2009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、2009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(2008年-2010年)》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2009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2009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方法
  • 大型和非线性问题的快速算法
  • 生物与材料科学中的数学模型及其算法
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 数学物理反问题的理论及其数值计算        
  • 黄建国 组合弹性结构的有限元方法研究       
  • 岳荣先 高维积分的拟蒙特卡洛新算法及其误差分析         
  • 田红炯 常微分系统的数值方法及其应用               
  • 徐承龙 信用风险分析及产品定价中的随机与确定性算法    
  • 王元明 拟线性抛物和椭圆型方程的高精度紧致差分方法     
  • 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             
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268.1万元。
A.国家科研项目9个,本年度到达经费67.4万元。
  • 郭本瑜   国家自然科学基金,谱方法若干问题研究。
  • 黄建国   国家自然科学基金,组合弹性结构的自适应有限元方法研究。
  • 黄建国   国家973计划项目子课题,高性能科学计算研究。
  • 岳荣先   国家自然科学基金,高维积分的伪蒙特卡洛算法及其在巴拿赫空间上的误差分析。
  • 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。
  • 田红炯   国家自然科学基金,时滞泛函微分动力系统的计算方法。
  • 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。
  • 王中庆   国家973计划项目子课题,高性能科学计算研究。
  • 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。
B.上海市及教育部科研项目9个,本年度到达经费200.7万元。
  • 郭本瑜   国家教育部博士点基金,谱方法中的若干前沿问题研究。
  • 郭本瑜   上海市科技攻关项目,高性能计算方法研究。
  • 程  晋   国家外专局和教育部111引智计划,复旦大学现代应用数学创新引智基地。
  • 岳荣先   国家教育部博士点基金,伪Monte Carlo积分法及其在Banach空间上的误差估计。
  • 田红炯   上海市科委重点项目,泛函微分动力系统的计算方法及其应用。
  • 田红炯   上海市教委曙光计划,泛函微分动力系统的数值模拟及其应用。
  • 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。
  • 郭  谦   上海市自然科学基金,癌症研究关键问题中的分歧与计算方法研究。
  • 郭  谦   上海市教委科研创新项目,前列腺癌治疗模型中的数值模拟方法。
5.最近申请并获准主持3个国家和上海市科研项目(2010年开始执行),总经费34万元。
  • 程  晋   国家基金委中美合作项目,数学物理方程的反问题及其应用。
  • 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。
  • 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。
 
三、科学研究成果
本年度在奇异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、常微分方程的数值研究、随机化伪Monte-Carlo方法及其应用、非线性特征值问题的理论分析及计算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35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。此外,出版学术著作1部,获得上海市级教学成果奖一等奖(排名第二)。

1.科学与工程中的高性能算法(郭本瑜,王中庆)
  • 提出广义Jacobi正交逼近方法,建立了基本结果,并应用于高阶微分方程的数值解,数值结果显示了这些新方法的优越性。
  • 提出Legendre和Laguerre拟正交逼近和插值方法,建立了基本结果,并应用于无限区域上非典则微分方程和外部问题数值解。
  • 建立了长方形障碍物外部问题和圆外Navier-Stokes方程的混合谱方法,数值结果显示了这些新方法的优越性。
  • 提出一类拟合端点导数值的Jacobi插值方法,为计算Neumann问题的拟谱方法和配置法提供了一个有效的工具。
  • 给出了由Auteri, Parolini 和 Quartapelle提出的精确满足Neumann边界条件的Legendre 谱方法的误差估计。
有关论文:
[1]Benyu Guo and Tianjun Wang, Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an infinite channel, Math. Comp., 78(2009), 129-151.
[2]Tianjun Wang and Benyu Guo, Composite Laguerre-Legendre pseudospectral method for exterior problems, Comm. Comp. Phys., 5(2009), 350-375.
[3]Benyu Guo and Yujian Jiao, Mixed generalized Laguerre-Fourier spectral method for exterior problem of Navier-Stokes equations, Disc. Cont. Dyna. Syst. B, 11(2009), 315-345.
[4]Xiaoyong Zhang, Benyu Guo and Yujian Jiao, Spectral method for three- dimensional nonlinear Klein-Gordon equation by generalized Laguerre and spherical harmonic functions, Numer. Math. TMA, 2(2009), 43-64.
[5]Benyu Guo, Jie Shen and Lilian Wang, Generalized Jacobi polynomials/ functions and their applications, Appl. Numer. Math., 59 (2009), 1011-1028.
[6]Yujian Jiao and Benyu Guo, Mixed spectral method for exterior problem of Navier-Stokes equations by using generalized Laguerre functions, Appl. Math.  Mech., 30(2009), 561-574.
[7]Zhongqing Wang, Benyu Guo and Yanna Wu, Pseudospectral method using generalized Laguerre functions for singular problems on unbounded domains, Disc. Cont. Dyna. Syst. B, 11(2009), 1019-1038.
[8]Tianjun Wang and Zhongqing Wang, Error analysis of Legendre spectral method with essential imposition of Neumann boundary condition, Appl. Numer. Math., 59(2009), 2444-2451.
[9]Lilian Wang and Benyu Guo, Interpolation approximations based on Gauss-Lobatto- Legendre-Birkhoff quadrature,J. Appro. Theor., 161 (2009), 142-173.

2.数学物理反问题的理论和数值方法(程晋)
  • 首次讨论了一维分数次微分方程的反问题,证明分数次微分次数和未知扩散系数可以被边界上的测量数据唯一决定。从理论上回答了实际领域学者所关心的问题,为反问题的数值模拟打下了坚实的基础。
  • 讨论探测方法对多个散射体的逆散射问题的应用,给出了理论分析和数值模拟的方法。
  • 针对求解3维Herlmholtz方程中一类具有超奇性的边界积分方程提出了一种新的有效的数值方法,理论分析和计算结果均表明方法的有效性,得到了国内外同行的关注。该文章名列2009年IMA J. Appl. Math.全文下载第三名。
有关论文:
[1]Jin Cheng, Junichi Nakagawa, Masahiro Yamamoto and Tomohiro Yamazaki,  Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems, 25 (2009), 115002, 16 pp.
[2]Jin Cheng, Jijun Liu, Gen Nakamura and Shengzhang Wang, Recovery of multiple obstacles by probe method, Quart. Appl. Math., 67 (2009), 221-247.
[3]Ke Chen, Jin Cheng and Paul J. Harris, A new study of the Burton and Miller method for the solution of a 3D Helmholtz problem, IMA J. Appl. Math., 74 (2009), 163-177.
[4]Xiangtuan Xiong, Chuli Fu and Jin Cheng, Spectral regularization methods for solving a sideways parabolic equation within the framework of regularization theory, Math. Comp. Simu., 79 (2009), 1668-1678.
[5]Wenbin Chen, Jin Cheng, Junshan Lin and LiFeng Wang, A level set method to reconstruct the discontinuity of the conductivity in EIT, Sci. China Ser. A, 52 (2009), 29-44.

3.组合弹性结构问题的有限元方法(黄建国)
  • 获得了求解Kirchhoff板弯曲问题的一类混合元方法(H-H-J混合元法)的后验误差估计和相应的自适应有限元方法,得到了后验误差估计子的上、下界估计,建立了二阶对称张量场的一个离散Helmholtz分解及稳定性估计和一个关键的离散inf—sup估计,获得了方法拟最优收敛率和最优复杂性分析。
  • 以最低次H-H-J元和修正Morley元的等价性为桥梁,获得求解Kirchhoff板弯问题的修正Morley元自适应有限元方法,并建立拟最优收敛性理论。
  • 系统建立Kirchhoff板和组合弹性结构振动问题的有限元算法和误差估计理论并进行数值模拟。
有关论文:
[1]Chengze Chen, Jianguo Huang and Xuehai Huang, A P1-P3-NZT FEM for solving general elastic multi-structure problems, J. Comp. Anal. Appl., 11(2009), 728-747.
[2]Junjiang Lai and Jianguo Huang, A finite element method for vibration analysis of elastic plate-plate structures, Disc. Cont. Dynam. Syst. Ser. B, 11 (2009), 387-419.
[3]Junjiang Lai, Jianguo Huang and Chuanmiao Chen, Vibration analysis of plane elasticity problems by the C -continuous time stepping finite element method, Appl. Numer. Math., 59(2009), 905-919.

4.常微分方程的数值方法(郭本瑜、王中庆、田红炯)
  • 提出了一阶和二阶常微分动力系统的Legendre- Gauss -Lobatto新配置法,该方法计算稳定并保持谱精度。
  • 提出了求解常微分动力系统的连续块 -方法,建立了方法的收敛性和数值稳定性理论,并应用于时滞微分方程的求解。数值结果表面该方法具有良好的长时间稳定性。
  • 通过分析特征方程的零点分布,建立了多时滞(中立型)和有界时滞线性偏泛函微分动力系统的渐近稳定性理论,研究了线性多步法的渐近稳定性,并给出方法保持稳定的充要条件。
  • 基于时滞微分动力系统时滞相关渐近稳定的等价条件,统一给出了一般线性方法数值稳定的充分性条件,并应用于线性多步法的稳定性分析。
有关论文:
[1]Benyu Guo and Jianping Yan, Legendre-Gauss collocation methods for initial value problems of second ordinary differential equations, Appl. Numer. Math., 59(2009), 1386-1408.
[2]Benyu Guo and Zhongqing Wang, Legendre-Gauss collocation methods for ordinary differential equations, Adv. Comp. Math., 30(2009), 249-280.
[3]Hongjiong Tian, Kaiting Shan and Jiaoxun Kuang, Continuous block  -methods for ordinary and delay differential equations, SIAM J. Sci. Comp., 31 (2009) 4266-4280.
[4]Hongjiong Tian, Dongyue Zhang and Yeguo Sun, Asymptotic stability analysis of the linear  -method for linear parabolic differential equations with delay, J. Diff. Equa. Appl., 15(2009) 473-487.
[5]Hongjiong Tian and Ni Guo, Dissipativity of delay functional differential equations with bounded lag, J. Math. Anal. Appl., 355(2009) 778-782.
[6]Jiaoxun Kuang, Hongjiong Tian and Taketomo Mitsui, Asymptotic and numerical stability of systems of neutral differential equations with many delays, J. Comp. Appl. Math.,  223(2009) 614-625.
[7]Hongjiong Tian, Dongyue Zhang and Yeguo Sun, Delay-independent stability of Euler method for nonlinear one-dimensional diffusion equation with constant delay, Fron. Math. China, 4(2009) 169-179.
[8]Chengming Huang, Yangzi Hu and Hongjiong Tian, Stability analysis of multistep methods for delay differential equations, Acta Math. Appl. Sinica, 25(2009) 607-616.

5.随机伪蒙特卡罗方法的理论与应用(岳荣先)
  • 对于定义在s维单位方体上的响应函数,考虑非参数Bayes预测的最优设计问题,得到给定离散设计域上设计的优良性准则。
  • 对于U-型设计,导出了设计准则的下界,给出多种情况下的最优U-型设计。
有关论文:
[1]Rongxian Yue and Kashinath Chatterjee, Bayesian U-type design for nonparametric response surface prediction., Metrika, DOI 10.1007/s00184- 009-0249-0.
[2]Rong-Xian Yue, Model-Robust Design for multiresponse linear model with possible Bias, FSKD, Sixth Int. Conf. Fuzzy Systems Knowl. Disc., vol. 2, 572-576, 2009.

6.非线性初(边)值问题的高精度有限差分方法(王元明)
  • 对一类非线性反应扩散方程组得有限差分方程组建立了高阶单调迭代方法, 该方法具有单调收敛性, 而且收敛率达到p+2阶。
  • 对一类两维非线性反映扩散方程组建立了高阶紧有限差分方法, 该方法在时间和空间方向上都具有四阶精度, 并保持了原始问题的一些主要性质.
  • 对一类Michaelis-Menten 类反应扩散方程组的有限差分解给出了数值分析, 包括单调算法及长时间解的渐近收敛性。
  • 用上下解方法, 对一类Lotka-Volterra 互助模型的长时间解的渐近收敛性给出了定性刻划, 给出了一些简单易验证的渐近稳定性条件。
有关论文:
[1]Yuanming Wang and Xiaolin Lan, Higher-order monotone iterative methods for finite difference systems of nonlinear reaction–diffusion–convection equations, Appl. Numer. Math., 59 (2009), 2677–2693.
[2]Yuanming Wang and Hongbo Zhang, Higher-order compact finite difference method for systems of reaction_diffusion equations, J. Comp. Appl. Math., 233 (2009), 502–518
[3]Yuanming Wang, Numerical solutions of a Michaelis–Menten-type ratio- dependent predator–prey system with diffusion, Appl. Numer. Math., 59 (2009), 1075–1093
[4]Yuanming Wang, Asymptotic behavior of solutions for a Lotka-Volterra mutualism reaction diffusion system with time delays, Comp. Math. Appl., 58 (2009), 597–604.

7.金融衍生物的偏微分方程定价及计算(徐承龙)
  • 基于控制变量技巧,对随机波动率情形下的一类方差互换产品的定价问题,提出了一种有效的Monte Carlo加速计算方法,大大减小了模拟误差。该方法为其它多因子金融衍生产品模型下的定价提供一种有效框架。
  • 建立了方差互换金融衍生产品的定价模型,提出了一种利用控制变量进行方差减小的新计算框架,并通过数值模拟研究了该方法的计算效率与模型参数之间的关系。该方法可为其他方差互换衍生产品定价提供了一种新的有效思路。
有关论文:
[1]徐承龙主编,《现代数值计算》,ISBN:978-7-115-21399-0,人民邮电出版社,2009。
[2]马俊美,徐承龙,周晶,方差衍生产品定价与控制变量蒙特卡罗方法,同济大学学报, 37(2009),1700-1705。
[3]Junmei Ma and Chenglong Xu, Modeling of variance swap and improved control variate for Monte Carlo method, 2009 Int. Conf.n Business Intel. Fina. Eng., 735-740.

8.非线性特征值问题的理论分析及计算(苏仰锋)
  • 研究了超大规模集成电路中非线性器件互连分析中的模型降阶。算法获得国内唯一集成电路设计分析软件厂家—华大九天EDA软件公司的认可。经该公司测试论证,算法在速度、精度上均全面超过当今世界上最大的EDA软件开发商Synopsis的软件HSPICE2008中的对应软件。
有关论文:
[1]F. Yang, X. Zeng, Y. Su, and W. Cai. NHAR: A non-homogeneous Arnoldi method for fast simulation of RCL circuits with large number of terminals, Int. J. Circuit Theory Appl., 2009. DOI: 10.1002/cta.

9.生物数学模型及其算法(郭谦)
  • 建立描述具有突变、竞争性质的两类肿瘤细胞的前列腺癌间歇治疗模型,给出间最优抑制剂量与实施时间的参考数据。
有关论文:
[1]Youshan Tao, Qian Guo and Kazuyuki Aihara, A model at the macroscopic scale of prostate tumor growth under intermittent androgen suppression, Math. Models Meth. Appl. Sci., 19(2009), 2177-2201.
 
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。
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开设学术讲座。
  • 邀请10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
  • 2009年1月,   与复旦大学联合举办“International Conference on Contemporary Applied  Mathematics”国际会议。
  • 2009年6月,   与香港城市大学联合举办“International Conference on Applied Analysis and Scientific Computation”国际会议。
  • 2009年8月,   与复旦大学联合举办全国“数学物理反问题及其应用”暑期班。
  • 2009年11月,  与复旦大学联合举办上海市第五届“科学与工程中的计算方法”研讨会。
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.计算数学学科被评为上海市第三期重点建设学科。
2.研究院成员共指导了3名博士后,18名博士生(其中毕业2名)。指导国内访问学者2名。
3.郭本瑜教授指导的1名博士生获全国优秀博士论文提名奖和上海市优秀博士论文奖,程晋教授指导的1名硕士生获得上海市优秀硕士论文。
4.徐承龙教授获宝钢优秀教师奖和上海市教学成果一等奖(排名第二,2009年3月)。

六、工作环境及实验室建设
在上海市计算数学重点建设学科和上海高校计算科学E—研究院的基础上,上海市教委在上海师范大学成立了“科学计算”上海高校重点实验室。目前,
1.拥有SGI工作站(32个CPU,内存为16GB,硬盘容量达到730GB)和2台多核高性能计算机及配套设施。
2.新订购了Elsevier ScienceDirect 数学学科电子期刊数据库。
3.新增培育人员办公室2间(面积约50平方米),博士生办公室1间(面积约50平方米)。
七、2009年论文摘要
A P1-P3-NZT FEM for solving general elastic multi-structure problems
Chengze Chen, Jianguo Huang and Xuehai Huang
Journal of Computational Analysis and Applications, 11(2009), 728-747.
Abstract
A new finite element method is introduced for solving general elastic multi-structure problems, where displacements on bodies, longitudinal displacements on plates, longitu-dinal displacements and rotational angles on rods are discretized by conforming linear elements, transverse displacements on rods and plates are discretized respectively by Hermite elements of third order and Zienkiewicz-type elements due to Wang, Shi, and Xu, and the discrete generalized displacement fields in individual elastic members are coupled together by some feasible interface conditions. The optimal error estimate in the energy norm is established for the method, which is also validated by some numerical examples.

Recovery of multiple obstacles by probe method
Jin Cheng, Jijun Liu, Gen Nakamura and Shengzhang Wang
Quart. Appl. Math. 67 (2009), 221-247.
Abstract
We consider an inverse scattering problem for multiple obstacles  with different types of boundary for . By constructing an indicator function from the far-field pattern of the scattered wave, we can firstly reconstruct the shape of all obstacles, then identify the type of boundary for each obstacle, as well as the boundary impedance in the case that obstacles have the Robin-type boundary condition. The novelty of our probe method compared with the existing probe method is that we succeeded in identifying the type of boundary condition for multiple obstacles by analyzing the behavior of both the imaginary part and the real part of the indicator function. The numerical realizations are given to show the performance of this inversion method.

Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation
Jin Cheng, Junichi Nakagawa, Masahiro Yamamoto and Tomohiro Yamazaki
Inverse Problems 25 (2009), 115002, 16 pp.
Abstract
We consider a one-dimensional fractional diffusion equation:   , where  and  denotes the Caputo derivative in time of order α. We attach the homogeneous Neumann boundary condition at  and the initial value given by the Dirac delta function. We prove that α and  , are uniquely determined by data  . The uniqueness result is a theoretical background in experimentally determining the order α of many anomalous diffusion phenomena which are important, for example, in environmental engineering. The proof is based on the eigenfunction expansion of the weak solution to the initial value/boundary value problem and the Gel'fand–Levitan theory.

A new study of the Burton and Miller method for the solution of a 3D Helmholtz problem
Ke Chen, Jin Cheng and Paul J. Harris
IMA J. Appl. Math. 74 (2009), 163--177.
Abstract
The exterior Helmholtz problem can be efficiently solved by reformulating the differential equation as an integral equation over the surface of the radiating and/or scattering object. One popular approach for overcoming either non-unique or non-existent problems which occur at certain values of the wave number is the so-called Burton and Miller method which modifies the usual integral equation into one which can be shown to have a unique solution for all real and positive wave numbers. This formulation contains an integral operator with a hypersingular kernel function and for many years, a commonly used method for overcoming this hypersingularity problem has been the collocation method with piecewise-constant polynomials. Viable high-order methods only exist for the more expensive Galerkin method. This paper proposes a new reformulation of the Burton–Miller approach and enables the more practical collocation method to be applied with any high-order piecewise polynomials. This work is expected to lead to much progress in subsequent development of fast solvers. Numerical experiments on 3D domains are included to support the proposed high-order collocation method.

A level set method to reconstruct the discontinuity of the conductivity in EIT.
Wenbin Chen, Jin Cheng, Junshan Lin and Lifeng Wang
Sci. China Ser. A 52 (2009), 29--44.
Abstract
In this paper, one level set method is applied to finding the interface of discontinuity of the conductivity in EIT(electrical impedance tomography) problem. By choosing one suitable velocity function, a level set reconstruction algorithm is proposed. The theoretical results for EIT problem and regularization are given. Finally the numerical examples demonstrate that the reconstruction algorithm is efficient and stable.

Mixed generalized Laguerre-Fourier spectral method for exterior problem of Navier-Stokes equations
Benyu Guo and Yujian Jiao
Disc. Cont. Dyna. Syst. B, 11 (2009), 315-345.
Abstract
In this paper, we investigate the mixed generalized Laguerre-Fourier spectral method and its applications to exterior problems of partial differential equations of fourth order. Some basic results on the mixed generalized Laguerre-Fourier orthogonal approximation are established, which play important roles in designing and analyzing various spectral methods for exterior problems of fourth order. As an important application, a mixed spectral scheme is proposed for the stream function form of the Navier-Stokes equations outside a disc. The numerical solution fulfills the compressibility automatically and keeps the same conservation property as the exact solution. The stability and convergence of proposed scheme are proved. Numerical results demonstrate its spectral accuracy in space, and coincide with the analysis very well.

Generalized Jacobi polynomials/functions and their applications
Benyu Guo, Jie Shen and Lilian Wang
Appl. Numer. Math., 59 (2009), 1011-1028.
Abstract
We introduce a family of generalized Jacobi polynomials/functions with indexes α, β ∈ R which are mutually orthogonal with respect to the corresponding Jacobi weights and which inherit selected important properties of the classical Jacobi polynomials. We establish their basic approximation properties in suitably weighted Sobolev spaces. As an example of their applications, we show that the generalized Jacobi polynomials/functions, with indexes corresponding to the number of homogeneous boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the use of generalized Jacobi polynomials/functions leads to much simplified analysis, more precise error estimates and well conditioned algorithms.

Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an infinite channel
Benyu Guo and Tianjun Wang
Math. Comp., 78 (2009), 129-151.
Abstract
In this paper, we propose a composite generalized Laguerre- Legendre spectral method for partial differential equations on two-dimensional unbounded domains, which are not of standard types. Some approximation results are established, which are the mixed generalized Laguerre-Legendre approximations coupled with domain decomposition. These results play an important role in the related spectral methods. As an important application, the composite spectral scheme with domain decomposition is provided for the Fokker-Planck equation in an infinite channel. The convergence of the proposed scheme is proved. An efficient algorithm is described. Numerical results show the spectral accuracy in the space of this approach and coincide well with theoretical analysis. The approximation results and techniques developed in this paper are applicable to many other problems on unbounded domains. In particular, some quasi-orthogonal approximations are very appropriate for solving PDEs, which behave like parabolic equations in some directions, and behave like hyperbolic equations in other directions. They are also useful for various spectral methods with domain decompositions, and numerical simulations of exterior problems.

Legendre-Gauss collocation methods for ordinary differential equations
Benyu Guo and Zhongqing Wang
Adv. Comp. Math., 30(2009), 249-280.
Abstract
In this paper, we propose two efficient numerical integration processes for initial value problems of ordinary differential equations. The first algorithm is the Legendre-Gauss collocation method, which is easy to be implemented and possesses the spectral accuracy. The second algorithm is a mixture of the collocation method coupled with domain decomposition, which can be regarded as a specific implicit Legendre-Gauss Runge-Kutta method, with the global convergence and the spectral accuracy. Numerical results demonstrate the spectral accuracy of these approaches and coincide well with theoretical analysis.

Legendre-Gauss collocation methods for initial value problems of second ordinary differential equations
Benyu Guo and Jianping Yan
Appl. Numer. Math., 59 (2009), 1386-1408.
Abstract
In this paper, we develop a new collocation method for solving initial value problems of second order ODEs. We approximate the solutions by the Legendre–Gauss interpolation directly. The numerical solutions possess the spectral accuracy. We also propose a multi-step version of Legendre–Gauss collocation method, which works well for long-time calculations. Numerical results demonstrate the effectiveness of proposed methods and coincide well with analysis.

Stability analysis of multistep methods for delay differential equations
Chengming Huang, Yangzi Hu and Hongjiong Tian
 Acta Math. Appl. Sini., 25(2009), 607-616.
Abstract
This paper deals with the delay-dependent stability of numerical methods for delay differential equations. First, a stability criterion of Runge-Kutta methods is extended to the case of general linear methods. Then, linear multistep methods are considered and a class of τ(0)-stable methods are found. Later, some examples of τ(0)-stable multistep multistage methods are given. Finally, numerical experiments are presented to confirm the theoretical results.

Mixed spectral method for exterior problem of Navier-Stokes equations by using generalized Laguerre functions
Yujian Jiao and Benyu Guo
Appl. Math. and Mech., 30 (2009), 561-574.
Abstract
In this paper, we investigate the mixed spectral method using generalized Laguerre functions for exterior problems of fourth order partial differential equations. A mixed spectral scheme is provided for the stream function form of the Navier-Stokes equations outside a disc. Numerical results demonstrate the spectral accuracy in space.

Asymptotic and numerical stability of systems of neutral differential equations with many delays
Jiaoxun Kuang, Hongjiong Tian and Taketomo Mitsui
J. Comp. Appl. Math.,  223(2009), 614-625.
Abstract
We are concerned with the asymptotic stability of a system of linear neutral differential equations with many delays in the form
 
where  ( ) are constant complex matrices,  ( ) are constant delays and  is an unknown vector-valued function for t > 0.We first establish a new result for the distribution of the roots of its characteristic function, next we obtain a sufficient condition for its asymptotic stability and then we investigate the corresponding numerical stability of linear multistep methods applied to such systems. One numerical example is given to testify our numerical analysis.

A finite element method for vibration analysis of elastic plate-plate structures
Junjiang Lai and Jianguo Huang
Disc. Cont. Dynam. Syst., Ser. B, 11(2009), 387-419.
Abstract
The semi and fully discrete finite element methods are proposed for investigating vibration analysis of elastic plate-plate structures. In the space directions, the longitudinal displacements on plates are discretized by conforming linear elements, and the corresponding transverse displacements are discretized by the Morley element, leading to a semi-discrete finite element method for the problem under consideration. Applying the second order central difference to discretize the time derivative, a fully discrete scheme is obtained, and two approaches for choosing the initial functions are also introduced. The error analysis in the energy norm for the semi and fully discrete methods are established, and some numerical examples are included to validate the theoretical analysis.

Vibration analysis of plane elasticity problems by the C0-continuous time stepping finite element method
Junjiang Lai, Jianguo Huang and Chuanmiao Chen
Appl. Numer. Math., 59(2009), 905-919
Abstract
This paper proposes a -continuous time stepping finite element method to solve vibration problems of plane elasticity. In the time direction, unlike the existing methods, this method does not use the discontinuous Galerkin (DG) method to simultaneously discretize the displacement and velocity fields, but only use the  -continuous Galerkin method to discretize the displacement field instead. This greatly reduces the size of the linear system to be solved at each time step. The finite element in the space directions is taken as the usual  -conforming element. It is proved that the error of the method in the energy norm is  , where h and k denote the mesh sizes of the subdivisions in the space and time directions, respectively. Some numerical tests are included to show the computational performance of the method.

Modeling of variance swap and improved control variate for Monte Carlo method
Junmei Ma and Chenglong Xu
2009 Int. Conf. Business Intel. Fina. Engi., 2009, 735-740
摘要
基于控制变量技巧,对随机波动率情形下的一类方差互换产品的定价问题,提出了一种有效的Monte Carlo加速计算方法。通过进一步的理论分析和高效率的控制变量的选取,大大减小了模拟误差。文中的计算方法也可以为其它多因子金融衍生产品模型下的定价提供一种有效框架。

方差衍生产品定价与控制变量蒙特卡罗方法
马俊美,徐承龙,周晶
同济大学学报, 37(2009) 1700-1705
摘要
建立了方差互换金融衍生产品的定价模型,提出了一种利用控制变量进行方差减小的新的计算框架:对随机波动率下的证券价格用确定性波动率下的证券价格作控制变量,而确定性波动率函数的选取的依据是原来标的资产价格模型的一阶矩、二阶矩与近似模型的矩近似相等。并通过数值模拟研究了该方法的计算效率与模型参数之间的关系。该计算方法可为其他方差互换衍生产品, 如Corridor 方差互换、Gamma 方差互换和Conditional 方差互换等产品以及其他多因子模型假设下的衍生产品定价提供了一种新的有效思路。

A model at the macroscopic scale of prostate tumor growth under intermittent androgen suppression
Youshan Tao, Qian Guo and Kazuyuki Aihara
Math. Models Meth. Appl. Sci., 19 (2009), 2177-2201.
Abstract
The relapse of tumor is a crucial problem in hormonal therapy of prostate cancer. The so-called androgen-independent cells are considered to be responsible for such a recurrence. These cells are not sensitive to androgen suppression but rather apt to proliferate even in an androgen-poor environment. Bruchovsky et al. in their experimental and clinical studies suggested that intermittent androgen suppression may delay or prevent the relapse when compared with continuous androgen suppression. This paper proposes a model at the macroscopic scale of prostate tumor growth under intermittent androgen suppression. Qualitative analysis shows that the tumor relapse cannot be avoided under continuous androgen suppression for typical parameter values. Numerical simulation supports the above-mentioned experimental and clinical suggestion, and implies an optimal medication scheme of intermittent androgen suppression therapy.

Dissipativity of delay functional differential equations with bounded lag
Hongjiong Tian and Ni Guo
J. Math. Anal. Appl., 355(2009), 778-782.
Abstract
Delay functional differential equations are essentially different from ordinary differential equations because their phase space is infinite dimensional. We first establish a sufficient condition for delay functional differential equations with bounded lag to be dissipative. Then we construct a one-leg θ-method to solve such dissipative equations and prove that it is dissipative if θ = 1. One numerical example is given to confirm our theoretical result.

Continuous block  -methods for ordinary and delay differential equations
Hongjiong Tian, Kaiting Shan and Jiaoxun Kuang
SIAM J. Sci. Comp., 31 (2009) 4266-4280.
Abstract
Continuous numerical methods have many applications in the numerical solution of discontinuous ordinary differential equations (ODEs), delay differential equations, neutral differential equations, integro-differential equations, etc. This paper deals with a continuous extension for the discrete approximate solution of ODEs generated by a class of block  -methods. Existence and uniqueness for the continuous extension are discussed. Convergence and absolute stability of the continuous block  -methods for ODEs are studied. As an application, we adopt the continuous block  -methods to solve delay differential equations and prove that the continuous block  -methods are  -stable if and only if they are  -stable for ODEs. Several numerical experiments are given to illustrate the performance of the continuous block  -methods.

Asymptotic stability analysis of the linear  -method for linear parabolic differential equations with delay
Hongjiong Tian, Dongyue Zhang and Yeguo Sun
J. Diff. Equa. Appl., 15(2009), 473-487.
Abstract
This paper is concerned with asymptotic stability property of linear  -method for partial functional differential equations with delay. A sufficient condition for the underlying partial functional differential equations to be asymptotically stable is presented. We investigate numerical stability of the linear  -method by using the spectral radius condition. When  [0, 1/2), a sufficient and necessary condition for the linear u-method to be asymptotically stable is established. When  [1/2, 1], the linear u-method is unconditionally asymptotically stable. The behaviour of the norm of the iteration matrix when the linear u-method is asymptotically stable is studied by using Kreiss resolvent condition. Numerical experiments have been implemented to confirm the derived stability properties of the numerical method.

Delay-independent stability of Euler method for nonlinear one-dimensional diffusion equation with constant delay
Hongjiong Tian, Dongyue Zhang and Yeguo Sun
Fron. Math. China, 4(2009), 169-179.
Abstract
This paper is concerned with delay-independent asymptotic stability of a numerical process that arises after discretization of a non-linear one-dimensional diffusion equation with a constant delay by the Euler method. Explicit sufficient and necessary conditions for the Euler method to be asymptotically stable for all delays are derived. An additional restriction on spatial stepsize is required to preserve the asymptotic stability due to the presence of the delay. A numerical experiment is implemented to confirm the results.

Interpolation approximations based on Gauss-Lobatto-Legendre-Birkhoff quadrature
Lilian Wang and Benyu Guo
J. Appro. Theor., 161 (2009), 142-173.
Abstract
We derive in this paper the asymptotic estimates of the nodes and weights of the Gauss-Lobatto-Legendre-Birkhoff (GLLB) quadrature formula, and obtain optimal error estimates for the associated GLLB interpolation in Jacobi weighted Sobolev spaces. We also present a user-oriented implementation of the pseudospectral methods based on the GLLB quadrature nodes for Neumann problems. This approach allows an exact imposition of Neumann boundary conditions, and is as efficient as the pseudospectral methods based on Gauss-Lobatto quadrature for PDEs with Dirichlet boundary conditions.

Composite Laguerre-Legendre pseudospectral method forexterior problems
Tianjun Wang and Benyu Guo
Comm. in Comp. Phys., 5 (2009), 350-375.
Abstract
In this paper, we propose a composite Laguerre-Legendre pseudospectral method for exterior problems with a square obstacle. Some results on the composite Laguerre-Legendre interpolation, which is a set of piecewise mixed interpolations coupled with domain decomposition, are established. As examples of applications, the composite pseudospectral schemes are provided for two model problems. The convergence of proposed schemes are proved. Efficient algorithms are implemented. Numerical results demonstrate the spectral accuracy in space of this new approach.

Error analysis of Legendre spectral method with essential imposition of Neumann boundary condition
Tianjun Wang and Zhongqing Wang
Appl. Numer. Math., 59(2009), 2444-2451.
Abstract
In this paper, we present error estimates of Legendre spectral method with essential imposition of Neumann boundary condition. The algorithm was firstly proposed by Auteri,Parolini and Quartapelle. This method differs from the classical spectral methods for Neumann boundary value problems. The homogeneous boundary condition is satisfied exactly. Moreover, a double diagonalization process is employed, instead of the full stiffness matrices encountered in the classical variational formulation of the problem with a weak natural imposition of the derivative boundary condition. We also consider nonhomogeneous Neumann data by means of a lifting. In particular, the lifting in this paper is expressed explicitly and is different from that by Auteri, Parolini and Quartapelle. For analyzing the numerical errors, some basic results on Legendre quasi-orthogonal approximations are established. The convergence of proposed schemes is proved.

Numerical solutions of a Michaelis-Menten-Type ratio-dependent predator-prey system with diffusion
Yuanming Wang
Appl. Numer. Math., 59 (2009), 1075–1093.
Abstract
This paper is concerned with finite difference solutions of a Michaelis-Menten-type ratio-dependent Predator-Prey system with diffusion. The system is discretized by the finite difference method, and the investigation is devoted to the finite difference system for the time-dependent solution and its asymptotic behavior in relation to the various steady-state solutions. Three monotone iterative schemes for the computation of the time-dependent solution are presented, and the sequences of iterations are shown to converge monotonically to a unique positive solution. A simple and easily verifiable condition on the rate constants is obtained, which ensures that for every nontrivial nonnegative initial function the corresponding time-dependent solution converges either to a unique positive steady-state solution or to a semitrivial steady-state solution. The above results lead to computational algorithms for the solution as well as the global asymptotic stability of the system. Some numerical results are given. All the conclusions are directly applicable to the finite difference solution of the corresponding ordinary differential system.

Asymptotic behavior of solutions for a Lotka-Volterra mutualism reaction-diffusion system with time delays
Yuanming Wang
Comp. Math. Appl., 58 (2009), 597–604.
Abstract
This paper is to investigate the asymptotic behavior of solutions for a time-delayed Lotka-Volterra N-species mutualism reaction-diffusion system with homogeneous Neumann boundary condition. It is shown, under a simple condition on the reaction rates, that the system has a unique bounded time-dependent solution and a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function the corresponding time-dependent solution converges to the constant positive steady-state solution as time tends to infinity. This convergence result implies that the trivial steady-state solution and all forms of semitrivial steady-state solutions are unstable, and moreover, the system has no nonconstant positive steady-state solution. A condition ensuring the convergence of the time-dependent solution to one of nonnegative semitrivial steady-state solutions is also given.

Higher-order monotone iterative methods for finite difference systems of nonlinear reaction-diffusion-convection equations
Yuanming Wang and Xiaolin Lan
Appl. Numer. Math., 59 (2009), 2677–2693.
Abstract
This paper is concerned with the computational algorithms for finite difference discretizations of a class of nonlinear reaction-diffusion-convection equations with nonlinear boundary conditions. A higher-order monotone iterative method is presented for solving the finite difference discretizations of both the time-dependent problem and the corresponding steady-state problem. This method leads to an efficient linear iterative algorithm which yields two sequences of iterations that converge monotonically to a unique solution of the system. The monotone property of the iterations gives concurrently improved upper and lower bounds of the solution in each iteration. It is shown that the rate of convergence for the sum of the two produced sequences is of order p+2, where p≥1 is a positive integer depending on the construction of the method, and under an additional requirement, the higher-order rate of convergence is attained for one of these two sequences. An application is given to an enzyme-substrate reaction-diffusion problem, and some numerical results are presented to illustrate the effectiveness of the proposed method.

Higher-order compact finite difference method for systems of reaction-diffusion equations
Yuanming Wang and Hongbo Zhang
J. Comp. Appl. Math., 233 (2009), 502–518.
Abstract
This paper is concerned with a compact finite difference method for solving systems of two-dimensional reaction-diffusion equations. This method has the accuracy of fourth-order in both space and time. 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. Three monotone iterative algorithms are provided for solving the resulting discrete system efficiently, and the sequences of iterations converge monotonically to a unique solution of the system. A theoretical comparison result for the various monotone sequences is given. The convergence of the finite difference solution to the continuous solution is proved, and Richard extrapolation are used to achieve fourth-order accuracy in time. An application is given to an enzyme-substrate reaction-diffusion problem, and some numerical results are presented to demonstrate the high efficiency and advantages of this new approach.

Pseudospectral method using generalized Laguerre functions for singular problems on unbounded domains
Zhongqing Wang, Benyu Guo and Yanna Wu
Disc. Cont. Dynam. Syst., Ser. B, 11(2009), 1019-1038
Abstract
In this paper, we develop a pseudospectral method for differential equations defined on unbounded domains. We first introduce Gauss-type interpolations using a family of generalized Laguerre functions, and establish basic approximation results. Then we propose a pseudospectral method for differential equations on unbounded domains, whose coefficients may degenerate or grow up. As examples, we consider two model problems. The proposed schemes match the underlying problems properly and exhibit spectral accuracy. Numerical results demonstrate the efficiency of this new approach.

Spectral regularization methods for solving a sideways parabolic equation within the framework of regularization theory
Xiangtuan Xiong, Chuli Fu and Jin Cheng
Math. Comp. Simu. 79 (2009), 1668-1678.
Abstract
We introduce three spectral regularization methods for solving a general sideways parabolic equation. For these three spectral regularization methods, we give some stability error estimates with optimal order under a-priori and a-posteriori regularization parameter choice rules. Numerical results show that these spectral methods are effective.

A non-homogeneous Arnoldi method for fast simulation of RCL circuits with a large number of ports
Fan Yang, Xuan Zeng, Yangfeng Su and Wei Cai
Int. J. Circuit Theory Appl., 2009. DOI: 10.1002/cta
Abstract
Large-scale RCL circuits with a large number of ports have been widely employed to model interconnect circuits, such as the power/ground networks, clock distribution networks and large data buses in VLSI. The input-dependent moment-matching technique, which takes the input excitations into account when constructing the projection matrices for the reduced-order systems, has been proposed to simulate this type of circuits. The existing input-dependent moment-matching methods suffer from either numerical instability in the case of extended Krylov subspace (EKS) and improved extended Krylov subspace (IEKS) methods, or unbearable memory consumption and CPU cost for the EXPanded LINearization (EXPLIN) method. In this paper, a Non-Homogeneous ARnoldi (NHAR) process, which consists of a memory-saving and computation-efficient linearization scheme and a numerical stable partial orthogonalization Arnoldi method, is proposed for the generation of the orthonormal projection matrix. By applying the obtained projection matrix to generate the reduced-order model, we derive the NHAR method for the model-order reduction of large-scale RCL circuits with a large number of ports. The proposed NHAR method can guarantee moment matching, numerical stability and passivity preserving. Compared with the EXPLIN method, NHAR can remarkably reduce the size of the linearized system and therefore can greatly save the memory consumption and computational cost with almost the same accuracy. Moreover, NHAR is numerically stable and can achieve higher accuracy with approximately the same computational cost compared with the EKS and IEKS methods. Copyright q 2009 John Wiley & Sons, Ltd.

Model-robust design for multiresponse linear model with possible Bias
Rongxian Yue
2009 Sixth Int.Conf.n Fuzzy Syst. Knowl. Disc., FSKD, 2, 572-576.
Abstract
This paper studies the design problem for the multi response linear model with possible bias. It is assumed that the fitted model for each response is polynomial of degree up to two, and the model bias includes the effects due to higher degree terms of multivariate Hermite polynomials. A criterion for choosing designs is proposed based on averaging the mean squared error over all possible bias. Several examples are given to illustrate the designs in Rs.

Bayesian U-type design for nonparametric response surface prediction
Rongxian Yue and Kashinath Chatterjee
Metrika, DOI 10.1007/s00184-009-0249-0.
Abstract
This paper deals with Bayesian design over U-type designs of n runs and s factors with q levels for nonparametric response surface prediction. The criterion is developed in terms of the asymptotic approach of Mitchell et al. (Ann Statist 22: 634–651, 1994) for a specific covariance kernel. An optimal design is given in approximate design theory over the all level combinations. A connection with orthgonality and aberration is established. A lower bound for the criterion is provided, and numerical results show that this lower bound is tight.

Spectral method for three-dimensional nonlinear Klein-Gordon equation by generalized Laguerre and spherical harmonic functions
Xiaoyong Zhang, Benyu Guo and Yujian Jiao
Numerical Mathematics TMA, 2 (2009), 43-64.
Abstract
In this paper, a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution. The stability and convergence of the proposed scheme are proved. Numerical results demonstrate the efficiency of this approach. We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation, which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.


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发布日期: 2009/12/31
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