上海交通大学计算与应用数学博士生论坛 Ph.D. Candidate Forum on Computational and Applied Mathematics

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1 上海交通大学计算与应用数学博士生论坛 Ph.D. Candidate Forum on Computational and Applied Mathematics 2012 年 10 月 10 日 13 日 This Forum aims to provide a suitable platform for Ph.D. Candidates to show their research and exchange ideas with each other. The topic of this forum concentrates on computational and applied mathematics and Ph.D. candidates are invited from top universities and institutes all over China. The Forum is supported by Department of Mathematics, Ministry of Education Key Laboratory of Scientific and Engineering Computing, and Institute of Natural Sciences, Shanghai Jiao Tong University. The check-in time is Oct. 10 and the forum last for three days (from Oct. 11 to Oct. 13). Welcome to the forum!

2 组委会成员 : David Cai,Shanghai Jiao Tong University Dan Hu,Shanghai Jiao Tong University Shi Jin,Shanghai Jiao Tong University Yachun Li,Shanghai Jiao Tong University Yaguang Wang,Shanghai Jiao Tong University Dongzhuo Zhou,Shanghai Jiao Tong University 论坛成员 : 1. Xingju Cai ( 蔡邢菊 ), Nanjing University 2. Zhenning Cai ( 蔡振宁 ), 3. Lihui Chai ( 柴利慧 ), Tsinghua University 4. Hong Cheng ( 程宏 ), Shanghai Jiao Tong University 5. Tao Du ( 杜涛 ), Shanghai Jiao Tong University 6. Yuxi Hu ( 胡玉玺 ), Shanghai Jiao Tong University 7. Shixiao Jiang ( 蒋诗晓 ), Shanghai Jiao Tong University 8. Yuling Jiao ( 焦雨领 ), Wuhan University 9. Bin Li ( 李彬 ), Institute of Computing Technology, Chinese Academy of Sciences 10. Haibo Li ( 李海波 ), Tsinghua University 11. Songting Li ( 李松挺 ), Shanghai Jiao Tong University 12. Xiaoguang Li ( 李筱光 ), 13. Bin Min ( 闵斌 ), 14. Xinghua Shi ( 侍兴华 ), Fudan University

3 15. Guangwei Si ( 司光伟 ), 16. Fengyang Tang ( 唐凤阳 ), 17. Yukun Wang ( 王昱焜 ), Shanghai Jiao Tong University 18. Sunan Wu ( 吴朔男 ), 19. Xiaojie Wu( 吴小杰 ), Wuhan University 20. Fangfang Xu ( 徐芳芳 ), Shanghai Jiao Tong University 21. Wenqi Yao ( 姚文琦 ), 22. Jian Zhao ( 赵健 ), 23. Duanmei Zhou ( 周端美 ), East China Normal University

4 注册时间 :2012 年 10 月 10 日 注册地点 : 上海交通大学闵行校区学术活动中心 注册日晚餐 : 留园餐厅 ( 校园内 ) 会议学术报告时间 :2012 年 10 月 11 日 -13 日上午 会议学术报告地点 : 上海交通大学包玉刚图书馆 601 交流活动 :2012 年 10 月 12 日晚 费用 : 免注册费, 上海交通大学数学系 科学工程计算教育部重点实验室和自然 科学研究院资助论坛成员往返机票 ( 经济舱 ) 或火车票和论坛期间的食宿, 但活 动期间, 学员的其它生活费及交通费自行解决

5 Date: Oct. 11; Lecture Room: 601 Pao Yue-kong Library Time Speaker Title Chair Dan Hu 8:30 9:10 Lihui Chai, 柴利慧 Tsinghua University 9:10 9:50 Bin Min, 闵斌 9:50 10:10 Take photo &Tea Break Chair David Cai 10:10 10:50 Tao Du, 杜涛, Shanghai Jiao Tong University 10:50 11:30 Xinghua Shi, 侍兴华 Fudan University 11:30 12:10 Guangwei Si, 司光伟,Peking University 12:10 14:00 Buffet at Maco Polo Chair Dongzhuo Zhou 14:00 14:40 Songting Li, 李松挺 Shanghai Jiao Tong University 14:40 15:20 Yuling Jiao, 焦雨领 Wuhan University 15:20 16:00 Zhenning Cai, 蔡振宁 16:00 16:20 Tea Break Chair Wenjun Ying 16:20 17:00 Bin Li, 李彬 Chinese Academy of Sciences 17:00 17:40 Fangfang Xu, 徐芳芳 Shanghai Jiao Tong University 18:00 Dinner at TianYuan Semi-classical Models for Schrödinger Equations with Periodic Potentials and Band-crossings On breakdown of Anderson localization due to time-dependent disorder and interaction Outlet Boundary Conditions for Blood Flow in Arterial Trees Perturbation Bound and Condition Numbers for the Tensor and Matrix Sylvester Equation & Convergence of General Nonstationary Iterative Methods for Solving Singular Linear Equations APathway-Based Mean-Field Model for E.coliChemotaxis A phenomenological integrate-and-fire model incorporating dendritic integration effects Multi-parameter Regularization Image Restoration Model Based on Framelet and Total Variation 动理学中数值正则化矩方法的新进展超声速气体化学反应流的数值模拟 Robust Linear Optimization Under Matrix Completion

6 Date: Oct. 12; Lecture Room: 601 Pao Yue-kong Library Time Speaker Title Chair Lei Zhang 8:30 9:10 Xingju Cai, 蔡邢菊 Nanjing University 9:10 9:50 Jian Zhao, 赵健 9:50 10:10 Tea Break Chair Xiaoqun Zhang 10:10 10:50 Haibo Li, 李海波 Tsinghua University 10:50 11:30 Yukun Wang, 王昱焜 Shanghai Jiao Tong University 11:30 12:10 Wenqi Yao, 姚文琦 12:10 13:30 Buffet at Maco Polo Chair Jinglai Li 13:30 14:10 Yuxi Hu, 胡玉玺 Shanghai Jiao Tong University 14:10 14:50 Duanmei Zhou, 周端美 East China Normal University 14:50 15:10 Tea Break Chair Min Tang 15:10 15:50 Shixiao Jiang, 蒋诗晓 Shanghai Jiao Tong University 15:50 16:30 Xiaoguang Li, 李筱光 16:30 Visit Huangpu River ADM based customized PPA for separable convex programming 相对论流体力学方程组的 RKDG 方法 Strong Predictor-Corrector Euler-Maruyama Methods for Stochastic Differential Equations with Markovian Switching Transmembrane Permeation of Charged and Polar Groups 一维 Boltzmann 方程的矩封闭模型及其在真一维 Si 半导体器件上的模拟 Neumann type problem for strongly degenerate parabolic-hyperbolic equations on a bounded rectangle Some properties of the nonlinear matrix equation Coupling between internal waves and surface waves Transition Path, Quasi-potential Energy Landscape and Stability of Genetic Switches

7 Date: Oct. 13; Lecture Room: 601 Pao Yue-kong Library Time Speaker Title Chair Dan Hu 9:00 9:40 Sunan Wu, 吴朔男 $H$-adaptive Mesh Method with Double Tolerance Adaptive Strategy for Hyperbolic Conservation Laws 9:40 10:20 Xiaojie Wu Wuhan University Calculation of some continuum variables in molecular dynamics 10:20 10:40 Tea Break Chair David Cai 10:40 11:20 Fengyang Tang 唐凤阳 An efficient Higher Order Heterogeneous Multiscale Method for Elliptic Problems 11:20 12:00 Hong Cheng, 程宏 Shanghai Jiao Tong University Object Extraction By Superpixel Grouping 12:00 14:00 Lunch at Liuyuan Restaurant

8 ADM based customized PPA for separable convex programming Xingju Cai Nanjing University The alternating direction method of multipliers (ADMM) becomes a bench-mark method for solving convex programming problems with separable objective functions. In the literature, it has been illustrated as an application of the proxi-mal point algorithm (PPA) to the dual problem of the model under consideration. Via a variational approach, this paper shows that ADMM is simply an application of PPA to the primal model with a customized choice of the proximal parameter. This primal illustration of ADMM is thus complemental to ADMM's dual illustration in the literature. The PPA illustration of ADMM from primal perspective also enables the possibility of accelerating ADMM by the relaxation technique in PPA literature, and thus the generalized ADMM proposed by Eckstein and Bert-sekas is recovered. A worst-case convergence rate is established for Eckstein and Bertsekas's generalized ADMM. 动理学中数值正则化矩方法的新进展蔡振宁北京大学 我们发展了玻尔兹曼方程的一个全新的数值求解器 该求解器基于 1949 年 Grad 所提出的矩方法, 即通过对分布函数的 Hermite 展开得到半离散方程组 我们在数值上实现了可求解展开至任意阶的矩方程组的高效求解器 同时, 为了使得到的矩方程组得到更广泛的应用, 我们考虑了粘性正则化和双曲正则化两套正则化方案 数值例子显示该方法在速度空间中具有极高的逼近效率 Semi-classical Models for Schrödinger Equations with Periodic Potentials and Band-crossings Lihui Chai, Shi Jin, and Qin Li Tsinghua University We study the linear Schrödinger equation with a periodic potential in the semi-classical limit. When the so called Bloch band gap is small, the inter-band transition is significant but cannot be described by the classical transport model. We derive a semi-classical Liouville system to capture the inter-band transition phenomena. This system can be seen as a first order approximation to the Wigner equation. A semi-classical hybrid model and the corresponding domain decomposition method are presented to solve this type of system efficiently.

9 Object Extraction By Superpixel Grouping Hong Cheng Shanghai Jiao Tong University Detecting a number of unknown objects in cluttered scenes is an important and difficult problem in computer vision research. Several previous approaches have been proposed by adopting powerful perceptual grouping cues, such as the gestalt laws of proximity and continuity etc, to anage object detection. In this paper, we propose a fragments grouping method to extract object contour via superpixel, which has strong edge upport in the image. Moreover, a new cost function has been introduced to promote spatially coherent sets of superpixels with object boundary. We evaluate the proposed method against two leading contour closure pproaches in the literature on the BSDS500. The results demonstrate that the proposed object detection method performs economic,efficient in time and favorable against state-of-the-art methods in terms of accuracy. Outlet Boundary Conditions for Blood Flow in Arterial Trees Tao Du Shanghai Jiao Tong University One-dimensional models have been well used in modeling blood flow. The pulse profile of blood flow is largely affected by the boundary conditions at the inlets and outlets of a truncated arterial tree. I will introduce the existing results on outlet boundary condition. By comparing the impedance derived from the Winkessel model (WK) and the structured tree model, a systemic method is designed to extract parameters of the three-element Windkessel model from the impedances of a structured tree. In particular, a three-element Winderkessel model with complex capacitance (CWK) is developed to capture the phase lag better. This method can also be used to extract parameters from experimental measurements of the profiles of blood flow rates and blood pressures. At the same time, we study the relation among the impedances of structured trees with the same bifurcation ratio but different root radii. As we will show, the impedances have approximately the same profile under a precise normalization. Based on the normalization property of the impedances, we show that the difference between the structured-tree- and the Windkessel- boundary conditions is insignificant when the radii of the outlet vessels of the truncated arterial tree are small (e.g., smaller than 1.5mm for real capacitance and 2.0mm for complex capacitance in the three element Winderkessel boundary condition). By studying the wave propagation in the arterial tree analytically, we found that the error of pressure and impedance in the arterial tree can be controlled by the error of impedance at the downstream outlets. This result, together with the normalization property, implies that the CWK model gives us good approximation of the outlet boundary condition.

10 Neumann type problem for strongly degenerate parabolic-hyperbolic equations on a bounded rectangle Yuxi Hu Shanghai Jiao Tong University We study a Neumann type initial-boundary value problem for strongly degenerate parabolic-hyperbolic equations under the nonlinearity-diffusivity condition. We suggest a notion of entropy solutions for this problem and prove the uniqueness of entropy solutions. The existence of entropy solutions is also discussed. Coupling between internal waves and surface waves Shixiao Jiang Shanghai Jiao Tong University Large amplitude internal solitons, as well as the rips at the surface, have been observed experimentally and accurately measured in many oceanic places. However, the phenomenon for coupling between the internal waves and the surfaces waves is still cannot be completely understood. In our analysis, we derive a one dimensional weakly nonlinear model, which describe the evolution of large amplitude nonlinear motions of an interface between two internal fluid layers, and its coupling with the motion of an overlying free surface. For solitary waves part, the model succeeds to capture the broadening phenomenon as the amplitude increases. For ripples part, we will give an explanation for the surface waves behavior. Multi-parameter Regularization Image Restoration Model Based on Framelet and Total Variation Yuling Jiao Wuhan University Image restoration based on variational method is one of the hot topics in digital image processing. The well-known ROF model preserves edges but causes stair case in the flatten part. For the remedy of this defect, Several works proposing high order regularization method have been done in the literature, but which suffered numerical inefficiency due to high nonlinearity and non-smoothness of the Euler-Lagrange equation. Considering this we propose a new multi-parameter regularization model based on framelet and total variation to restore image from blurry and noisy or corrupted observations, which utilizes edge preserving property and the adaptivity of the framelet transform to the underlying regularity of image. We solved the proposed model by alternative direction minimization algorithm with an efficient and automatical multi-regularization parameter choosing strategy based on balancing principle which overcome the difficulty of toning the regularization parameter manually in numerical simulation. Complexity of the algorithm was analyzed and convergence of the algorithm was proved. Performance of this new model was shown by several artificial and benchmark test images.

11 超声速气体化学反应流的数值模拟李彬中国科学院 超声速气体化学反应流的研究对于高超声速推进系统的设计, 空间飞行器的气动热防护及布局优化等有重要的指导作用. 在超声速气体化学反应流的数值模拟中, 存在着复杂的化学反应机理, 刚性问题及由流动和化学反应相互作用引起的广泛的时空尺度等众多困难. 如何有效地模拟化学反应流, 是计算科学研究者致力解决的一个难题 本文研究高分辨率格式在计算超声速气体燃烧流中的模拟能力问题. 在结构网格有限体积法和 LU-SGS 格式求解多组分反应 Navier-Stokes 方程基础上, 分别应用基于 Roe 近似 Riemann 求解器的 MUSCL 格式和 WENO 格式, 基于局部 Lax-Friedrichs 分裂通量的 NND 格式和 WENO 格式, 以及 Harten-Yee 迎风 TVD 格式等五种高分辨率方法离散对流项, 数值模拟了 H2/Air 和 H2/O2 预混合气体以马赫数 M = 3 55 M = 4 48 和 M = 6 46 绕球头体外流和 M = 6 7 绕冲压加速器内流的激波诱导燃烧流场. 数值结果表明 : 对于马赫数 M = 3 55 激波与燃烧波分离的情况, 二阶 NND MUSCL 和 TVD 格式都可以很好地收敛到定常解, 而 WENO 格式很难收敛到残差很小的定常解. 当网格数较密时, 所有格式计算 M = 6 46 球头体和 M = 6 7 冲压加速器的燃烧流场都得不到收敛的定常解, 尽管它们在粗网格或在无化学反应项的密网格下能得到收敛的结果. 文中探讨了气体化学反应流数值模拟中不收敛的原因. 此外, 本文还将计算格式和程序推广到非定常燃烧和三维问题的数值模拟, 获得了初步的结果. 关键词 : 有限体积法 ; 高分辨率格式 ; 超声速燃烧 ; 网格收敛性. Strong Predictor-Corrector Euler-Maruyama Methods for Stochastic Differential Equations with Markovian Switching 报告人 : 李海波清华大学数学科学系导师 : 白峰杉, 联合导师 : 叶俊,George Yin In this paper numerical methods for solving stochastic differential equations with Markovian switching (SDEwMSs) are developed by pathwise approximation. The proposed family of strong predictor-corrector Euler-Maruyama methods is designed to overcome the propagation of errors during the simulation of an approximate path. This paper not only shows the strong convergence of the numerical solution to the exact solution but also reveals the order of the error under some conditions on the coefficient functions. A natural analogue of p-stability criterion is studied. Numerical examples are given to illustrate the computational efficiency of the new predictor-corrector Euler-Maruyama approximation. 关键字 :Strong Predictor-Corrector Euler-Maruyama methods, Markovian switching, Numerical solutions

12 A phenomenological integrate-and-fire model incorporating dendritic integration effects Songting Li Shanghai Jiao Tong University A single neuron may receive and integrate thousands of excitatory and inhibitory synaptic inputs from dendrites. In order to understand how the brain works, it is important to understand the rules that govern dendritic integration which remain to be fully elucidated. It has been discovered recently in experiments that the dendritic integration of excitatory glutamatergic inputs and inhibitory GABAergic inputs in hippocampus CA1 pyramidal neurons obeys a simple arithmetic rule as V_S V_E+V_I+ kv _E V_I, where V_S, V_E and V_I are the voltage values of the summed somatic potential, the excitatory postsynaptic potential (EPSP) and the inhibitory postsynaptic potential measured at the time when the EPSP reaches its peak value, respectively. Moreover, the shunting coefficient k in this rule only depends on the spatial location but not the amplitude of the excitatory or inhibitory input on the dendrite. In this talk, I will first address the theoretical issue of how much the above dendritic integration rule can be accounted for using subthreshold somatic membrane potential dynamics as characterized in the conductance-based integrate-and-fire (I&F) model. Then, we propose a simple I&F neuron model that incorporates the spatial dependence of the shunting coefficient k by a phenomenological parametrization. Our analytical and numerical results show that this dendritic-integration-rule-based I&F (DIF) model is able to capture many experimental observations and it also gives rise to predictions that can be used to verify the validity of the DIF model experimentally. In addition, the DIF model incorporates the dendritic integration effects dynamically and is applicable to more general situations than that in experiment in which the excitatory and inhibitory inputs are concurrent. Finally, we extend the DIF neuronal model to incorporate multiple inputs and obtain a similar dendritic integration rule that is consistent with the results obtained by using a realistic neuronal model with multiple compartments. This extended DIF model can be potentially used to study network dynamics that may involve effects arising from dendritic integration. Transition Path, Quasi-potential Energy Landscape and Stability of Genetic Switches Xiaoguang Li One of the fundamental cellular processes governed by genetic regulatory networks in cells is the transition among different states under the intrinsic and extrinsic noise. Based on a two-state genetic switching model with positive feedback, we develop a framework to understand the metastability in gene expressions. This framework is comprised of identifying the transition path, reconstructing the global quasi-potential energy landscape, analyzing the uphill and downhill transition paths, etc. It is successfully utilized to investigate the stability of genetic switching models and fluctuation properties in different regimes of gene expression with positive feedback. The quasi-potential energy landscape, which is the rationalized version of Waddington potential, provides a quantitative tool to understand the metastability in more general biological processes with intrinsic noise.

13 On breakdown of Anderson localization due to time-dependent disorder and interaction Bin Min As predicted by Anderson in 1958, static/time-independent disorder would arrest transport and lead to Anderson localization (AL) in the non-interacting quantum system. In this report, I will focus on two issues related to AL. The first is about how dynamic/time-dependent disorder affects AL in the continuum. By developing a first-principle unified approach, we provide a complete classification of behaviors when AL is destroyed by time-dependent disorder. Different regimes found by different approaches in previous work are unified in our theory. Moreover, some novel regimes unexplored before in the physics community are predicted. The second issue I will address is about how interaction affects AL in continuous models. By long-time numerical simulation, we demonstrate that short-range interactions would destroy AL in a subdiffusive way. A heuristic argument is developed to explain this subdiffusion phenomenon. Perturbation Bound and Condition Numbers for the Tensor and Matrix Sylvester Equation & Convergence of General Nonstationary Iterative Methods for Solving Singular Linear Equations Xinghua Shi Fudan University This talk is divided into two parts. First we investigate the backward error and perturbation bounds for the high order Sylvester tensor equation (STE) and the effective condition numbers for the generalized Sylvester equation. We also apply the small sample statistical method for the fast condition estimation of the generalized Sylvester equation, which requires O(m2n + mn2) flops, comparing with O(m3 + n3) flops for the generalized Schur method and generalized Hessenberg-Schur method for solving the generalized Sylvester equation. We extend classic perturbation results of the Sylvester equation to the high order case. Numerical examples illustrate the sharpness of our perturbation bounds. Second we analyze the convergence of the general nonstationary iterative methods for solving singular linear equations. We apply our results to solve singular Hermitian positive semidefinite linear systems and discuss relations of general stationary results and ours. We utilize the quotient convergence to prove the convergence of the two-stage iterative algorithms for solving the singular Hermitian positive semidefinite linear equations.

14 APathway-Based Mean-Field Model for E.coliChemotaxis Guangwei Si Center for Quantitative Biologyand Physics School,, Beijing To understand the behavior of living organisms from the underlying interaction of biomolecules is one of the main goals of modern biological science. The problem is challenging as there are many scales(time and space)spanning form molecules to behaviors. Here, we develop amulti-scale modeling approach to describe the population level behavior of Escherichia coli (E.coli) chemotaxis based on the underlying signaling pathway dynamics. We show that our model not only leads to insight into the molecular origins of important chemotaxis behavior, it also allows us to predict novel chemotaxis behavior in complex environments. The swimming behavior of an E.colicell follows an alternating run and tumble pattern with a bias toward higher concentration of attractant which results in the cell s chemotaxis behavior. At the population level such biased-random-walk motion has be described by a drift-diffusion equation, proposed by Keller and Segel(KS)more than 40 years ago. However, KS equation fails in fast-varying environments since it does not take into account the intracellular signaling dynamics. In the past, other groups also attempted to construct a population level model for chemotaxis, but were not successful due to lack of understanding of detail signaling pathway dynamics and the complexity of the problem. Our key innovation is to focus on relevant internal degree of freedom in the problem, namely the receptor methylation level, which has a long relaxation time and serves as the cell s memory of its chemical environment. In particular, we constructed a field of the internal variable, i.e. the methylation field, which is then coupled with the cell density field. We came up with a rigorous approach based on the master equation for the cell population in space, velocity, and the key internal variable, average receptor methylation level. This leads to the first pathway-based mean-field-theory(pb-mft)describing the population level bacterial chemotaxis behaviors. Subsequent analysis of the PB-MFT model shows excellent agreement with exiting experiments, it reveals a remarkable scaling relation between the macroscopic chemotaxis velocity and the microscopic adaptation rate, and it also predicts interesting new phenomena that can be tested by future experiments. Our work provides a good example of bridging molecular mechanisms with population-level behaviors. This integrative multi-scale approach may be used to study other biological systems, such as quorum sensing and biofilm formation. *This work is assisted by Tailin Wu,an undergraduate student frompku, advised byprof.qi Ouyang(PKU)and Prof. Yuhai Tu fromibmt.j. Watson research center. Part of this workhasbeenpublished: Pathway-Based Mean-Field Model for Escherichia colichemotaxis, Phys. Rev. Lett (2012).

15 AN EFFICIENT HIGHER ORDER HETEROGENEOUS MULTISCALE METHOD FOR ELLIPTIC PROBLEMS Fengyang Tang We propose an efficient heterogeneous multiscale finite element method based on a local least-squares reconstruction of the effective matrix using the data retrieved from the solution of cell problems posed on the vertices of the triangulation. The method achieves high order accuracy for high order macroscopic solver with essentially the same cost as the linear macroscopic solver. Optimal error bounds are proved for the elliptic problem. Numerical results demonstrate that the new method significantly reduces the cost without loss of accuracy. Transmembrane Permeation of Charged and Polar Groups Yukun Wang Shanghai Jiao Tong University Charged methyl guanidine (MguanH+) is used as a model to study the transmembrane permeation of charged or polar groups. A proper collective variable for general transmembrane permeation is used as the reaction coordinate to run umbrella sampling with molecular dynamics simulation. The energy barrier obtained through our simulation doesn't differ much from those reported in previous works. However, in contrast to previous studies, a continuous trajectory is obtained through our umbrella sampling. This suggests the validity of the reaction coordinate in describing a class of transmembrane dynamics. The continuous trajectory clearly demonstrates the transmembrane pathway: 1) Thermal fluctuation leads to the formation of a small-thickness region of the membrane near the charged group; 2) A water bridge connecting the two head-group layers is formed at this region; 3) The charged group moves to the opposite layer and the water bridge disappears. Our study also demonstrate that when the size of the simulation box is too small, the large deformation of the membrane cost more energy. As a result, the energy barrier is overestimated using such a small simulation box.

16 $H$-adaptive Mesh Method with Double Tolerance Adaptive Strategy for Hyperbolic Conservation Laws Sunan Wu The numerical method to solve hyperbolic conservation laws is often an explicit scheme. As the commonly used technique to improve the quality of numerical simulation, $h$-adaptive mesh method is adopted to resolve the structure of the solution. Since the computational cost in altering the mesh and solving the PDEs are comparable, too often the mesh adaption triggered may bring down the overall efficiency of solving hyperbolic conservation laws using $h$-adaptive mesh method. In this talk, we propose a so-called "double tolerance adaptive strategy" to optimize the overall numerical efficiency by reducing the number of mesh adaption, as well as the quality of the numerical solution is preserved. Some numerical results are shown to demonstrate the robustness and effectiveness of our $h$-adaptive algorithm using the double tolerance adaptive strategy. Calculation of some continuum variables in molecular dynamics Xiaojie Wu Wuhan University We derive the stress from Molecular Dynamics(MD) simulation and reduce the gaps between continuum variables and atomistic variables, such as displacement, momentum and stress. The conservation law and a generalized Irving-Kirkwood formula with time and space average are employed. Then, numerical simulations are taken to test the expressions. In addition, the generalized function is discussed for improving the consistency of the variables from MD and continuum variables in nonlinear situation. Robust Linear Optimization Under Matrix Completion Fangfang Xu Shanghai Jiao Tong University Linear programming models have been widely used in input-output analysis for analyzing the interdependence of industries in an economy and in environmental science. In these applications, some of the entries of the coefficient matrix cannot be measured physically or there exists sampling errors. On the other hand, the coefficient matrix can often be low-rank. We characterize the robust counterpart of these types of linear programming problems with uncertainty set described by the nuclear norm. Simulations for the input-output analysis show that the new paradigm can be helpful.

17 一维 Boltzmann 方程的矩封闭模型及其在真一维 Si 半导体器件上的模拟姚文琦北京大学 简单介绍矩方法的思想和其运用在一维 Boltzmann 方程上的主要结论, 在原有的理论基础上给出了新的碰撞项以及其矩模型, 设计合适的器件参数进行一维半导体 Si 器件的模拟, 通过考察各宏观量的特征, 验证了矩方法的正确性和收敛性, 最后得到了符合经典模型的电流电压特性曲线 相对论流体力学方程组的 RKDG 方法赵健北京大学相对论流体力学方程组在天体物理等领域有着诸多应用. 与非相对论情形的 Euler 方程相比, 其非线性更强, 有关其数值方法的研究是个重要的课题. 在众多数值方法中,RKDG 方法具有局部紧致性, 容易达到高精度, 且易实现并行的优点. 我们尝试把 RKDG 方法推广到了相对论流体力学方程组的求解中. 在问题的解有比较强的间断时,RKDG 的近似解可能出现振荡甚至非线性不稳定, 需要使用限制器. 我们选择使用 WENO limiter 做限制器. 通过若干数值试验表明了该方法的精度和稳定性. Some properties of the nonlinear matrix equation s * t X AX A Q 周端美 华东师范大学计算数学 2011 级博士 In this paper, the nonlinear matrix equation s * t X AX A Q is investigated, where Q is an Hermitian positive definite matrix. We consider three cases of this equation: the case s, t > 0, the case s 1, 0 t 1 and the case 0 s 1, t 1. In the case st, 0, we derive necessary conditions and sufficient condition for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. As compared to earlier works on these topics, the results we present here are more general, and the analysis here is much simpler. In the case s 1, 0 t 1 and the case 0 s 1, t 1, necessary conditions for the existence of a Hermitian positive definite solution is given, which is sharper than that of [J. Cai, G. Chen, On the Hermitian positive definite solutions of nonlinear matrix equation s * t X AX A Q. Appl. Math. Comput. 217 (2010) ].

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