solution of error-relocation truncated to fit: R_X86_64_PC32 解决办法： The default memory model for the PGI compilers is the “small” model. This requires that the object be smaller than 2 GB in size. The PGI compilers support the “medium” memory model, which allows objects to be larger than 2 GB. Unfortunately, for a code to use the medium memory model, all objects and static libraries must be compiled under the medium memory model. 在编译选项中加如选项：-mcmodel=medium，重新编译程序
Size-Matching/Size-Calling Algorithm Size-Matching Size-Calling Algorithm This algorithm uses a dynamic programming approach that is efficient (runs in low polynomial time and space) and guarantees an optimal solution. It first matches a list of peaks from the electropherogram to a list of fragment sizes from the size standard. It then derives quality values statistically by examining the similarity between the theoretical and actual distance between the fragments. Size-Matching Algorithm Example Figure 3-14 shows an example of how the size-matching/calling algorithm works using contaminated GeneScan™ 120 size standard data. Detected peaks (standard and contamination) are indicated by blue lower bars along the x-axis. The size standard fragments as determined by the algorithm (and their corresponding lengths in base pairs) are designated by the upper green bars. Note that there are more peaks than size standard locations because the standard was purposely contaminated to test the algorithm. The algorithm correctly identifies all the size standard peaks and removes the contamination peaks (indicated by the black triangles) from consideration. The large peak is excluded from the candidate list by a filter that identifies the peak as being atypical with respect to the other peaks. Figure 3-14 Size-matching example
word 单词分开： 解决办法： 1.自己： 选中text， Paragraph - Asian Typography - 取消勾选“Allow Latin text to wrap in the middle of a word” 2.网络： Ctrl+A全选或自行点选某段。 在选取的文字上右击===〉段落==〉中文版式===〉取消 。 如果不行，则应该是你在单词中间多打了空格
本人写的一个针对硫化铅量子点太阳能电池的最新综述，对2011-2012年间量子点太阳能电池的研究做了一些总结，欢迎感兴趣的朋友下载讨论： Recent development in colloidal quantum dots photovoltaics Frontiers of Optoelectronics 2012, DOI 10.1007/s12200-012-0285-7. Li Peng , Jiang Tang*, Mingqiang Zhu 摘要：The increasing demanding for sustainable and green energy supply spurred the surging research on high- ef fi ciency, low-cost photovoltaics. Colloidal quantum dot solar cell (CQDSC) is a new type of photovoltaic device using lead chalcogenide quantum dot film as the absorber materials. It not only has the potential to break the 33 % Shockley-Queisser efficiency limit for single junction solar cell, but also possesses the low-temperature, high-throughput solution processing. Since its first report in 2005, CQDSCs experienced rapid progress achieving acertified 7% efficiency in 2012, an averaged 1% efficiency gain per year. In this paper, we reviewed the research progress reported in the last two years. We started with background introduction and motivation for CQDSC research. We then briefly introduced the evolution history of CQDSC development as well as multiple exciton generation effect. We further focused on the latest efforts in improving the light absorption and carrier collection efficiency, including the bulk-heterojunction structure, quantum funnel concept, band alignment optimization and quantum dot passivation. Afterwards, we discussed the tandem solar cell and device stability, and concluded this article with a perspective. Hopefully, this review paper covers the major achievement in this field in year 2011 –2012 and provides readers with a concise and clear understanding of recent CQDSC development.
Can the complex motions in fluid, such as Brownian motion and diffusion, be described with the exact solution of the motion equations of fluid? This problem is closely related to the famous " Millennium Prize Problems " established by the Clay Mathematics Institute of Cambridge, Massachusetts，for celebrating mathematics of new millennium. One of them is about the Navier-Stokes equation. This problem was introduced shortly and vividly in the website of the Clay Institute as follows: Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations. Obviously, one of the possible explorations to the this problem is to try to give an exact solution to the Euler equation (which is the simplest case of Navier-Stokes equations) for describing some complex fluid motions. In the past thirty years, several exact solutions of such kind were given, such as in , , and . But these solutions usually need some complex and unnatural external force in the Euler equation, i.e., the corresponding complex motions were driven by the complex and unnatural external force (here, the "unnatural external force" means a non-potential force). So, these solutions are somehow not quite satisfactory. From 2006 to 2008, this problem was also studied by me and my graduate students Weiwei Yu and Minghui Liu. Based on the "pseudo-potential" conception proposed by Weiwei Yu, a kind of exact solution of the Euler equation was found out. This kind of exact solution contains two arbitrary given functions and three arbitrary given parameters, and the external force of the corresponding Euler equation could be zero or any given potential force. Based on the choice of the two functions and three parameters contained in the solution, and based on the KAM theory and Melnikov Method, it is proven that the Brownian motion and diffusion of the fluid can described by the chosen exact solution. The concrete exact solutions and the sketch of the related proofs are introduced in my blog paper A Series along the Nature and Beauty in Chinese. The exact solutions and the obtained second order Melnikov function are also listed on the attached pdf file Main Mathematical Formulae in English. Main Mathematical Formulae.pdf To show the complex motion (diffusion), an animation (click on the animation to watch it) was made with the software Mathematica. In this animation, 40000 fluid particles are initially distributed to four small circles, and the four groups of particles are each dyed with a different color, so that each circle has their own unified color. The animation shows how the 40000 particles move according to the chosen exact solution, and how the four colored circles develop into four different closed curves following the fluid particles on it. It is a well known fact that if infinitely many particles are continuously distributed on the four circles, following the motions of the fluid particles, the shapes of the four circles will develop into four closed curves (homotopic to the original four circles), while the areas surrounded by them are maintained respectively, and the four closed curves will never intersect each other. This means the true diffusion (or osmosis) can not really happen if the continuity of the curves is not destroyed. However, for a practical fluid, the fluid particles are always with finite number, no matter how large the number is. So, each circles are formed with only a finite number fluid particles. When the "pseudo-continuous" curves are stretched and deformed drastically, the "continuity" of these curves will be destroyed, obviously, and the diffusion (or osmosis) will really happen between the particles distributed on the four closed curves, shown this way by the animation. The velocity field described by the chosen exact solution used for the animation is periodic both in time and in the coordinates of the two dimensional plane. It is proven by calculation that the mean value of the velocity over time and over space is zero, while the mean value of the square of the velocity is a positive number if the motion exists. Clearly, the larger the mean value of the square of the velocity is, the stronger the complex motion of the fluid is. Therefore, if the period of time and period of space are small from the view point of macro-scope, then the exact solution obtained can be treated as a module of static water with temperature which is proportional the mean value of the square of the velocity. References: T.H.Solomon and J.P. Gollub, Chaotic particle transport in time-dependent Rayleigh-Benard convection , Physical Review A. Vol.38 No. 12, (1988) 6280-6286 S. Wiggins, The dynamical systems approach to Lagrangian transport in oceanic flows , Annu. Rev. Fluid Mech. 37, (2005) 295–328. N. Malhotra and S. Wiggins, Geometric Structures, Lobe Dynamics, and Lagrangian Transport in Flows with Aperiodic Time-Dependence, with Applications to Rossby Wave Flow ,J. Nonlinear Sci. Vol. 8: pp. 401–456 (1998) Author: Keying Guan (Science College, Beijing Jiaotong University) email: keying.guan@gmail.com
Assembly of Graphene Sheets into Hierarchical Structures for High-Performance Energy Storage The graphene oxide (GO) solution was subjected to another 30 min of centrifugation at 14000 rpm. We then adjusted the pH value of the top clean solution to about 9 with 1 mol/L NaOH aqueous solution. After that, we titrated the GO aqueous solution with 1 mg/mL DODA.Br chloroform solution. When the color of aqueous solution changed to light yellow, we stopped adding DODA.Br chloroform solution. The organic phase was then separated and washed by DI water. Finally, the composite material GO/DODA was obtained by evaporating the chloroform to dryness. Typically, the honeycomb thin films were prepared by direct casting GO/DODA complex chloroform solution (1 mg/mL) onto the glass substrates under a moist airflow (relative humidity ca. 85%). The brown thin films covering an area of ca. 2 cm2 were left behind after the complete evaporation of the solvent and water within 30-60 s.The control experiments without humid airflow had been performed in the desiccators (relative humidity ca. 30%) and no macroporous structures developed, leaving only unpatterned flat films as a result. To keep the macroporous structure, we used the hydrazine vapor to reduce the film. The films were then put into the autoclave, and then 10 μL of hydrazine monohydrate was added into the autoclave. Finally, the autoclave was heated at 90 ℃ for 16 h to get the reduced films.
Assembly of Graphene Sheets into Hierarchical Structures for High-Performance Energy Storage Graphite oxide was prepared from natural graphite powder (Sigma, 45 μm) via a modified Hummers method. Typically, graphite powder (0.3 g) was put into an 80 ℃ solution of concentrated H 2 SO 4 (2.4 mL), K 2 S 2 O 8 (0.5 g), and P 2 O 5 (0.5 g). The mixture was kept at 80 ℃ for 4 h and then cooled to room temperature, diluted with deionized water (DI, 0.5 L), and then filtrated. The powder produced was preoxidized product. The preoxidized graphite was further oxidized in concentrated H 2 SO 4 (12 mL) and KMnO 4 (1.5 g). After the addition of KMnO 4 , the mixture was stirred at 35 ℃ for 2 h. Then, the mixture was diluted with DI water (25 mL) in an ice bath to keep the temperature below 50 ℃. After it was stirred for 2 h, it was further diluted with DI water (70 mL). Then 30% H 2 O 2 (2 mL) was added to the mixture, and a brilliant yellow product was formed. The product was filtrated and washed with HCl aqueous solution (1:10, 1 L) and DI water (1 L). Purified graphite oxide suspensions were then dispersed in water to create a 0.1 wt % dispersion. Exfoliation of graphite oxide was achieved by ultrasonication for 2 h. The obtained brown dispersion was then subjected to 30 min of centrifugation at 3000 rpm to remove any unexfoliated graphite oxide (usually present in a very small amount), using a themo-centrifuge with a rotor radius of 14 cm.
2001 Monodispersed hard carbon spherules with uniform nanopores In this work, sugar was selected as the precursor. The typical preparation process can be described simply in two steps, namely dewatering at low temperatures and carbonization at high temperatures. In the dewatering process,aqueous sugar solution of 1.5 mol/ l was filled in a stainless steel autoclave with a fill rate of 90%. After 5 h hydrothermal treatment at 190℃, the obtained black powder was further carbonized in a tube furnace in argon atmosphere. The argon flow rate, final temperature and heating rate of the furnace were 25 ml/min, 1000℃ and 1℃/min below 500℃, 5℃/min above 500℃, respectively.
Abstract: Catalan conjecture is well-known elementary number theory mathematical problems, although already in 2004 by the Swiss mathematician proof, but proof of his use of cyclotomic fields such as mathematics tools. This article attempts to use elementary mathematical methods, give a proof of the conjecture. Key words: Number Theory; Indeterminate equation; Three cubic equation; lnteger solution 摘要： 卡特兰（Catalan）猜想是初等数论中著名的数学难题，虽然已于2004年被瑞士数学家证明，但在他的证明中运用了分圆域等高等数学工具。本文试图运用初等数论的方法给出一个该猜想成立的证明。 关键词： 数论，不定方程，连续数猜想，整数解 卡特兰猜想的证明.pdf
Published by China Daily on 31 October, 2011 Recycled cooking oil is proliferating because illegal companies make money at every stage of the process. Two experts have opposing views on a waste-to- energy solution. Here is some of opinions, see others please enter: http://www.chinadaily.com.cn/opinion/2011-10/31/content_14004907.htm Jiang Gaoming Joint efforts can cut profit chain A nationwide crackdown on "gutter oil", or recycled cooking oil, is on. To prevent recycled cooking oil from appearing on our dinner tables, some people suggest a waste-to-energy transfer is better than a crackdown by police. But that is easier said than done. The government first launched a crackdown on "gutter oil" 10 years ago, but the situation, against all expectations, has only deteriorated. As the cooking oil scandal shows, a strong profit chain has been formed in the illegal oil processing industry: illegal operators collect waste cooking oil from sewages and restaurants and sell it at 4,000 yuan ($629) a ton to underground factories. The illegal factories then process the oil and sell it for 6,000 yuan a ton to middlemen, who then sell it to restaurants for about 8,000 yuan a ton, which is still a lot cheaper than normal cooking oil that cost about 12,000 yuan a ton. Lured by the huge profits that can be made at every stage of the illegal process, many individuals and enterprises have joined the illegal oil recycling sector. The result: the vicious industry is growing by the day. Some people suggest that we learn from the practices of developed countries, most of which have established a mechanism for dealing with waste oil. For instance, a law passed in Germany in the 1970s makes it mandatory for all restaurants to sign a contract with the government and keep an accurate record of every drum of kitchen garbage they produce, thus preventing waste cooking oil from returning to the dinner table. In the US and Japan, it is mandatory for restaurants to sell (or give) garbage containing used cooking oil to only certain collectors so that it can be dealt with in an environmentally friendly way. Japanese collectors even add inedible castor oil to the waste oil they sell to prevent it from being reused as cooking oil. It, still, can be processed as biodiesel that can be used in the garbage trucks, which certainly is a healthy cycle. Some people advise that we should emulate these examples, and help the biodiesel industry replace the underground factories. They say that because biodiesel is sold at almost the same price as petrochemical products - about 7,500 yuan a ton - the government should offer subsidies to biodiesel producers to keep illegal "gutter oil" collectors at bay. That may be a piece of good advice but people who advise it forget how much cooking oil waste is produced in China everyday. The general consumption habit in China is to order more food in a restaurant than we can eat and create unnecessary waste, owing to which an amazingly high amount - 2 to 3 million tons - of waste cooking oil is generated every year. The latest figure I found, shows that in April 2006, Japan's garbage dumps collected less than 1,500 tons of waste cooking oil. Small amounts of subsidy can work in countries like Japan, but China has to provide huge amounts of subsidy - a heavy burden on taxpayers - to just get things going. Besides, even with government subsidies, it would be difficult for biodiesel producers to fight a price war with illegal oil recycling factories, for there is too big a gap between their profit margins. To raise their sales, underground factories generally sell recycled cooking oil at 8,000 yuan a ton, still about 2,000 to 4,000 yuan less than fresh cooking oil. Once biodiesel producers enter the market, underground factories can easily raise the collection price by 1,500 yuan a ton to compete with them, something that can debilitate biodiesel producers that rely on government subsidies for survival. So promoting a healthy industry to recycle waste cooking oil for non -edible use may be a good practice in other countries, but not so practical in China. What we should learn from the countries that have laws or regulations on recycling used cooking oil to pre-empt public health hazards is stricter supervision and detection. Instead of passing laws for collecting of waste cooking oil from restaurants, the government should strengthen supervision over the oil they use to prevent waste cooking oil from returning to the dinner table. That would be more effective and more convenient both. No less important than detecting waste cooking oil is curtailing over-ordering, given people's consumption habit in China. That would require consumers to change their habits. Every consumer should bear in mind that each morsel or bit of food he/she leaves on the table could be used to reprocess cooking oil - that's why he/she should take away the uneaten food instead of wasting it. Perhaps the government should introduce some measures - fines, for instance - to punish people who waste food. In other words, the problem of waste cooking oil is deep-rooted in China and can be solved only with the help and efforts of all. (The author is a researcher from Institute of Botany, the Chinese Academy of Sciences)
谈谈计算数学- 从计算数学的字面来看，应该与计算机有密切的联系，也强调了实践对于计算数学的重要性。也许Parlett教授的一段话能最好地说明这个问题：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 that, with suitable pivoting, the results are 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 did Wilkinson keep the QR algorithm as a backup to a Laguerre-based method for the unsymmetric eigenproblem for at least two years after the appearance of QR? Why did more than 20 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.(引自 www.siam.org/siamnews/11-03/matrix.pdf) 既然是计算数学专业的学生，就不能对自己领域内的专家不有所了解。早些年华人在计算数学领域里面占有一席之地是因为冯康院士独立于西方，创立了有限元方法，而后又提出辛算法。这里只是列出几位比较年轻的华人计算数学专家，因为他们代表了当前计算数学的研究热点，也反映华人对计算数学的发展的贡献。 侯一钊（加州理工） 研究方向：计算流体力学、多尺度计算与模拟、多相流 http://www.acm.caltech.edu/~hou/ 鄂维南（Princeton大学） 北京大学长江学者，研究方向：多尺度计算与模拟 http://ccse.pku.edu.cn/staff/weinane.htm 包刚（Michigan州立大学） 吉林大学长江学者，研究方向：光学与电磁场中的计算等 http://www.mth.msu.edu/~bao/ 金石(Wisconsin大学) 清华大学长江学者，研究方向：双曲守恒律、计算流体力学、 动力学理论等 http://www.math.wisc.edu/~jin/ 汤涛（香港浸会大学） 中科院，研究方向：移动网格法等 http://www.math.hkbu.edu.hk/~ttang/ 舒其望（Brown大学） 中科大长江学者，研究方向：计算流体力学、谱方法 http://www.dam.brown.edu/people/shu/home.html 陈汉夫（香港中文大学） 研究方向：数值线性代数 http://www.math.cuhk.edu.hk/~rchan/ 许进超（Pennsylvania州立大学） 北京大学长江学者，研究方向：有限元、多重网格法 http://www.math.psu.edu/xu/ 袁亚湘 中科院，研究方向为非线性最优化 http://lsec.cc.ac.cn/~yyx/ 张平文（北京大学） 北京大学长江学者，研究方向为复杂流体的模拟、多尺度计算与 模拟、移动网格法等 http://www.math.pku.edu.cn/pzhang/index.html 陈志明（中科院） 研究方向：科学计算与数值分析，主要为有限元法 http://lsec.cc.ac.cn/~zmchen/index-c.html 其他还有黄维章、吴宗敏、Xu Kun、程今等人也非常突出 作为计算数学专业的学生，经常阅读本专业中的主要杂志也许 是颇有裨益的。 理论： 最好的基本是 Mathematics of Computation Numerische Mathematik SIAM Journal on Numerical Analysis SIAM Journal on Matrix Analysis Applications SIAM Journal on Scientific Computing 较好的有： BIT IMA Journal of Numerical Analysis Advances in Computational Mathematics Inverse Problems 还有应用性质的杂志： 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 还有很多带有Computational字眼的其他学科的期刊：Journal of Computational Chemistry，Computational Material Sciences 也可以浏览。 但是作为入门来说，大家的综述特别能帮助我们这些新人迅速把握了解、把握一个领域，因而值得特别重视。这方面最好的是剑桥大学出版社出版的Acta Numerica连续出版物。Acta Numerica每年出版一本，作者均是该领域的顶尖人物。比如说最近几年水平集方法非常热门，05年就有一篇水平集方法创始人之一的Stanley Osher写的Level Set Method in Image Science。其他论题有：entropy stability (Tadmor E)，radial basis function (Buhmann MD)等 等。该出版物我们学校没有订，不过可以从网上可以找到不少。我 这里大概也有二三十篇，可以提供上载。 另外一本就是SIAM Review。SIAM Review的每一期里面都有几篇文 章关于计算数学的内容的，经常从实际问题引伸出计算的问题，或 者是介绍每一个领域的最新进展等。 SIAM News的每一期也有关于 计算的有意思的短文，不妨浏览浏览。 作为数学系的学生，无疑是需要读很多数学书。计算数学的书可以 称得上是汗牛充栋。以前在系版上提到过几本。现在再补充一些。 微分方程数值解是计算数学中的核心论题。传统的方法有有限差分 法、有限元法、边界元法和谱方法。 有限差分法想法最为简单，比较容易理解。李荣华的那本《微分方程 数值解》就介绍了最基本的东西：收敛性、相容性和稳定性。 Richtmeyer Morton的《Difference Methods for Initial-Value Problems》则是差分法方面的经典著作。R. LeVeque最近也有一本 《Finite Difference Method for Differential Equations》也很有意思，介绍了差分方法的新的现代概念。LeVeque的书可以在他的主页（ http://www.amath.washington.edu/~rjl/ ）上下载，他的另外一本书《Numerical Methods for Conservation Laws》是守恒律数值方法方面非常出色的著作。 有限元法方面自然是推荐使用Ciarlet的《The Finite Element Method for Elliptic Problems》。这也是系里专业科的教材，另外Brenner Scott的《Mathematical Theory of the Finite Element Method》据说也是不错的。 谱方法对于规则区域上的问题往往是最为有效的方法。华东师大的 郭本瑜教授在这方面做过很好的工作，他的《Spectral Methods and Their Applications》广受好评。Purdue大学的沈捷教授也有 很出色的工作，他的一个讲义可从他的主页（ http://www.math.purdue.edu/~shen/ ） 上下载，同时还有相关的Matlab和Fortran程序。谱方法方面最好的 入门书为Trefethen的《Spectral Methods in Matlab》，其他的还 有Canuto等人的《Spectral Methods in Fluid Dynamics》，不过 不知道能不能再学校里找到。除了上面这些方法之外，还有近年来比较热门的无网格方法，这些可以参考张雄和刘岩的《无网格方法》（清华大学出版社，2003，50￥）。计算数学的主要工具是泛函分析。一般推荐的Yoshida的《Functional Analysis》（有中译本：吉田耕作，《泛函分析》）或者Rudin的 《Functional Analysis》。这两本书都是非常难的，但是也是非常经典的书，可能当字典比较合适。但是，泛函分析里面重要的定理在计算里面并不见得特别有用，所以我们要甄别那些可能有用的东西，Sawyer的《数值泛函分析引论》也许是比较合适的入门读物。这本书里面介绍了一些泛函分析概念的来由，如 Holder不等式的导出，也有泛函分析在计算数学中的应用，比如Kantorovich迭代收敛性准则的解释。张恭庆的《泛函分析》强调泛函分析的应用，里面 也有一些应用于数值计算的例子，比如Lax等价定理，值得读一下。 计算数学还有其他许多重要的分枝，如矩阵计算、反问题、计算流体力学、最优化、逼近论等。由于这方面本人涉略甚少，这里也没有什么好说的了。希望计算数学这些方向的其他同许能补充上去。最后补充一句，订阅mailing list也是不错的，可以迅速获得关于计算数学会议、新出版文章等的信息。中文的推荐使用CAM，可在下面的网址注册 http://www.math.hkbu.edu.hk/cam-net/indexcn.html 英文的推荐订阅Clever Moler的NA Digest，可在下面的网址注册 http://www.netlib.org/na-net 先订正一个错误：Sawyer的那本书的题目我 记错了，应该叫《数值泛函分析初览》，系资料室和图书馆 都有中译本的。接下来介绍几本矩阵计算方面的书的。浙大的张振跃老师在这方面有很出色的工作，中科院的白中治，北京大学的徐树方，复旦的魏益民和曹志浩，澳门大学的金小庆都是这方向的，还有复旦出去的柏兆俊。肯定还有许多学者在这方面有很突出的工作，可惜我基本上没什么涉略，这里也不能列出来。 国外的大牛有Golub，很多这个方向的大家都是他的学生。 Kahan, James Demmel, Peter Stewart, L N Trefethen, Higham,这个名单可以列的很长，这些人是矩阵计算方面 的大家。 矩阵计算方面最经典的书应该是J H Wilkinson的《The Algebraic Eigenvalue Problem》（有中译本，石钟慈等 人译，《代数特征值问题》，科学出版社，学校图书馆有， 系里有英文版的）。这本书虽然老，但是据说读一下还是 很有启发的。现在的经典是Golub和van Loan的《Matrix Computation》（有中译本，袁亚湘译，《矩阵计算》，科学出版社），英文版的电子版可以在网上找到的。其他的书有Demmel的《Applied Numerical Linear Algebra》，Trefethen Bau 的《Numerical Linear Algebra》据说也是很好的。Yousef Saad有两本书《Iterative methods for sparse systems》和 《Numerical methods for large eigenvalue problems》，写的挺有意思的，在他的主页 （ http://www-users.cs.umn.edu/~saad/ ）上可以down。说到矩阵计算，还得提到Householder的一本老书，《The theory of matrices in numerical analysis》(有中译本，系里中英文版的都有)。 LN Trefethen现在是剑桥大学的教授，他写的每一本书都很经典，前面已经到过他的几本书了，《Spectral Method in Matlab》，《Numerical Linear Algebra》，还有《Finite Difference and Spectral methods》（在他的主页上可以down， http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/ ）。读他的书和文章感觉也是人生的一大享受。他在Cornell大学任教时，曾上过一门课，就是阅读数值计算的经典文献。为此他写过一个短文，列举了数值计算中的十三篇经典文献，也许对大家有点启发。 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 10. Fletcher Powell (1963)optimization via quasi-Newton updates 11. Wanner, Hairer Norsett (1978) order stars and applications to ODE 12. Karmarkar (1984)interior pt. methods for linear prog. 13. Greengard Rokhlin (1987)multipole methods for particles 他的remark也很有意思，We were struck by how young many of the authors were when they wrote these papers (averageage: 34), and by how short an influential paper can be (Householder: 3.3 pages, Cooley Tukey: 4.4).这说明大家 都还是很有希望的，呵呵。 反问题无疑是计算数学中最热门的方向之一。该方向现在有如下 几本杂志：Inverse Problems，Journal of Inverse and Ill-posed 买Problems, Inverse Problems in Sciences and Engineering(以前叫Inverse Problems in Engineering).第一本杂志最好，第二本杂志上面有很多苏联人的工作，第三本偏向于应用。在很多高档次的杂志中都有反问题方面的文章，比如SIAM Journal on Numerical Analysis，SIAM Journal on Mathematical Analysis, SIAM Journal on Matrix Analysis and Applications，SIAM Journal on Scientific Computing上也有不少反问题方面的文章。在国内做反问题做的最好的应该是复旦大学的程晋老师，他在反问题的理论估计方面有不少工作，南京大学的金其年老师也有不少好的结果（很年轻！），哈工大有几个人是做应用方面的工作的（他们的前校长就是做地球物理中的反问题的）。国际上知名的有HW Engl（澳大利亚），Yamamoto（日本）， Kress（德国）， Martin Hanke（德国）， Isakov（美国）等。反问题的一个重要特点就是与实际问题联系特别紧密，往往需要根据问题的特点设计专门的算法，这也是反问题的难点所在。很多应用领域与反问题结合之后成为一个单独的研究领域,如EIT。 水平集方法应用于反问题似乎是当前反问题算法研究中的一个热点。明尼苏达大学 的Fadil Santosa最早将水平集方法应用于求解反问题，但是没有很大的反响。Engl的学生Martin Burger在2000年将水平集方法应用于反问题（发表在Inverse Problems上），在国际上 有很大的反响。Martin Burger在博士毕业后就被邀请到UCLA的Osher的小组作研究，并和Osher一起就水平集方法在反问题的应用作了一个综述和展望，值得参考。反问题反面最为经典的当属Tikhonov和Arsenin的《Solutions of Ill-posed Problems》（有中译本，《不适定问题的解法》，学校里有，英文版的系里有）。现在反问题反面每篇重要的文章基本上都要引用这本书。这本书比较抽象，算法方面有所涉及，但是不多。后来Tikhonov和Yogola等人一起写过非线性反问题反问题理论方面的书，还写过一本算法方面的书，可惜书名我已经忘记的。个人感觉Groetsch的《The theory of Tikhonov regularization for Fredholm equation of the first kind》是比较好的入门书，这本书比较薄，也比较容易读懂。读了这本 书之后，阅读反问题理论方面应该不会有很大问题。Kress的《Linear Integral Equations》和Kirsch的《An Introduction to the Mathematical Theory of Inverse Problems》也是不错 的入门书。这些书在系资料室里都能找到。Engl等人的《Regularization of Inverse Problems》广受好评，应该可以作为进一步阅读的材料。专门的著作有很多，如Isakov的《Inverse problems for partial differential equations》，Martin Hanke的《Conjugate Gradient Type Methods for Ill-posed Problems》应该也是不错的。在反问题的数值算法方面的书籍不多，只有Hansen的《Rank-deficient and discrete ill-posed problems》和 Vogel的《Computational Methods for Inverse Problems》。两本书都是非常棒的，要求的基础基本上类似，对矩阵计算的基本概念非常熟悉。但是侧重点有所不同，Hansen的书容易阅读，所以在工程师里面也是很 popular。Vogel的书稍微数学化，涉及的范围也稍微广一点，比如说很重要的Total Variation regularization在Hansen的书里就不讨论，但是Vogel的书里做了非常详细的讨论。Tikhonov的算法书应该也有很大的参考价值，可惜我没办法搞到，所以也没法评论了。 反问题的reading list 可以在下面的链接中找到： http://infohost.nmt.edu/~borchers/geop529/readings/readings.html 计算的热点似乎有两个特点：一个是与具体的应用结合形成新的学科，比如说计算流体力学、计算空气动力学、计算力学、计算物理。这里强调的是为新的学科的发展做出贡献，也就是所谓的作为除实验和理论之外的第三种研究手段。材料和生物中的计算问题似乎将是以后的计算数学中的一个热点，可以参考鄂维南老师的评论文章。一个是应用新的数学工具。比如说应用Lie群理论构造保格式的微分方程数值解法，拓扑引出的continuation method。其缘由可能是基于某种物理上的考虑，但是可以通过引入新的数学工具来解决。这也应该是一个值得注意的地方。
B.W. Li, Y.S. Sun . Iterative and direct Chebyshev collocation spectral methods for one-dimensional radiative heat transfer. International Journal of Heat and Mass Transfer ,2008, 51(25-26): 5887-5894. download Abstract: In this paper, the Chebyshev collocation spectral method for one-dimensional radiative heat transfer equation with participating media is presented; and sequentially the iterative and direct solvers are developed. Implementation of the new method shows its flexibility to complex problems: highly anisotropic and space-dependent scattering. The new solvers can provide exponential convergence in space and can capture large oscillations. Numerical results verified the high accuracy of the new method, and its competitive ability compared with other newly appeared methods.
Y.S. Sun , B.W. Li. A direct spectral collocation method for radiative heat transfer inside a plane-parallel participating medium with a graded index. 6th International Symposium on Radiative Transfer , (to be prepared). download Abstract: A spectral collocation method based on discrete-ordinates equation is employed to directly solve one dimensional radiative heat transfer in an absorbing, emitting and scattering medium with a graded index. Numerical results by the direct spectral collocation method are compared with those available data in references. The results show that the direct spectral collocation method has good accuracy for one dimensional radiative heat transfer even with space-dependent anisotropic scattering and graded index medium. The CPU time cost comparisons against the resolutions between the direct and iterative solvers are made using MATLAB computer languages. The CPU time cost of direct solver is shorter than that of iterative solver for the same nodes.
标签: solution
