2011年工作年报



一、机构设置和聘任工作
二、科学研究方向、科研项目和经费
三、科学研究成果
四、学术活动
五、学科建设和人才培养
六、工作环境及实验室建设
七、2011年论文摘要

一、机构设置和聘任工作
1.根据《上海高校计算科学E—研究院建设总体规划书》、《上海高校计算科学E—研究院建设发展规划书》和《上海高校计算科学E—研究院管理章程》等文件,制订和执行2011年度工作计划。
2.郭本瑜教授为首席研究员,并聘请下列专家为特聘研究员:
  • 程  晋 复旦大学教授
  • 黄建国 上海交通大学教授
  • 岳荣先 上海师范大学教授
  • 王元明 华东师范大学教授
  • 徐承龙 同济大学教授
  • 田红炯 上海师范大学教授
  • 王中庆 上海师范大学教授
  • 苏仰锋 复旦大学教授
  • 郭  谦 上海师范大学副教授
  • 上海师范大学彭丽副教授、徐东博士、郭玲博士和徐海燕博士为上海高校计算科学E—研究院青年培育人员。
3.由下列专家组成学术委员会:
  • 主任:石钟慈 中国科学院院士
  • 委员:林  群 中国科学院院士
  • 姜礼尚 同济大学教授
  • 郭本瑜 上海师范大学教授
  • 张伟江 上海交通大学教授
  • 吴宗敏 复旦大学教授
  • 王翼飞  上海大学教授
  • 香港城市大学王世全教授为研究院顾问。
4.田红炯教授兼任业务秘书,谢丽同志任行政秘书。
二、科学研究方向、科研项目和经费
1.根据E—研究院科学研究方向,制订并资助2011年度研究课题,承担国家和上海市其它科研项目,积极申请新的科研项目。
2.主要研究方向:
  • 科学与工程中的高性能算法
  • 数学物理中的反问题计算方法
  • 复杂结构和复杂物理现象的数学模型及算法
  • 随机模型和随机算法
  • 常微分方程的高效数值解法
  • 生物与材料科学中的数学模型及其算法
  • 大型和非线性代数问题的快速算法
3.本年度资助下列研究课题
  • 郭本瑜   无界区域问题和外部问题的高精度算法    
  • 程  晋  数学物理反问题及其应用  
  • 黄建国 组合弹性结构问题的有限元方法研究               
  • 岳荣先 线性回归模型的最优设计与稳健设计  
  • 田红炯 常微分系统的数值方法及其应用                        
  • 徐承龙  金融中的随机算法的加速    
  • 王元明 拟线性抛物和椭圆边值问题的高精度紧有限差分方法     
  • 王中庆 数学物理问题的高精度算法
  • 苏仰锋  非线性特征值问题的理论分析及计算              
  • 郭  谦 癌症治疗模型分析与模拟
4.特聘研究员承担了18项国家和上海市科研项目,本年度到达总经费143.09万元。
A.国家科研项目12个,本年度到达经费126.09万元
  • 郭本瑜   国家自然科学基金,谱方法若干问题研究。
  • 程  晋   国家自然科学基金,抛物型偏微分方程中的系数辨识与算法。
  • 程  晋   国家自然科学基金中德芬国际合作项目,Banach空间非线性正则 化理论及条件稳定性估计。
  • 程  晋   国家自然科学基金中美合作项目,偏微分方程反问题的理论与算法。  
  • 程  晋   科技部973计划项目子课题,适应于千万亿次科学计算的新型计 算模式。
  • 黄建国   国家自然科学基金中德芬国际合作项目,生物医学成中反问题 的稀疏性约束正则化方法。
  • 岳荣先   国家自然科学基金,随机系数回归模型的最优设计与稳健设计。
  • 田红炯   国家自然科学基金,常微分方程初值问题若干新算法及其应用。
  • 徐承龙   国家973计划项目子课题,信用风险分析和信用衍生产品定价。
  • 王中庆   国家自然科学基金,外部问题的高精度算法及其应用。
  • 苏仰锋   国家自然科学基金,非线性特征值问题的理论分析及计算。
  • 郭  谦   国家自然科学基金,癌症研究关键问题中的分歧与计算方法研究。
B.上海市及教育部科研项目6个,本年度到达经费17万元
  • 郭本瑜 教育部博士点基金,谱方法中的若干前沿问题研究。
  • 田红炯 教育部科学技术研究重点项目,中立型微分代数系统的计算方法及其数值分析。
  • 岳荣先 教育部博士点基金,随机系数回归模型的稳健试验设计。
  • 岳荣先 上海市教委科研创新重点项目,随机系数里复测量模型的试验设计若干问题研究。
  • 王中庆 上海市教委曙光计划,Neumamm边值问题和外部问题的谱方法以及时间方向的配置法。
  • 王元明 上海市自然科学基金,拟线性边值问题的高精度紧有限差分方法。
5. 最近申请并获准主持6个国家和上海市科研项目(2012年开始执行),总经费193万元。
  • 郭本瑜 国家自然科学基金,高维非直角区域上的谱和谱元方法。
  • 黄建国 国家自然科学基金,组合弹性结构问题的混合DG有限元方法与高效求解。
  • 徐承龙 国家自然科学基金,蒙特卡罗加速方法及其在金融衍生产品定价中的应用。
  • 田红炯 教育部博士点基金,广义中立型微分代数系统的数值方法及其稳定性。
  • 王中庆 国家自然科学基金,若干数学物理问题的谱方法和配置方法研究。
  • 王中庆 上海市教委科研创新重点项目,数学物理问题的高精度数值方法研究。
三、科学研究成果

本年度在非直角和无界区域问题的高精度数值方法、数学物理反问题的数值解法、组合弹性结构的数学模型和计算、常微分方程的高效数值方法、线性回归模型的最优设计与稳健设计,非线性特征值问题的理论及计算和癌症治疗模型分析与模拟等方面取得了一批研究成果,在国内外重要学术刊物上发表了27篇论文,其中某些结果是原创性的,有关结果引起国内、外同行的高度评价。


1.科学与工程中的高性能算法(郭本瑜,王中庆)
建立了一般凸四边形区域上的Legendre插值逼近理论,在此基础上提出了多角形上椭圆型和抛物型方程混合非齐次边值问题的区域分解拟谱方法。这些方法克服了对所求解问题区域形状的严格限制,从而从根本上发展了拟谱方法理论及其应用。
构造了园外区域上非线性强阻尼波动方程的Fourier-Laguerre谱与拟谱方法,并建立了有关算法的收敛性结果。提出了精确满足第二类边值条件的椭圆型方程无界区域问题的Laguerre谱方法,建立了相应的逼近理论,数值结果显示了该方法的有效性。
有关论文:
[1]Benyu Guo and Hongli Jia, Pseuspectral method for quadrilaterals, J. Comp. Appl. Math., 236(2011), 962-979.
[2]Benyu Guo, Chao Zhang and Tao Sun, Some developments in spectral methods, Studies Adv. Math., 51(2011), 561-574.
[3]Yuanyuan Ji, Hua Wu, Heping Ma and Benyu Guo, Multidomain pseudospectral methods for nonlinear convection-diffusion equations, Appl. Math. Mech., 32(2011), 1255–1268.
[4]Zhongqing Wang and Rong Zhang, Mixed spectral and pseudospectral methods for a nonlinear strongly damped wave equation in an exterior domain, Numer. Math.: Theo., Meth. Appl., 4(2011), 255-282.
[5]Zhongqing Wang, The Laguerre spectral method for solving Neumann boundary value problems, J. Comp. Appl. Math., 235(2011), 3229-3237.

2.数学物理反问题的理论和数值方法(程晋)
对具有重要应用背景的分数次扩散方程,对分数阶为1/2的情况提出了一种局部化的方法,证明了Carelman估计,为后续研究打开了一条新的道路。在我们工作的基础上,国内外已经有了一些重要的进展。
有关论文:
[1]Xiang Xu, Jin Cheng and Masahiro Yamamoto, Carleman estimate for a fractional diffusion equation with half order and application. Appl. Anal., 90(2011), 1355–1371.

3.组合弹性结构问题的有限元方法(黄建国)
通过引进两类插值转移算子,借鉴许进超-周爱辉提出的两水平有限元方法的思想,建立了求解任意维空间上四阶椭圆型方程的基于Morley-Wang-Xu非协调元离散的并行求解算法,并给出最优误差估计、提供数值模拟结果。
给出求解Kirchhoff板弯问题的一类自适应混合有限元方法。通过获得一个离散Helmholtz分解和一个离散inf-sun条件,导出了弯矩场的拟正交性和离散可靠性估计,进而获得该自适应混合元解的拟误差衰减性和最优复杂性。对于区域规则的组合弹性结构定常问题,建立了基于Adini-P3-P1离散的有限元方法。
将分子发光图像处理问题描述为基于辐射传输方程(RTE)的源函数反演模型,提出带正则化项的最小二乘法重构源项方法,建立模型理论分析和算法收敛性分析结果并对算法进行数值模拟。
有关论文:
[1]Jianguo Huang and Xuehai Huang, Local and parallel algorithms for fourth order problems discretized by the Morley-Wang-Xu element method, Numer. Math., 119(2011), 667-697.
[2]Jianguo Huang, Xuehai Huang and Yifeng Xu, Convergence of an adaptive mixed finite element method for Kirchhoff plate bending problems, SIAM J. Numer. Anal., 49(2011), 574-607.
[3]Ling Guo and Jianguo Huang, Adini-Q1-P3 FEM for general elastic multi-structure problems, Numer. Meth. Part. Diff. Equa., 27(2011), 1092-1112.
[4]Weimin Han, Joseph A. Eichholz, Jianguo Huang and Jia Lu, RTE-based bioluminescence tomography: a theoretical study, Inv. Prob. Sci. Engi., 19(2011), 435-459.

4.常微分方程的数值方法(郭本瑜、田红炯)
把谱方法推广到常微分方程初值问题,提出二阶常微分方程初值问题的Laguerre配置法。这些方法直接逼近方程的整体真解,并基于正交逼近快速收敛性保证数值解的谱精度,特别是克服了高阶Runge-Kutta方法的长时间计算不稳定性。这些结果既拓广了谱方法应用,又提供常微分方程数值解的新方法。
提出了常微分系统的连续混合块并行算法,并应用于时滞微分方程的数值计算。研究具有时滞的中立型微分代数系统的渐近稳定性理论,给出了线性多步法为数值渐近稳定的充要条件。
有关论文:
[1]Jianping Yan and Benyu Guo, A collocation method for initial value problems of second-order ODEs by using Laguerre functions, Numer. Math. Theo. Meth. Appl., 4(2011), 282-294.
[2]Jianping Yan and Benyu Guo, Laguerre-Gauss collocation method for initial value problems of second-order ODEs, Appl. Math. Mech., 32(2011), 1541-1564.
[3]Hongjiong Tian, Quanhong Yu and Cilai Jin, Continuous block implicit hybrid one-step methods for ordinary and delay differential equations, Appl. Numer. Math., 61 (2011), 1289–1300.
[4]Jiaoxun Kuang, Hongjiong Tian and Kaiting Shan, Asymptotic stability of neutral differential systems with many delays, Appl. Math. Comp., 217 (2011), 10087-10094.
[5]Hongjiong Tian, Quanhong Yu and Jiaoxun Kuang, Asymptotic stability of linear neutral delay differential-algebraic equations and linear multistep methods, SIAM J. Numer. Anal., 49(2011), 608-618.

5.线性回归模型的最优设计与稳健设计(岳荣先)
对于多响应试验设计问题,分别利用多响应线性回归模型和多响应贝叶斯非参数回归模型,提出基于响应变量预测协方差阵的最优设计准则,建立了最优设计的判别条件。
有关论文:
[1]Rongxian Yue, Hong Qin, Kashinath Chatterjee,Optimal U-type design for Bayesian nonparametric multiresponse prediction, J. Stat. Plan. Infer., 141(2011), 2472-2479.
[2]Xin Liu, Rongxian Yue, Fred J. Hickernell, Optimality criteria for multiresponse linear models based on predictive ellipsoids, Stat. Sini., 21(2011), 421-432.
[3]Fei Zhao, Rongxian Yue, Hanxin Wang, A Markov risk model with two classes of insurance business, Stoch. Anal. Appl., 29(2011), 1102-1110.
[4]Hathaikan Chootrakool, Jian Qing Shi and Rongxian Yue, Meta-analysis and sensitivity analysis for multi-arm trials with selection bias, Stat. Medi., 30(2011), 1183-1198.

6.非线性初(边)值问题的高精度有限差分方法(王元明)
构造了一类拟线性边值问题的四阶紧有限差分方法,引入了一类新的单调迭代技巧,建立了相应的单调迭代方法。运用 离散能量分析技巧,对一类紧局部一维LOD方法给出了无穷范数误差估计。
引入了一类新的加速迭代技巧,设计了一种新的求解有限差分反应扩散方程组的加速单调迭代算法,简化了已知算法的计算过程。建立了求解非线性反应扩散方程组的紧交替方向隐式ADI方法。
有关论文:
[1]Yuanming Wang, A modified accelerated monotone iterative method for finite difference reaction-diffusion-convection equations, J. Comp. Appl. Math., 235 (2011), 3646-3660.
[2]Yuanming Wang, On Numerov's method for a class of strongly nonlinear two-point boundary value problems, Appl. Numer. Math., 61(2011) , 38-52.
[3]Yuanming Wang, Error and extrapolation of a compact LOD method for parabolic differential equations, J. Comp. Appl. Math., 235 (2011), 1367-1382.
[4]Yuanming Wang, Wenjia Wu and Ravi P. Agarwal, A fourth-order compact finite difference method for nonlinear higher-order multi-point boundary value problems, Comp. Math. Appl., 61 (2011), 3226-3245.
[5]Yuanming Wang, Cuixia Liang and Ravi P. Agarwal, A block monotone iterative method for numerical solutions of nonlinear elliptic boundary value problems, Numer. Meth. Part. Diff. Equa., 27(2011), 680-701.
[6]Yuanming Wang, Global asymptotic stability of Lotka-Volterra competition reaction-diffusion systems with time delays, Math. Comp. Model., 53(2011), 337-346.
[7]Yuanming Wang and Jie Wang, A higher-order compact ADI method with monotone iterative procedure for systems of reaction-diffusion equations, Comp. Math. Appl., 62 (2011), 2434-2451.

7.金融中的随机算法的加速(徐承龙)
研究了金融中一类欧式超高维定价的蒙特卡罗重点取样加速计算问题,提出了最佳参数的新计算框架。大大节省了计算时间。
研究了利用市场中债权实际报价反演短期利率模型(特别是推广的CIR与Vasicek模型)中的参数问题,与其它文献相比,首次考虑了CIR模型中的参数约束问题, 并证明了稳定性等结果。研究了一类逆高斯早偿下的抵押贷款衍生证券的定价计算分层与校正问题。

8.非线性特征值问题的理论分析及计算(苏仰锋)
发展了大规模有理特征值问题的线性化求解方法.该方法已经被瑞士联邦理工大学(ETH)的Daniel Kressner 教授成功应用于频率依赖的吸收光子晶体的计算模拟,被台湾清华大学的Wen-Wei Lin教授等应用于流体-固体耦合系统的数值模拟.
有关论文:
[1]Yangfeng Su and Zhaojun Bai, Solving rational eigenvalue problems via linearization, SIAM J. Matrix Anal. Appl., 32(2011), 201–216.

9.癌症治疗模型分析与模拟(郭谦)
构造了可以有效求解Ito型随机微分方程正解并且具有长时间保结构性质的数值格式,相对于已有的格式,该方法在均方稳定性上也有一定改进。

四、学术活动
遵循研究院管理章程进行日常学术活动,举办或合办了一些国内或国际学术会议,提升上海高校计算科学E-研究院的影响力。
1.日常学术活动
  • 每月召开特聘研究员工作会议,交流科学研究工作并部署下一步研究工作。
  • 每月举办一次面向全市的学术报告会,由特聘研究员或院外专家介绍科学计算的新进展。
  • 邀请著名专家来校参加计算数学学术年活动并开设学术讲座。
  • 邀请10余名国内、外专家来研究院讲学或合作研究。
  • 研究院成员参加国际、国内学术会议10多人次,并作邀请报告或报告。多名研究员到国外或境外讲学或短期合作研究。
2.举办或合办国内、外学术会议
  • 2011年4月 与浙江理工大学和复旦大学联合举办数学物理反问题及其应用研讨会。60人。
  • 2011年6月 参与组织同济大学和交通大学联合举办的Problems and Challenges in Financial Engineering国际研讨会。45人。
  • 2011年6月 参与组织在上海交通大学举办的The Fifth Workshop of Chinese Young Computational Mathematicians。 50人。
  • 2011年9月 与复旦大学联合举办International Conference on Statistical Inverse 2011年11月 与同济大学数学系联合举办第七届上海市科学与工程计算方法学术研讨会。310人。
  • 2011年11月 与中国现场统计研究会试验设计分会联合举办2011全国试验设计学术研讨会。40人。
五、学科建设和人才培养
根据上海高校E—研究院的建设宗旨,加速培养上海市各高校计算数学专业的学术带头人和高水平专业人才,促进有关高校计算数学学科的建设。
1.上海师范大学数学系被批准为数学一级学科博士点。
2.研究院成员共指导了3名博士后(2名已出站),32名博士生(其中毕业7名)。
3.黄建国教授指导的博士生黄学海获得2011年上海市优秀博士论文奖。
4.郭本瑜教授指导的博士生贾红丽的论文"Spectral method on quadrilaterals"获2011年中国计算数学学会第五届优秀青年论文二等奖。
5.称晋教授当选为中国数学会副理事长,并担任我国重大研究项目〈高性能科学计算中的基础算法与可计算建模〉专家组成员。
6.郭本瑜教授被聘为国家重点基础研究发展计划项目"适应于千万亿次科学计算的新型计算模式"专家委员会委员,教育部物质计算科学重点实验室学术委员会委员,及广东省计算科学重点实验室学术委员会委员。 
六、工作环境及实验室建设
加强上海高校计算科学E—研究院和"科学计算"上海高校重点实验室建设。

1.拥有SGI工作站(32个CPU,内存为16GB,硬盘容量达到730GB)。
2.购置了20台多核高性能计算机及配套设施。
3.启动高性能科学计算实验室建设项目,订购了高性能科学计算机群系统(配备40个瘦结点、2个胖结点、1个GPU计算结点、6个IO结点、2个IO存储系统。)



七、2011年论文摘要

Meta-analysis and sensitivity analysis for multi-arm trials with selection bias
Hathaikan Chootrakool, Jian Qing Shi and Rongxian Yue
Stat. Medi., 30(2011), 1183-1198.
Abstract
Multi-arm trials meta-analysis is a methodology used in combining evidence based on a synthesis of different types of comparisons from all possible similar studies and to draw inferences about the effectiveness of multiple compared-treatments. Studies with statistically significant results are potentially more likely to be submitted and selected than studies with non-significant results; this leads to false-positive results. In meta-analysis, combining only the identified selected studies uncritically may lead to an incorrect, usually over-optimistic conclusion. This problem is known as selection bias. In this paper, we first define a random-effect metaanalysis model for multi-arm trials by allowing for heterogeneity among studies. This general model is based on a normal approximation for empirical log-odds ratio. We then address the problem of publication bias by using a sensitivity analysis and by defining a selection model to the available data of a meta-analysis. This method allows for different amounts of selection bias and helps to investigate how sensitive the main interest parameter is when compared with the estimates of the standard model. Throughout the paper, we use binary data from Antiplatelet therapy in maintaining vascular patency of patients to illustrate the methods.

Pseuspectral method for quadrilaterals
Benyu Guo and Hongli Jia
J. Comp. Appl. Math., 236(2011), 962-979.
Abstract
In this paper, we investigate the pseudospectral method on quadrilaterals. Some results on Legendre-Gauss-type interpolation are established, which play important roles in the pseudospectral method for partial differential equations defined on quadrilaterals. As examples of applications, we propose pseudospectral methods for two model problems and prove their spectral accuracy in space. Numerical results demonstrate the efficiency of the suggested algorithms. The approximation results and techniques developed in this paper are also applicable to other problems defined on quadrilaterals.

Some developments in spectral methods
Benyu Guo, Chao Zhang and Tao Sun
Studies Adv. Math., 51(2011), 561-574.
Abstract
In this paper, we review some new developments in spectral methods. We first consider the generalized Jacobi spectral method. Then, we present the Jacobi quasi-orthogonal approximation and its applications. Next, we consider the generalized Laguerre spectral method. We also present the Laguerre quasi-orthogonal approximation and its applications.

Adini-Q1-P3 FEM for general elastic multi-structure problems
Ling Guo and Jianguo Huang
Numer. Meth. Part. Diff. Equa., 27(2011), 1092-1112.
Abstract
An Adini-Q1-P3 finite element method is introduced to solve general elastic multi-structure problems, where displacements on bodies, longitudinal displacements on plates, longitudinal displacements and rotational angles on rods are discretized by conforming linear (bilinear or trilinear) elements, transverse displacements on plates and rods are discretized by Adini elements and Hermite elements of third order, respectively. The unique solvability and optimal error estimates in the energy norm are established for the discrete method, whose numerical performance is illustrated by some numerical examples.

RTE-based bioluminescence tomography: A theoretical study
Weimin Han, Joseph A. Eichholzb, Jianguo Huang and Jia Lu
Inv. Prob. Sci. Engi., 19(2011), 435-459.
Abstract
Molecular imaging has become a most rapidly developing area in biomedical imaging. Bioluminescence tomography (BLT) is an emerging and promising molecular imaging technology. Light propagation within biological media is accurately described by the radiative transfer equation (RTE). However, due to the difficulties in theoretical investigation and numerical simulations, so far, the study of BLT problem has been largely based on a diffusion approximation of the RTE. In this paper, we provide a rigorous theoretical foundation for the study of the RTE based BLT. After a discussion of the forward problem of the RTE and its numerical approximation, we establish a comprehensive mathematical framework for the RTE based BLT problem through Tikhonov regularization. We show the solution existence, uniqueness and continuous dependence on the data for the regularized formulation. We then introduce stable numerical methods for the BLT reconstruction and show convergence of the numerical solutions. Finally, we present simulation results from a numerical example to demonstrate that reasonable numerical results can be expected from solving the RTE based BLT problem via regularization.

Local and parallel algorithms for fourth order problems discretized
by the Morley-Wang-Xu element method
Jianguo Huang and Xuehai Huang
Numer. Math., 119(2011), 667-697.
Abstract
This paper systematically studies numerical solution of fourth order problems in any dimensions by use of the Morley-Wang-Xu (MWX) element discretization combined with two--grid methods. Since the coarse and fine finite element spaces are nonnested, two intergrid transfer operators are first constructed in any dimensions technically, based on which two classes of local and parallel algorithms are then devised for solving such problems. Following some ideas by Xu and Zhou, the intrinsic derivation of error analysis for nonconforming finite element methods of fourth order problems, and the error estimates for the intergrid transfer operators, we derive error estimates for the two classes of methods. Numerical results are performed to support the theory obtained and to compare the numerical performance of several local and parallel algorithms using different intergrid transfer operators.

Convergence of an adaptive mixed finite element method
for Kirchhoff plate bending problems
Jianguo Huang, Xuehai Huang and Yifeng Xu
SIAM J. Numer. Anal., 49(2011), 574-607.
Abstract
Some reliable and efficient a posteriori error estimators are produced for a mixed finite element method (the Hellan-Herrmann-Johnson (H-H-J) method) for Kirchhoff plate bending problems. Based on these results with k=0,1, where k denotes the polynomial order of the discrete moment-field space, an adaptive mixed finite element method (AMFEM) is set up and its convergence and complexity are studied thoroughly. The key points of the theoretical analysis include achieving a discrete Helmholtz decomposition and a discrete inf-sup condition, which serve as the main tool to deduce the quasi-orthogonality of the moment field and the discrete reliability of the estimator. It is shown that the AMFEM is a contraction for the sum of the moment-field error in an energy norm and the scaled error estimator between two consecutive adaptive loops. Moreover, an estimate for the AMFEM's complexity via the number of elements is developed.

Multidomain pseudospectral methods for nonlinear convection-diffusion equations
Yuanyuan Ji, Hua Wu, Heping Ma and Benyu Guo,
Appl. Math. Mech., 32(2011), 1255–1268.
Abstract
Multidomain pseudospectral approximations to nonlinear convection-diffusion equations are considered. The schemes are formulated with the Legendre-Galerkin method, but the nonlinear term is collocated at the Legendre/Chebyshev-Gauss-Lobatto points inside each subinterval. Appropriate base functions are introduced so that the matrix of the system is sparse, and the method can be implemented efficiently and in parallel. The stability and the optimal rate of convergence of the methods are proved. Numerical results are given for both the single domain and the multidomain methods to make a comparison.

Asymptotic stability of neutral differential systems with many delays
Jiaoxun Kuang, Hongjiong Tian and Kaiting Shan
Appl. Math. Comp., 217 (2011), 10087-10094
Abstract
We are concerned with delay-independent asymptotic stability of linear system of neutral differential equations. We first establish a sufficient and necessary condition for the system to be delay-independently asymptotically stable, and then give some equivalent stability conditions. This paper improves many recent results on the asymptotic stability in the literature. One example is given to show that the sufficient and necessary condition is easy to verify.

Optimality criteria for multiresponse linear models
based on predictive ellipsoids
Xin Liu, Rongxian Yue and Fred J. Hickernell
Statistica Sinica, 21(2011), 421-432.
Abstract
This paper proposes a new class of optimum design criteria for the linear regression model with r responses based on the volume of the predictive ellipsoid. This is referred to as -optimality. The -optimality criterion is invariant with respect to different parameterizations of the model, and reduces to -optimality as proposed by Dette and O'Brien (1999) in single response situations. An equivalence theorem for -optimality is provided and used to verify -optimality of designs, and this is illustrated by several examples.

Solving rational eigenvalue problems via linearization
Yangfeng Su and Zhaojun Bai
SIAM J. Matrix Anal. Appl., 32(1) (2011), 201-216.
Abstract
The rational eigenvalue problem is an emerging class of nonlinear eigenvalue problems arising from a variety of physical applications. In this paper, we propose a linearization-based method to solve the rational eigenvalue problem. The proposed method converts the rational eigenvalue problem into a well-studied linear eigenvalue problem, and meanwhile, exploits and preserves the structure and properties of the original rational eigenvalue problem. For example, the low-rank property leads to a trimmed linearization. We show that solving a class of rational eigenvalue problems is just as convenient and efficient as solving linear eigenvalue problems.

Asymptotic stability of linear neutral delay differential-algebraic equations and linear multistep methods
Hongjiong Tian, Quanhong Yu and Jiaoxun Kuang
SIAM J. Numer. Anal., 49(2011), 608-618.
Abstract
This paper is concerned with delay-independent asymptotic stability of linear neutral delay differential-algebraic equations and linear multistep methods. We first give some sufficient conditions for the delay-independent asymptotic stability of these equations. Then we study and derive a sufficient and necessary condition for the delay-independent asymptotic stability of numerical solutions obtained by linear multistep methods combined with Lagrange interpolation. Finally, one numerical example is performed to confirm our theoretical result.

Continuous block implicit hybrid one-step methods forordinary and delay differential equations
Hongjiong Tian, Quanhong Yu and Cilai Jin
Appl. Numer. Math., 61 (2011), 1289–1300.
Abstract
A class of high order continuous block implicit hybrid one-step methods has been proposed to solve numerically initial value problems for ordinary and delay differential equations. The convergence and -stability of the continuous block implicit hybrid methods for ordinary differential equations are studied. Alternative form of continuous extension is constructed such that the block implicit hybrid one-step methods can be used to solve delay differential equations and have same convergence order as for ordinary differential equations. Some numerical experiments are conducted to illustrate the efficiency of the continuous methods.

A modified accelerated monotone iterative method for finite difference reaction-diffusion-convection equations
Yuanming Wang
J. Comp. Appl. Math., 235 (2011), 3646-3660.
Abstract
This paper is concerned with monotone algorithms for the finite difference solutions of a class of nonlinear reaction-diffusion-convection equations with nonlinear boundary conditions. A modified accelerated monotone iterative method is presented to solve the finite difference systems for both the time-dependent problem and its corresponding steady-state problem. This method leads to a simple and yet efficient linear iterative algorithm. It yields two sequences of iterations that converge monotonically from above and below, respectively, to a unique solution of the system. The monotone property of the iterations gives concurrently improving upper and lower bounds for the solution. It is shown that the rate of convergence for the sum of the two sequences is quadratic. Under an additional requirement, quadratic convergence is attained for one of these two sequences. In contrast with the existing accelerated monotone iterative methods, our new method avoids computing local maxima in the construction of these sequences. An application using a model problem gives numerical results that illustrate the effectiveness of the proposed method.

On Numerov's method for a class of strongly nonlinear two-point boundary value problems
Yuanming Wang
Appl. Numer. Math., 61(2011) , 38-52.
Abstract
The purpose of this paper is to give a numerical treatment for a class of strongly nonlinear two-point boundary value problems. The problems are discretized by fourth-order Numerov's method, and a linear monotone iterative algorithm is presented to compute the solutions of the resulting discrete problems. All processes avoid constructing explicitly an inverse function as is often needed in the known treatments. Consequently, the full potential of Numerov's method for strongly nonlinear two-point boundary value problems is realized. Some applications and numerical results are given to demonstrate the high efficiency of the approach.

Error and extrapolation of a compact LOD method for parabolic differential equations
Yuanming Wang
J. Comp. Appl. Math., 235 (2011) , 1367-1382.
Abstract
This paper is concerned with a compact locally one-dimensional (LOD) finite difference method for solving two-dimensional nonhomogeneous parabolic differential equations. An explicit error estimate for the finite difference solution is given in the discrete infinity norm. It is shown that the method has the accuracy of the second-order in time and the fourth-order in space with respect to the discrete infinity norm. A Richardson extrapolation algorithm is developed to make the final computed solution fourth-order accurate in both time and space when the time step equals the spatial mesh size. Numerical results demonstrate the accuracy and the high efficiency of the extrapolation algorithm.

A fourth-order compact finite difference method for nonlinear higher-order multi-point boundary value problems
Yuanming Wang, Wenjia Wu and Ravi P. Agarwal
Comp. Math. Appl., 61 (2011), 3226-3245.
Abstract
A fourth-order compact finite difference method is proposed for a class of nonlinear 2nth-order multi-point boundary value problems. The multi-point boundary condition under consideration includes various commonly discussed boundary conditions, such as the three- or four-point boundary condition, (n+2)-point boundary condition and 2(n-m)-point boundary condition. The existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. The convergence and the fourth-order accuracy of the method are proved. An efficient monotone iterative algorithm is developed for solving the resulting nonlinear finite difference systems. Various sufficient conditions for the construction of upper and lower solutions are obtained. Some applications and numerical results are given to demonstrate the high efficiency and advantages of this new approach.

A block monotone iterative method for numerical solutions of nonlinear elliptic boundary value problems
Yuanming Wang, Cuixia Liang and Ravi P. Agarwal
Numer. Meth. Part. Diff. Equa., 27(2011), 680-701.
Abstract
The aim of this article is to develop a new block monotone iterative method for the numerical solutions of a nonlinear elliptic boundary value problem. The boundary value problem is discretized into a system of nonlinear algebraic equations, and a block monotone iterative method is established for the system using an upper solution or a lower solution as the initial iteration. The sequence of iterations can be computed in a parallel fashion and converge monotonically to a maximal solution or a minimal solution of the system. Three theoretical comparison results are given for the sequences from the proposed method and the block Jacobi monotone iterative method. The comparison results show that the sequence from the proposed method converges faster than the corresponding sequence given by the block Jacobi monotone iterative method. A simple and easily verified condition is obtained to guarantee a geometric convergence of the block monotone iterations. The numerical results demonstrate advantages of this new approach.

Global asymptotic stability of Lotka-Volterra competition reaction-diffusion systems with time delays
Yuanming Wang
Math. Comp. Model., 53(2011), 337-346.
Abstract
This paper is concerned with a time-delayed Lotka-Volterra competition reaction-diffusion system with homogeneous Neumann boundary conditions. Some explicit and easily verifiable conditions are obtained for the global asymptotic stability of all forms of nonnegative semitrivial constant steady-state solutions. These conditions involve only the competing rate constants and are independent of the diffusion-convection and time delays. The result of global asymptotic stability implies the nonexistence of positive steady-state solutions, and gives some extinction results of the competing species in the ecological sense. The instability of the trivial steady-state solution is also shown.

A higher-order compact ADI method with monotone iterative procedure for systems of reaction-diffusion equations
Yuanming Wang and Jie Wang
Comp. Math. Appl., 62 (2011), 2434-2451.
Abstract
This paper is concerned with an existing compact finite difference ADI method, published in the paper by Liao et al. (2002) [3], for solving systems of two-dimensional reaction-diffusion equations with nonlinear reaction terms. This method has an accuracy of fourth-order in space and second-order in time. The existence and uniqueness of its solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear reaction terms. The convergence of the finite difference solution to the continuous solution is proved. An efficient monotone iterative algorithm is presented for solving the resulting discrete system, and some techniques for the construction of upper and lower solutions are discussed. An application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method.

Mixed spectral and pseudospectral methods for a nonlinear strongly damped wave equation in an exterior domain
Zhongqing Wang and Rong Zhang
Numer. Math. Theor. Meth. Appl., 4(2011), 255-282.
Abstract
The aim of this paper is to develop the mixed spectral and pseudospectral methods for nonlinear problems outside a disc, using Fourier and generalized Laguerre functions. As an example, we consider a nonlinear strongly damped wave equation. The mixed spectral and pseudospectral schemes are proposed. The convergence is proved. Numerical results demonstrate the efficiency of this approach.

The Laguerre spectral method for solving Neumann
boundary value problems
Zhongqing Wang
J. Comp. Appl. Math., 235(2011), 3229-3237.
Abstract
In this paper, we propose a Laguerre spectral method for solving Neumann boundary value problems. This approach differs from the classical spectral method in that the homogeneous boundary condition is satisfied exactly. Moreover, a tridiagonal matrix is employed, instead of the full stiffness matrix encountered in the classical variational formulation of such problems. For analyzing the numerical errors, some basic results on Laguerre approximations are established. The convergence is proved. The numerical results demonstrate the efficiency of this approach.

Carleman estimate for a fractional diffusion equation with half order and application
Xiang Xu, Jin Cheng and Masahiro Yamamoto
Appl. Anal., 90(2011), 1355–1371.
Abstract
We consider a fractional diffusion equation in where the derivative in time t is of half order in the sense of Caputo and we establish a Carleman estimate. Since the derivatives of non-natural number orders do not satisfy the integration by parts, which is essential for establishing a Carleman estimate, we twice apply the Caputo derivative to convert the original fractional diffusion equation to a system with a usual partial differential operator: . Next we apply the Carleman estimate to prove the conditional stability in a Cauchy problem with data , .

A collocation method for initial value problems of second-order ODEs by using Laguerre functions
Jianping Yan and Benyu Guo
Numer. Math. Theo. Meth. Appl., 4(2011), 282-294.
Abstract
We propose a collocation method for solving initial value problems of secondorder ODEs by using modified Laguerre functions. This new process provides global numerical solutions. Numerical results demonstrate the efficiency of the proposed algorithm.

Laguerre-Gauss collocation method for initial value problems of second-order ODEs
Jianping Yan and Benyu Guo
Appl. Math. Mech., 32(2011), 1541-1564.
Abstract
This paper proposes a new collocation method for initial value problems of second order ODEs based on the Laguerre-Gauss interpolation. It provides the global numerical solutions and possesses the spectral accuracy. Numerical results demonstrate its high efficiency.

Optimal U-type design for Bayesian nonparametric multiresponse prediction
Rongxian Yue, Hong Qin and Kashinath Chatterjee
J. Stat. Plan. Infer., 141(2011), 2472-2479.
Abstract
This paper presents an extension of the work of Yue and Chatterjee (2010) about U-type designs for Bayesian nonparametric response prediction. We consider nonparametric Bayesian regression model with p responses. We use U-type designs with n runs, m factors and q levels for the nonparametric multiresponse prediction based on the asymptotic Bayesian criterion. A lower bound for the proposed criterion is established, and some optimal and nearly optimal designs for the illustrative models are given.

A Markov risk model with two classes of insurance business
Fei Zhao, Rongxian Yue and Hanxin Wang
Stoch. Anal. Appl., 29(2011), 1102-1110.
Abstract
A Markov risk model with two classes of insurance business is studied. In this model, the two classes of insurance business are independent. Each of the two independent claim number processes is the number of jumps of a Markov jump process from time 0 to t, whichever has not independent increments in general. An integral equation satisfied by the ruin probability is obtained and the bounds for the convergence rate of the ruin probability are given by using a generalized renewal technique





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发布日期: 2012/4/20
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