beat·365(中国)-官方网站

beat365官方网站将于2018年7月15日至8月3日（根据基金委天元数学专家小组的建议，时间已作修改）主办“应用数学”暑期学校，旨在充分利用beat365官方网站数学学科的优势和学术资源，邀请海内外一流专家学者，为国内相关院校的研究生和青年教师，讲授应用数学领域的基础核心课程和特色课程。学员通过学习与交流，夯实数学基础，把握前沿研究成果，拓宽科研思路，提高自主创新能力，实现资源共享，增进彼此友谊。

【组织机构】

组织人员：黄云清，汤华中，杨银，聂家旺，易年余，杨伟

资助单位：国家自然科学基金委；

beat365官方网站；

科学工程计算与数值仿真湖南省重点实验室

主办单位：beat365官方网站

【课程设置】

2018年度beat365官方网站“应用数学”暑期学校将开设8门课程，涉及科学工程计算、优化等领域。具体如下：

课程一：《非线性方程组数值方法》

授课教师：范金燕教授，上海交通大学

教学内容：非线性方程组的求解是数值代数和数值优化的重要问题，也是科学计算和计算数学的核心问题，它在国防、经济、工程、管理等许多领域有着广泛的应用。本课程将介绍求解非线性方程组的牛顿法、拟牛顿法、高斯牛顿法、Levenberg-Marquardt方法、信赖域方法、子空间方法等重要方法，以及近年来国内外关于非线性方程组的最新前沿研究成果。

参考文献：

1. Bjorck, Ake, Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.

2. Dennis, J. E., Jr.; Schnabel, Robert B. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Classics in Applied Mathematics, 16. SIAM, Philadelphia, PA, 1996.

3. 范金燕，袁亚湘. 非线性方程组数值方法. 北京：科学出版社，2018.

4. 李庆扬，莫孜中，祁立群. 非线性方程组的数值解法. 北京: 科学出版社, 1999.

5. 袁亚湘，非线性优化计算方法. 北京：科学出版社, 2014.

6. 袁亚湘，孙文瑜. 最优化理论与方法. 北京：科学出版社, 1997.

课程二：《Mathematical analysis and simulation of metamaterial Maxwell's equations》

授课教师：李继春教授，University of Nevada & beat365官方网站

教学内容：In this series of lectures, I'll first talk about the short history of metamaterials with related physics and potential applications. Then I'll present some popular modeling PDEs for metamaterials, followed by the wellposedness study of those PDEs. After this, I'll go over the H(div) and H(curl) finite elements used to solve Maxwell's equations. Then I'll present some time-domain finite element methods using edge elements to solve the metamaterial models. To simulate practical wave propagation problems, we need to use the perfectly matched layers (PMLs), which I'll explain some PML models we studied. Finally, I'll present some interesting simulation results for metamaterials such as the invisibility cloaks and optical black holes. If time permits, I'll introduce some DG methods we developed for the metamaterial models.

参考文献：

1. J. Li and Y. Huang, Time-Domain Finite Element Methods for Maxwell's Equations in Metamaterials, Springer Series in Computational Mathematics, vol.43,Springer, 2013.
2. Y. Huang, J. Li and W. Yang, Modeling backward wave propagation in metamaterials by the finite element time domain method, SIAM Journal on Scientific Computing, vol.35, no.1 (2013) B248–B274.
3. J. Li, C. Meng and Y. Huang, Improved Analysis and Simulation of a Time-Domain Carpet Cloak Model, Comput. Methods Appl. Math. (in press).

课程三：《Structure-preserving high order numerical methods for time-dependent problems》

授课教师：刘海亮教授，Iowa State University & beat365官方网站

教学内容：The course discusses the high order structure-preserving numerical methods for several time-dependent problems, modeled by partial differential equations balanced with diffusion, convection, and various interactions. The main topic focuses on the development of stable, accurate approximate algorithms for these problems, and the interplay between analytical theory and computational aspects of such algorithms with real applications. Here is an outline of the main topics:

1. Background and canonical mathematical models

2. Critical threshold phenomena

3. Direct DG methods for diffusion with drifts

4. Alternating evolution methods for nonlinear equations

参考文献：

1. Mathematical models in Biology: Benoit Perthame, Transport equations in Biology, Birkaeuser Verlag, 2007.

2. Critical threshold phenomena: S. Engelberg, H. Liu and E. Tadmor, Critical threshold phenomena in Euler-Poisson equations , Indiana University Mathematics Journal, 50 (1) (2001), 109-157.

3. DDG methods: H. Liu and J. Yan, The Direct Discontinuous Galerkin (DDG) method for diffusion with interface. corrections, Commun. Comput. Phys. 8(3) (2010), 541-564.

4. AEDG methods: H. Liu and M. Pollack, Alternating evolution discontinuous Galerkin methods for convection-diffusion equations , J. Comp. Phys. 307 (2016), 574-592.

课程四：《随机非线性优化的算法简介》

授课教师：张洪超教授，Louisiana State University

教学内容：近些年随机非线性优化方法广泛出现在机器学习、统计和大数据分析等领域。在这些模型中目标函数往往是随机变量的数学期望。传统的优化算法需要精确计算目标函数的函数值和梯度，因此很难有效处理这类问题。本课程将介绍求解这类问题的两大类随机优化算法，包括 Stochastic Approximation (SA) 方法和 Sample Average Approximation (SAA) 方法。我们将讨论这些算法的收敛性质和计算复杂度，以及关于这些算法的前沿研究结果。

参考文献：

1. Yurii Nesterov, Introduction lectures on convex optimization: A basic course, Kluwer Academic Publishers, 2004.

2. S. Ghadimi, G. Lan and H. Zhang, Mini-batch Stochastic approximation methods for nonconvex stochastic composite optimization, Math Programming, 155 (2016), pp. 267-305.

3. S. Ghadimi and G. Lan, Accelerated gradient methods for nonconvex nonlinear and stochastic programming, Math Programming, 156 (2016), pp. 59-99.

4. L. Xiao and T. Zhang, A proximal stochastic gradientmethod with progressive variance reduction, SIAM J. Optim. 24 (2014), 2057–2075.

5. X. Wang, S. Wang and H. Zhang, In exact proximal stochastic gradient method for convex composite optimization, Comput. Optim. Appl. 68 (2017), pp. 579-618.

6. S. J. Reddi, A. Hefny, S. Sra and B. Poczos, Stochastic Variance Reduction for Nonconvex Optimization, CoRR, abs/1603.06160, 2016.

课程五：《多项式全局最优化》

授课教师：支丽红研究员，中国科学院数学与系统科学研究院

教学内容：介绍基于符号和数值方法求解由多项式等式和不等式定义的可行域上的多项式全局最优问题。我们将主要介绍实代数几何基础、半正定规划方法及其在多项式全局最优问题中的应用。本课程也将介绍多项式优化方面的一些新的成果及其在图像处理、程序验证、量子计算中的应用。

参考文献：

1. Semidefinite Optimization and Convex Algebraic Geometry，SIAM, 2012.

2. J.B. Lasserre. Moments, Positive Polynomials and Their Applications. Imperial College Press, 2009.

3. M. Laurent. Sums of squares, moment matrices and optimization over polynomials. In Emerging applications of algebraic geometry, volume 149 of IMA Vol. Math. Appl., pages 157{270. Springer, New York, 2009.

4. J. Bochnak, M. Coste, and M.F. Roy. Real algebraic geometry. Springer Verlag, 1998.

课程六：《Fractional PDEs and their numerical solutions》

授课教师：周知博士，Hong Kong Polytechnic University

教学内容：Fractional calculus was introduced in seventeenth century and almost fully-developed by the middle of the twentieth century. It captured and held people’s attention again due to the wide applications of fractional PDEs in modelling anomalous diffusion, which describes a diffusion process where the mean square displacement of a particle grows faster (super-diffusion) or slower (sub-diffusion) than that in the normal diffusion process. In many cases, the appearance of fractional operators in PDEs may bring in new consequences for underlying physical processes, as well as new challenges for mathematical and numerical analysis . A major part of the lectures is devoted to numerical methods for the fractional PDEs with a rigorous theoretical analysis. An introduction on fractional calculus and fractional PDEs will also be included. The lectures will be organized as follows:
1. some background of fractional calculus, its history, basic features and some applications

2. fractional partial differential equations, dynamics and motivations

3. time-fractional diffusion equations, regularity and time-stepping schemes

4. space-fractional diffusion equations, regularity and finite element approximations

参考文献：

1. Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. MR 2218073.

2. Igor Podlubny, Fractional Differential Equations, Academic Press, Inc., San Diego, CA, 1999. MR 1658022.

课程七：《Finite element construction》

授课教师：张上游教授，University of Delaware

教学内容： This course summarizes the constructions of scalar finite elements, mixed finite elements for scalar equations, mixed finite elements for vector equations, and mixed tensor finite elements for vector equations. Namely, we present various finite elements for the Laplace equation, the Stokes equations, and the Navier equation. The main topics include:

1. Conforming and nonconforming finite elements on simplicial and tensor grids

2. Mixed finite elements and the theory

3. Divergence-free finite elements and open problems

4. Conforming symmetric tensor finite elements on simplicial and tensor grids

参考文献：

1. Brenner, Susanne C.; Scott, L. Ridgway. The mathematical theory of finite element methods. Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008.

2. Hu, Jun; Huang, Yunqing; Zhang, Shangyou. The lowest order differentiable finite element on rectangular grids. SIAM J. Numer. Anal. 49 (2011), no. 4, 1350-1368.

3. Hu, Jun; Man, Hongying; Wang, Jianye; Zhang, Shangyou. The simplest nonconforming mixed finite element method for linear elasticity in the symmetric formulation on n-rectangular grids. Comput. Math. Appl. 71 (2016), no. 7, 1317-1336.

4. Hu, Jun; Man, Hongying; Zhang, Shangyou. A simple conforming mixed finite element for linear elasticity on rectangular grids in any space dimension. J. Sci. Comput. 58 (2014), no. 2, 367-379.

5. Hu, Jun; Zhang, Shangyou. Nonconforming finite element methods on quadrilateral meshes. Sci. China Math. 56 (2013), no. 12, 2599-2614.

6. Hu, Jun; Zhang, Shangyou. The minimal conforming Hk finite element spaces on Rn rectangular grids. Math. Comp. 84 (2015), no. 292, 563-579.

7. Hu, Jun; Zhang, Shangyou. A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids. Sci. China Math. 58(2015), no. 2, 297–307.

8. Zhang, Shangyou. Divergence-free finite elements on tetrahedral grids for k >= 6. Math. Comp. 80 (2011), no. 274, 669-695.

9. Zhang, Shangyou. On the P1 Powell-Sabin divergence-free finite element for the Stokes equations. J. Comput. Math. 26 (2008), no. 3, 456-470.

10. Zhang, Shangyou A P4 bubble enriched P3 divergence-free finite element on triangular grids. Comput. Math. Appl. 74 (2017), no. 11, 2710-2722.

课程八：《张量计算简介》

授课教师：聂家旺教授，University of California, San Diego & beat365官方网站

教学内容：张量计算是现代计算数学的最新前沿方向，也是现代应用与计算数学研究的重点、热点。本课程将介绍张量空间的基本概念和基本理论，各种张量秩的定义及其理论性质（比如cp-秩，多重线性-秩，边界-秩，对称-秩），张量分解的基本理论和算法，低秩逼近问题，及其它相关专题。

参考文献：

1. P. Comon, G. Golub, L.-H. Lim and B. Mourrain. Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. Appl., 30, no. 3, 1254-1279, 2008.

2. T. Kolda and B. Bader. Tensor decompositions and applications. SIAM Rev. vol. 51, no. 3, 455-500, 2009.

3. J.M. Landsberg. Tensors: geometry and applications. Graduate Studies in Mathematics, 128. American Mathematical Society, Providence, RI, 2012.

4. L.-H. Lim. Tensors and hypermatrices, in: L. Hogben (Ed.), Handbook of linear algebra, 2nd Ed., CRC Press, Boca Raton, FL, 2013.

5. J. Nie. Generating Polynomials and Symmetric Tensor Decompositions. Foundations of Computational Mathematics, Vol. 17, No. 2, pp. 423-465, 2017.

【前沿讲座】

已邀请的前沿报告人（按姓氏首字母排序）：

1. 戴书洋，武汉大学；

2. 龚伟，中国科学院数学与系统科学研究院；

3. 胡光辉，澳门大学；

4. 李卫华，复旦大学；

5. 刘歆，中国科学院数学与系统科学研究院；

6. 卢宇源，中科院长春应用化学研究所；

7. 沈捷，Purdue University & 厦门大学；

8. 史安昌，McMaster University；

9. 王涵，北京应用物理与计算数学研究所；

10. 武海军，南京大学；

11. 杨志坚，武汉大学；

12. 张辉，北京师范大学；

13. 张磊，北京大学；

14. 张振，南方科技大学；

15. 张继伟，北京计算科学研究中心。

【招生】

招生对象：青年教师、博士后、博士生、硕士生

招生人数：120人

报名：

1. 申请人需提交申请表（含专家推荐意见及单位盖章）扫描版及纸质版。扫描版以“学校+姓名”的方式命名，发送至：icam@xtu.edu.cn；纸质版通过顺丰快递寄至：湖南省湘潭市雨湖区beat365官方网站刘天凤（收），电话：15773289156。

2. 报名截止日期为2018年6月15日（申请表纸质版需在此日期前寄出，以邮戳时间为准）。

学员待遇：

为正式学员购买暑期学校期间在湘潭的意外保险（中国平安-团体短期综合意外险，起止时间：2018.07.15-2018.08.04），提供免费住宿及基本生活用品（包括整套床上用品、水桶和脸盆）、一定的生活补助、教材或授课讲义、互联网、计算机、自习室、图书阅览等必需辅助学习条件。注：湘潭地区的学员不提供住宿。

【录取】

1. 录取结果已于2018年7月2日前，以邮件的方式告知预录取学员，收到邮件并确认参加即表示录取；

2. 报到时间为7月15日8:00-22:00，地点为beat365官方网站数学楼北楼一楼大厅。逾期未报到且未提前告知者，视为放弃。从机场、高铁站、火车站到beat365官方网站的乘车方式见附件。

3. 学员须通过结业考核，方可获得结业证书。

【日程安排】

每天授课或讲座时间为：上午08:30-09:50，10:10-11:30；下午14:30-15:50，16:10-17:30。地点为图书馆报告厅。具体日程安排见附件。

【联系方式】

杨银，0731-58292763；

刘天凤，0731-58292742，icam@xtu.edu.cn。

附件一：2018年度beat365官方网站“应用数学”暑期学校申请表.doc