing", Science Press), the English version of the electronic version can be found online. Other books have Demmel "Applied numerical Linear Algebra", Trefethen & Bau numerical Linear Algebra is said to be very good. Yousef Saad has two books "iterative Methods for sparse systems" and "numerical methods for large eigenvalue problems", which is very interesting to write on his homepage
(http://www-users.cs.umn.edu/~saad/) can be down. When it comes to matrix computing, it is also necessary to mention an old book by Householder, the theory of matrices in numerical (with Chinese translation, in both English and Chinese versions). LN Trefethen is now a professor at Cambridge University, and every book he writes is classic, with several books in front of him, "Spectral Method in Matlab", "Numerical Linear Algebra" and "finite Difference and spectral methods down, http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/). Reading his book and article feeling is also a great enjoyment of life. When he was teaching at Cornell University, he took a course in reading the classical literature of numerical calculation. For this he wrote a short article, listed the numerical calculation of the 13 classic documents, perhaps a little inspiration to everyone.

1. Cooley & Tukey (1965) The Fast Fourier Transform
2. Courant, Friedrichs & Lewy (1928) Finite difference methods for PDE
3. Householder (1958) QR factorization of matrices
4. Curtiss & Hirschfelder (1952) stiffness of odes; BD formulas
5. De Boor (1972) Calculations with B-splines
6. Courant (1943) Finite element methods for PDE
7. Golub & Kahan (1965) The singular value decomposition
8. Brandt (1977) Multigrid algorithms
9. Hestenes & Stiefel (1952) The Conjugate gradient iteration
Fletcher & Powell (1963) optimization via Quasi-Newton updates
Wanner, Hairer & Norsett (1978) Order stars and applications to ODE
Karmarkar (1984) Interior Pt. Methods for linear Prog.
Greengard & Rokhlin (1987) Multipole Methods for particles

His remark is also interesting, We were struck by what young many of the authors were when they wrote these papers (AVERAGEAGE:34), and by How short a influential paper can be (householder:3.3 pages, Cooley & tukey:4.4). This shows that everyone is still very hopeful, hehe. "(On his homepage, you can

The inverse problem is undoubtedly one of the hottest directions in computational mathematics. The direction now has the following magazines: Inverse problems,journal of inverse and ill-posed problems, inverse problems in Sciences and Engineering (formerly called Inverse problems in Engineering). The first magazine is the best, the second magazine has many Soviet jobs, and the third one is biased towards application. In many high-grade magazines, there are anti-problem articles, such as SIAM Journal on numerical analysis, SIAM Journal in mathematical analysis, SIAM Journal on Matrix Analysis and applications, SIAM Journal on scientific Computing There are also many anti-problem aspects of the article.

In the domestic do anti-problem do the best should be Fudan University's Chengjin teacher, he has a lot of work on the theory of anti-Problem estimation, Nanjing University's Golden Years teachers also have a lot of good results (very young!) ), there are several people who are doing application work (their former principal is doing geophysical anti-problem). Internationally well-known are HW Engl (Australia), Yamamoto (Japan), Kress (Germany), Martin Hanke (Germany), Isakov (USA) etc. An important feature of the inverse problem is that it is very close to the actual problem, often need to design a special algorithm according to the characteristics of the problem, which is also the difficulty of anti-problem. Many applications have been combined with inverse problems to become a separate research area, such as EIT.

The application of level set method to inverse problem seems to be a hotspot in the research of inverse problem algorithm. Fadil Santosa of the University of Minnesota first applied the level set method to solving the inverse problem, but there was no big response. Engl's student, Martin Burger, applied the level-set approach to counter-issues (published on Inverse problems) in 2000, and had a great international response. Martin Burger was invited to study at UCLA's Osher Group after his doctorate, and with Osher a review and prospect of the application of the level set method in the inverse problem is worth reference. The inverse problem is the most classic of the Tikhonov and Arsenin "Solutions of Ill-posed Problems" (with Chinese translation, "The solution of ill-posed problem"). Now the inverse of the problem, each important article is basically to cite this book. This book is more abstract, algorithmic aspects are involved, but not much. Later Tikhonov and Yogola and other people wrote a non-linear inverse problem theory of the book, but also wrote a book on algorithms, unfortunately the title I have forgotten.

Personal feeling Groetsch's "The Theory of Tikhonov regularization for Fredholm equation of the first kind" is a relatively good introductory book, this book is relatively thin, it is easier to read. After reading this book, it should not be a big problem to read the theory of anti-problem. Kress's "Linear Integral equations" and Kirsch's "an Introduction to the mathematical theory of Inverse problems" are also good introductory books.

Engl and other people's "regularization of Inverse problems" widely praised, should be able to be used as a further reading material. There are many specialized works, such as Isakov's "inverse problems for partial differential equations", Martin Hanke's conjugate Gradient Type Methods fo R ill-posed Problems "should also be good. There are not many books on the numerical algorithm of inverse problem, only Hansen "rank-deficient and discrete ill-posed problems" and Vogel "computational Methods for Inverse Pr Oblems ". Both books are very good, the basis of the requirements is basically similar, the basic concept of matrix computing is very familiar. But the focus is different, Hansen book is easy to read, so in the engineer is also very popular. Vogel's book is a little more mathematical, involving a slightly wider range, such as the very important total variation regularization in Hansen's book is not discussed, but Vogel's book did a very detailed discussion.

The reading list of anti-problems can be found in the following links:
Http://infohost.nmt.edu/~borchers/geop529/readings/readings.html

The calculated hotspots seem to have two features: one is to combine with specific applications to form new disciplines, such as computational fluid dynamics, computational aerodynamics, computational mechanics, computational physics. The emphasis here is to contribute to the development of new disciplines, the so-called third method of research besides experimentation and theory. Computational problems in materials and organisms seem to be a hot spot in computational mathematics later, and can be consulted in the comments of the Wimaan teacher. One is to apply a new mathematical tool. For example, the Lie group theory is used to construct the numerical solution of the differential equation, the continuation method of topological derivation. The reason may be based on some kind of physical consideration, but it can be solved by introducing new mathematical tools. This should also be a notable place.

(ext.) Online Excerpt: Computational mathematics research direction and online information