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## 2008年工作年报一、机构设置和聘任工作 ## 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2008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 上海师范大学副教授
- 苏仰锋 复旦大学教授
- 郭 谦 上海师范大学副教授
**聘请彭丽副教授、徐东博士、郭玲博士和徐海燕博士担任上海高校计算科学E—研究院青年培育人员**。
- 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。
## 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，制订并资助本年度研究课题，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
- 动力系统的数值研究
- 生物与材料科学中的数学模型及其算法
- 大型问题的快速算法
- 郭本瑜 无界区域问题和外部问题的高精度算法 4万
- 程 晋 数学物理反问题的理论及其数值计算 4万
- 黄建国 组合弹性结构的有限元方法研究 3.5万
- 岳荣先 高维积分的拟蒙特卡洛新算法及其误差分析 3.5万
- 田红炯 常微分系统的数值方法及其应用 3.5万
- 徐承龙 信用风险分析及产品定价中的随机与确定性算法 3万
- 王元明 拟线性抛物和椭圆型方程的高精度紧致差分方法 3万
- 王中庆 数学物理问题的高精度算法 3万
A.国家科研项目8个，本年度到达经费57.9万元。
- 郭本瑜 国家自然科学基金项目，奇异问题及非矩形和无界区域问题的谱方法。
- 黄建国 国家自然科学基金，组合弹性结构问题的自适应有限元方法研究。
- 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。
- 王元明 国家自然科学基金，非线性椭圆边值问题的高精度紧致有限差分 方法。
- 田红炯 国家自然科学基金，滞时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。
B.上海市及教育部科研项目10个，本年度到达经费312.8万元。
- 郭本瑜 上海市科委重大项目，若干数学物理复杂问题的计算方法。
- 郭本瑜 国家教育部博士点基金，奇异问题和无界区域问题的谱方法及
- 其应用。
- 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。
- 程 晋 上海市科委重点项目，飞行器设计中的数学模型与算法。
- 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。
- 彭 丽 上海市科委自然基金，脑影像EEG/MEG反问题及其数值计算。
- 徐海燕 上海市教委晨光计划，加速寿命试验设计中若干问题的研究。
- 郭本瑜 国家自然科学基金，谱方法若干问题研究。
- 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。
- 程 晋 国家自然科学基金，抛物型偏微分方程的参数辨识问题及其算法。
- 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。
- 彭 丽 上海市教委科研创新项目，脑影像EEG/MEG反问题及其科学计算。
- 郭 玲 上海市教委科研创新项目，具有随机参数的弹性结构问题的算法
- 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、动力系统的数值研究、随机化伪Monte-Carlo方法及其应用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34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，出版学术专著1部，获得黑龙江省科学技术二等奖（排名第二）和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，徐承龙）
- 建立了Fourier—广义Laguerre函数的混合正交逼近和插值理论，并由此提出了计算二维外部问题的混合谱方法。
- 建立了球面调和—广义Laguerre函数的混合正交逼近理论，并由此提出了计算三维外部问题的混合谱方法。
- 提出了基于Laguerre-Radau插值逼近的数值计算方法，并应用于常微分方程的数值模拟。
- 提出了半直线上的Jacobi有理和无理谱方法，并应用于一类具退化系数问题的数值模拟。
有关论文：
[1]Benyu Guo, Zhongqing Wang，Hongjiong Tian and Lilian Wang, Integration processes of ordinary differential equations based on Laguerre-Radau interpolations, Math. Comp., 77(2008), 181-199. [2]Zhongqing Wang and Benyu Guo, Jacobi rational approximation and spectral method for differential equations of degenerate type, Math. Comp., 77(2008), 883-907. [3]Benyu Guo and Jie Shen, Irrational approximations and their applications to partial differential equations in exterior domains, Adv. Comp. Math., 28(2008), 237-267. [4]Tianjun Wang and Benyu Guo, Composite generalized Laguerre-Legendre pseudospectral method for Fokker-Planck equation in an infinite channel, Appl. Numer. Math., 58(2008), 1448-1466. [5]Benyu Guo, Navier-Stokes equations with slip boundary conditions, Math. Methods Appl. Sciences，31(2008), 607-626. [6]Rong Zhang, Zhongqing Wang and Benyu Guo, Mixed Fourier-Laguerre spectral and pseudospectral methods for exterior problems using generalized Laguerre functions, J. Sci. Comput., 36(2008), 263-283. [7]Zhongqing Wang, Benyu Guo and Wei Zhang, Mixed spectral method for three-dimensional exterior problems using Harmonic and generalized Laguerre functions, J. Comput. Appl. Math., 217(2008), 277-298. [8]Benyu Guo and Keji Zhang, On non-isotropic Jacobi pseudospectral method, J. Comput. Math., 26(2008), 511-535. [9]Wei Huang and Benyu Guo, Fully discrete Jacobi-spherical harmonic spectral method for Navier-Stokes equations, Appl. Math. Mech., 29(2008), 453-476. [10]Chenglong Xu and Benyu Guo, Modified Laguerre spectral and pseudospectral methods for nonlinear partial differential equations in multiple dimensions，Appl. Math. Mech., 29(2008), 311-331. 2.数学物理反问题的理论和数值方法（程晋、黄建国）
- 讨论二维逆向热传导问题，基于Green函数的单层位势表达式，提出了拟正则化方法，具有计算量少和稳定性好的优点。
- 讨论了热成像中系数决定的反问题，证明在适当的约束条件下，该问题具有一定的稳定性，为设计稳定的数值算法提供了可靠的理论基础。
- 使用作者独创的虚拟区域正则化方法，获得任意维空间情形基于带噪声局部平均数据重构函数的有效方法。该方法在数据处理，图像去噪，和计算力学等领域有直接应用。给出方法的误差分析，并提供基于不同正则化参数选取策略的系列数值实验结果。
有关论文：
[1]Jin Cheng and Jijun Liu, A quasi Tikhonov regularization for a two-dimensional backward heat problem by a fundamental solution, Inverse Problems, 24 (2008) 065012 (18pp) doi: 10.1088/0266-5611/24/6/065012. [2]Tao Tang, Xiang Xu and Jin Cheng, On spectral methods for Volterra integral equations and the convergence analysis. J. Comput. Math., 26 (2008), 825-837. [3]Gufan Yang, Masahiro Yamamoto and Jin Cheng, Heat transfer in composite materials with Stefan-Boltzmann interface conditions. Math. Methods Appl. Sci., 31 (2008), 1297-1314. [4]Mourad Bellassoued, Jin Cheng and Mourad Choulli, Stability estimate for an inverse boundary coefficient problem in thermal imaging. J. Math. Anal. Appl., 343 (2008), 328-336. [5]Gen Nakamura, Yu Jiang, Sei Nagayasu and Jin Cheng, Inversion analysis for magnetic resonance elastography, Appl. Anal., 87 (2008), 165-179. [6]Jin Cheng, Mourad Choulli and Junshan Lin, Stable determination of a boundary coefficient in an elliptic equation, Math. Models Methods Appl. Sci., 18 (2008), 107-123. [7]Yu Chen, Jianguo Huang and Weimin Han, Function reconstruction from noisy local averages, Inverse Problems, 24(2008), 024003 (14pp). 3.组合弹性结构问题的有限元方法（黄建国、郭玲）
- 通过极值原理等技巧研究了关于环状域腐蚀系数重构问题的一个数值算法的理论分析，给出误差估计和噪声数据对解的定量分析，并进行数值模拟。
- 构造C0－连续时间步有限元方法求解由任意多个弹性梁组成的组合弹性结构振动问题。用一次协调元历史梁的纵向位移和转角，用三次Hermite元离散垂直位移。通过细致的推导得到方法的能量范数误差。对时间方向使用有限差分方法进行模态分析，说明方法在低频情形的弥散性质和谱分析性质。该C0－连续时间步有限元方法还被用于求解几个典型的非线性问题。数值实验结果说明算法的有效性和稳定性。
- 构造C0－连续时间步有限元方法求解弹性板平面问题。在时间方向离散时，不用DG方法同时离散位移场和速度场，而是直接用C0二次连续有限元离散位移场，从而减少每次迭代时所需计算问题的规模，数值实验说明算法的有效性。
有关论文：
[1]Xuehai Huang, Jianguo Huang and Yu Chen, Error analysis of a parameter expansion for corrosion detection in a pipe, Comp. Math. Appl., 56(2008), 2539—2549. [2]Junjiang Lai, Jianguo Huang and Zhongci Shi, Vibration analysis for elastic multi-beam structures by the C0-continuous time-stepping finite element method, Comm. Numer. Methods Eng., 2008, DOI: 10.1002/cnm.1143. [3]Jianguo Huang，Ling Guo and Zhongci Shi，Vibration analysis of Kirchhoff plates by the Morley element method，J. Comp. Appl. Math., 213(2008), 14-34。 4.动力系统的数值方法（田红炯、郭谦）
- 构造了一类求解非自治时滞泛函微分系统的单支 -方法，分析了该系统的渐近稳定性、压缩性和耗散性等动力学性质，并研究了单支 -方法保持上述动力学性质的充分必要条件。
- 分析了一类时滞线性抛物型方程的渐近稳定性，研究了数值方法的稳定性说明其与时间和空间步长相关。
- 建立了前列腺癌间歇治疗偏微分方程模型，利用非线性分岔理论找到了最优治疗参数。
- 分析了用病毒和抑制剂治疗肿瘤的偏微分方程模型，利用非线性分岔理论给出了最优的抑制剂剂量以及最优的抑制剂实施时间。
有关论文：
[1]Hongjiong Tian, Asymptotic stability of numerical methods for linear delay parabolic differential equations, Comp. Math. Appl., 56(2008), 1758-1765. [2]Hongjiong Tian and Ni Guo, Asymptotic stability, contractivity and dissipativity of one-leg θ-method for non-autonomous delay functional differential equations, Appl. Math. Comp., 203(2008), 333-342. [3]Qian Guo, Youshan Tao and Aihara Kazuyuki, Mathematical Modelling of prostate tumor growth under intermittent androgen suppression. Int. J. Bifur. Chaos, 18(2008), 3789-3797. [4]Youshan Tao and Qian Guo，A mathematical model of combined therapies against cancer using viruses and inhibitors，Science in China, Series A: Mathematics，51(2008), 2315-2329. 5.随机伪蒙特卡罗方法的理论与应用（岳荣先、徐海燕）
- 对有界或无界矩形区域上的高维积分，利用适当的变量变换与积分格子点序列构造新的等权求积公式，并在Banach函数空间中单位球上估计这种求积法的极端误差。
- 对Banach函数空间上函数的高维积分，构造基于积分格子点序列的具有最优权系数的求积公式，并估计这种求积法在Banach空间中单位球上的极端误差。
- 运用拟蒙特卡洛方法研究统计模型Bayes非参数预测的最优设计，利用渐近预测方差建立设计准则，给出准则下界，建立均匀性与正交性之间的关系。
- 对多响应线性模型建立基于预测协方差矩阵的最优设计准则，对设计特性及构造进行讨论。
有关论文：
[1]Rongxian Yue and Xiaodong Zhou, Bayesian robust designs for linear models with possible bias and correlated errors, Metrika, 2008 (DOI 10.1007/s00184-008-0197-0). [2]Rongxian Yue and Xin Liu, Minimax Designs for Approximately Linear Multiresponse Models, Proceedings of Fifth International Conference on Fuzzy Systems and Knowledge Discovery, IEEE. 324-328. [3]Xin Liu and Rongxian Yue, P-optimal robust designs for multiresponse approximately linear regression, Appl. Math. J. Chinese Univ. 23(2008), 168-174. [4]Haiyan Xu, Yincai Tang and Heliang Fei, Bayesian Design Comparison for Accelerated Life Tests, 14th ISSAT International Conference on Reliability and Quality in Design , Florida, USA , pp.29-33, 2008. 6.非线性初（边）值问题的高精度有限差分方法（王元明）
- 对一类变系数四阶非线性椭圆边值问题建立了具有四阶精度的紧有限差分方法, 该方法保持了原始问题解的一些主要性质。讨论了差分格式解的存在唯一性和收敛性。构造了三种求解非线性差分格式的单调迭代算法, 给出了算法几何收敛的充分条件。
- 对一类高阶常系数 Lidstone 边值问题建立了具有四阶精度的紧有限差分方法, 该方法保持了原始问题解的一些主要性质。讨论了差分格式解的存在唯一性和收敛性。构造了一种求解非线性差分格式的单调迭代算法, 给出了算法几何收敛的充分条件。
- 对一类变系数非线性初边值问题建立了具有高阶精度的紧有限差分方法, 讨论了差分格式解的存在唯一性和收敛性。构造了求解非线性差分格式的具有二次收敛率的单调迭代算法，计算结果显示了方法的优越性。
- 对一类非线性反应扩散方程和相应的定常问题的标准有限差分解给出了一些数值分析,求解的单调迭代算法和非定常解的渐近收敛性, 所得结果不需要非线性项的任何单调性, 因而推广了已知的结果。
有关论文：
[1]Yuanming Wang and Benyu Guo, Fourth-order compact finite difference method for fourth-order nonlinear elliptic boundary value problems, J. Comp. Appl. Math., 221 (2008), 76–97. [2]Yuanming Wang, Haiyun Jiang and Ravi P. Agarwal, A fourth-order compact finite difference method for higher-order Lidstone boundary value problems, Comp. Math. Appl., 56 (2008), 499–521. [3]Yuanming Wang and Benyu Guo, A monotone compact implicit scheme for nonlinear reaction-diffusion equations, J. Comp. Math., 26(2008), 123–148. [4]Yuanming Wang, Global asymptotic stability of 3-species Lotka–Volterra models with diffusion and time delays, Appl. Math. Comp., 195 (2008), 34–48. [5]Chia Ven Pao and Yuanming Wang, Numerical solutions of a three-competition Lotka–Volterra system, Appl. Math. Comp., 204 (2008), 423–440. 7.金融衍生物的偏微分方程定价及计算（徐承龙）
- 对金融上较广泛的一类离散取样的随机波动率衍生物定价问题建立了方差减少Monte Carlo 算法，减少计算时间。
- 与姜礼尚教授等合作者出版了专著《金融衍生品定价的数学模型和案例分析》。
有关论文：
[1]姜礼尚、徐承龙、任学敏、李少华著，金融衍生品定价的数学模型和案例分析，高等教育出版社，2008年。 [2]任学敏，徐承龙，有违约风险的保底型基金的定价， 同济大学学报（自然科学版），36(2008), 849-853。 ## 四、学术活动遵循研究院管理章程进行日常学术活动，并举办或合办了一些国内或国际学术会议。1.日常学术活动 - 每月召开全体特聘研究员工作会议，相互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。
- 2008年4月， 与复旦大学联合举办“地震中的数学模型与方法”学术会议。
- 2008年10月， 与复旦大学联合举办“The International Conference on Inverse Problems and its Applications”国际会议。
- 2008年11月， 与华东师范大学联合举办上海市第四届“科学与工程中的计算方法”研讨会。
- 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。
## 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“计算数学”学科被评为上海市第三期重点建设学科。 2.上海市教委在上海师范大学成立“科学计算”上海高校重点实验室。 3.1名年轻成员新评为博士生导师。 4.研究院成员共指导了3博士后，23名博士生（其中毕业7名），49名硕士生（其中毕业12名）。指导国内访问学者1名。 5.郭本瑜教授指导的一名博士生获得上海市优秀博士论文，程晋教授指导的一名硕士生获得上海市优秀硕士论文。 ## 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，上海市教委批准在上海师范大学成立“科学计算”上海高校重点实验室。目前，拥有SGI工作站（32个CPU，内存为16GB，硬盘容量达到730GB）和多台多核高性能计算机（XW8600）及配套设施。同时，委托上海艮泰信息技术有限公司，组织集群式并行计算系统使用培训服务。 ## 七、2008年论文摘要Stability estimate for an inverse boundary coefficient problem in thermal imaging Mourad Bellassoueda, Jin Cheng and Mourad Choullic J. Math. Anal. Appl. 343 (2008), 328-336 Abstract We establish a stability estimate for an inverse boundary coefficient problem in thermal imaging. The inverse problem under consideration consists in the determination of a boundary coefficient appearing in a boundary value problem for the heat equation with Robin boundary condition (we note here that the initial condition is assumed to be a priori unknown). Our stability estimate is of logarithmic type and it is essentially based on a logarithmic estimate for a Cauchy problem for the Laplace equation. A quasi Tikhonov regularization for a two-dimensional backward heat problem by a fundamental solution Jin Cheng and Jijun Liu Inverse Problems, 24 (2008) 065012 (18pp). Abstract Consider a two-dimensional backward heat conduction problem for a general domain with a boundary. Based on the fundamental solution to the heat equation, we propose to solve this problem by the boundary integral equation method, which generates a coupled ill-posed integral equation. Then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proven. Our regularizing scheme can be considered a quasi Tikhonov regularization, with the advantage of a relatively small amount of computation compared with the classical Tikhonov regularization. Numerical performances are given to show the validity of our inversion method. Stable determination of a boundary coefficient in an elliptic equation Jin Cheng, Mourad Choulli and Junshan Lin Math. Models Methods Appl. Sci. 18 (2008), 107-123. Abstract We prove a logarithmic stability estimate for a Cauchy problem associated with a second order elliptic operator. Our proof is essentially based on a Carleman estimate by A. L. Bukhgeim. This result is applied to establish a stability estimate for the inverse problem of determining a boundary coefficient (or a boundary function) by a single boundary measurement. This kind of inverse problems is motivated by the corrosion detection problem. Function reconstruction from noisy local averages Yu Chen, Jianguo Huang and Weimin Han Inverse Problems, 24(2008), 024003 (14pp). Abstract A regularization method is proposed for the function reconstruction from noisy local averages in any dimension. Error bounds for the approximate solution in L2-norm are derived. A number of numerical examples are provided to show computational performance of the method, with the regularization parameters selected by different strategies. Integration processes of ordinary differential equations based on Laguerre-Radau interpolations Benyu Guo, Zhongqing Wang, Hongjiong Tian and Lilian Wang Math. Comp., 77(2008), 181-199. Abstract In this paper, we propose two integration processes for ordinary differential equations based on modified Laguerre-Radau interpolations, which are very efficient for long-time numerical simulations of dynamical systems. The global convergence of proposed algorithms are proved. Numerical results demonstrate the spectral accuracy of these new approaches and coincide well with theoretical analysis. On non-isotropic Jacobi Pseudospectral method Benyu Guo and Keji Zhang J. Comp. Math., 26(2008), 511-535. Abstract In this paper, a non-isotropic Jacobi pseudospectral method is proposed and its applications are considered. Some results on the multi-dimensional Jacobi-Gauss type interpolation and the related Bernstein-Jackson type inequalities are established, which play an important role in pseudospectral method. The pseudospectral method is applied to a twodimensional singular problem and a problem on axisymmetric domain. The convergence of proposed schemes is established. Numerical results demonstrate the efficiency of the proposed method. Irrational approximations and their applications to partial differential equations in exterior domains Benyu Guo and Jie Shen Adv. Comput. Math., 28(2008), 237-267. Abstract A family of orthogonal systems of irrational functions on the semi-infinite interval is introduced. The proposed orthogonal systems are based on Jacobi polynomials through an irrational coordinate transform. This family of orthogonal systems offers great flexibility to match a wide range of asymptotic behaviors at infinity. Approximation errors by the basic orthogonal projection and various other orthogonal projections related to partial differential equations in unbounded domains are established. As an example of applications, a Galerkin approximation using the proposed irrational functions to an exterior problem is analyzed and implemented. Numerical results in agreement with our theoretical estimates are presented. Navier-Stokes equations with slip boundary conditions Benyu Guo Mathematical Methods in the Applied Sciences, 31(2008), 607-626. Abstract In this paper, we consider incompressible viscous fluid flows with slip boundary conditions. We first prove the existence of solutions of the unsteady Navier-Stokes equations in n-spacial dimensions. Then, we investigate the stability, uniqueness and regularity of solutions in two and three spacial dimensions. In the compactness argument, we construct a special basis fulfilling the incompressibility exactly, which leads to an efficient and convergent spectral method. In particular, we avoid the main difficulty for ensuring the incompressibility of numerical solutions, which occurs in other numerical algorithms. We also derive the vorticity-stream function form with exact boundary conditions, and establish some results on the existence, stability and uniqueness of its solutions. Mathematical Modelling of prostate tumor growth under intermittent androgen suppression Qian Guo, Youshan Tao and Aihara Kazuyuki Int. J. Bifur. Chaos, 18(2008), 3789-3797. Abstract Since most prostate tumors are initially hormone-sensitive, hormonal therapy with androgen suppression is a major treatment for them. In this hormonal therapy, however, a tumor relapse is a crucial problem. Androgen-independent tumor cells are considered to be responsible for such a relapse. These cells are not sensitive to androgen suppression but rather apt to proliferate even in an androgen-poor environment. Bruchovsky et al. proposed intermittent androgen suppression (IAS), which may prolong the relapse time when compared with continuous androgen suppression (CAS). IAS therapy is based on switching of medication through monitoring of the serum prostate-specific antigen (PSA). Namely, the medication is suspended when the PSA concentration falls below the lower threshold during on-treatment periods and it is reinstituted when the concentration exceeds the upper threshold during off-treatment periods. In this paper, we propose a model of partial differential equations (PDE) for IAS therapy, on the basis of our previous model of ordinary differential equations, under the assumption that the prostate tumor is a mixed assembly of androgen-dependent (AD) and androgen-independent (AI) cells. Numerical analysis compares the effect of the IAS therapy with that of the CAS therapy for different growth rates of the AI cells, which suggests an optimal protocol of the IAS therapy. Vibration analysis of Kirchhoff plates by the Morley element method Jianguo Huang, Ling Guo, Zhongci Shi J. Comp. Appl. Math., 213 (2008), 14-34. Abstract Vibration analysis of Kirchhoff plates is of great importance in many engineering fields. The semi-discrete and the fully discrete Morley element methods are proposed to solve such a problem, which are effective even when the region of interest is irregular. The rigorous error estimates in the energy norm for both methods are established. Some reasonable approaches to choosing the initial functions are given to keep the good convergence rate of the fully discrete method. A number of numerical results are provided to illustrate the computational performance of the method in this paper. Fully discrete Jacobi-spherical harmonic spectral method for Navier-Stokes equations Wei Huang and Benyu Guo Appl. Math. Mech. 29(2008), 453-476. Abstract A fully discrete Jacobi-spherical harmonic spectral method is provided for the Navier-Stokes equations in a ball.Its stability and convergence are proved.Numerical results show efficiency of this approach.The proposed method is also applicable to other problems in spherical geometry. Error analysis of a parameter expansion for corrosion detection in a pipe Xuehai Huang, Jianguo Huang and Yu Chen Computers and Mathematics with Applications, 56(2008), 2539-2549 Abstract Error analysis is developed for a parameter expansion method of determining the corrosion coefficient in a pipe. For the two approximate solutions proposed, the magnitude of their errors is shown to be O(a) and O(a2), respectively, where a stands for the thickness of the pipe. Vibration analysis for elastic multi-beam structures by the C0-continuous time-stepping finite element method Junjiang Lai, Jianguo Huang and Zhongci Shi Comm. Numer. Methods Eng., 2008 Abstract Some C0-continuous time-stepping finite element method is proposed for investigating vibration analysis of elastic multi-beam structures. In the time direction, the C0-continuous Galerkin method is used to discretize the generalized displacement field. In the space directions, the longitudinal displacements and rotational angles on beams are discretized using conforming linear elements, while the transverse displacements on beams are discretized by the Hermite elements of third order. The error of the method in the energy norm is proved to be O(h+k3), where h and k denote the mesh sizes of the subdivisions in the space and time directions, respectively. The finite difference analysis in time is developed to discuss the spectral behavior of the algorithms as well as their dissipation and dispersion properties in the low-frequency regime. The method has also been extended to study some nonlinear problems. A number of numerical tests are included to illustrate the computational performance of the method. 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 C0-continuous time stepping finite element method to solve vibration problems of plane elasticity. In the time direction, unlike the existing methods [F. Costanzo, H. Huang, Proof of unconditional stability for a single-field discontinuous Galerkin finite element formulation for linear elasto-dynamics, Comput. Methods Appl. Mech. Engrg. 194 (2005) 2059–2076; D.A. French, A space–time finite element method for the wave equation, Comput. Methods Appl. Mech. Engrg. 107 (1993) 145–157; H. Huang, F. Costanzo, On the use of space–time finite elements in the solution of elasto-dynamic problems with strain discontinuities, Comput. Methods Appl. Mech. Engrg. 191 (2002) 5315–5343; T.J.R. Hughes, G. Hulbert, Space–time finite element methods for elastodynamics: Formulations and error estimates, Comput. Methods Appl. Mech. Engrg. 66 (1988) 339–363; G. Hulbert, T.J.R. Hughes, Space–time finite element methods for second-order hyperbolic equations, Comput. Methods Appl. Mech. Engrg. 84 (1990) 327–348; C. Johnson, Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 107 (1993) 117–129; X.D. Li, N.E. Wiberg, Structural dynamic analysis by a time-discontinous Galerkin finite element method, Int. J. Numer. Methods Engrg. 39 (1996) 2131–2152; X.D. Li, N.E. Wiberg, Implementation and adaptivity of a space–time finite element method for structural dynamics, Comput. Methods Appl. Mech. Engrg. 156 (1998) 211–229], this method does not use the discontinuous Galerkin (DG) method to simultaneously discretize the displacement and velocity fields, but only use the C0-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 Pr−1-conforming element with r 2. It is proved that the error of the method in the energy norm is O(hr−1+k3), 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. P-optimal robust designs for multiresponse approximately linear regression Xin Liu and Rongxian Yue Applied Mathematics - A Journal of Chinese Universities, 23(2008), 168-174 Abstract This paper deals with the problem of -optimal robust designs for multiresponse approximately linear regression models. Each response is assumed to be only approximately linear in the regressors, and the bias function varies over a given ℒ2-neighbourhood. A kind of bivariate models with two responses is taken as an example to illustrate how to get the expression of the design measure. Inversion analysis for magnetic resonance elastography Gen Nakamura, Yu Jiang, Sei Nagayasu and Jin Cheng Appl. Anal. 87 (2008), 165-179. Abstract We will propose a new reconstruction scheme to identify the viscoelasticity of a living body from MRE measurements. The reconstruction scheme consists of application of the oscillating-decaying solution, Taylor expansion, complex geometric optics solutions, and an iterative method for solving the Cauchy problem for elliptic equations. Numerical solutions of a three-competition Lotka–Volterra system Chia Ven Pao and Yuanming Wang Appl. Math. Comp., 204 (2008), 423–440. Abstract This paper is concerned with finite difference solutions of a Lotka–Volterra reaction–diffusion system with three-competing species. The reaction–diffusion 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 corresponding steady-state problem. 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. Also discussed is the asymptotic behavior of the time-dependent solution in relation to various steady-state solutions. A simple condition on the competing 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 one of the semitrivial steady-state solutions. The above results lead to the coexistence and permanence of the competing system as well as computational algorithms for numerical solutions. Some numerical results from these computational algorithms are given. All the conclusions for the reaction–diffusion equations are directly applicable to the finite difference solution of the corresponding ordinary differential system. 有违约风险的保底型基金的定价 任学敏 徐承龙 同济大学学报（自然科学版）, 36(2008), 849-853. 摘要 由于市场价格变化常常是跳跃式的,保底型基金的保底收益不一定能实现.利用无套利原理,当股指满足跳扩散模型时,用偏微分方程的方法给出了有违约可能的保底基金的定价和保底不成功概率的公式并作了一些定性分析. On spectral methods for Volterra type integral equations and the convergence analysis Tao Tang, Xiang Xu and Jin Cheng J. Comput. Math. 26 (2008), 825-837. Abstract The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach (with theoretical justification) for the Volterra type equations. A mathematical model of combined therapies against cancer using viruses and inhibitors Youshan Tao and Qian Guo Science in China Series A: Mathematics, 51(2008), 2315-2329. Abstract This paper deals with a procedure for combined therapies against cancer using oncolytic viruses and inhibitors. Replicating genetically modified adenoviruses infect cancer cells, reproduce inside them and eventually cause their death (lysis). As infected cells die, the viruses inside them are released and then proceed to infect other tumor cells. The successful entry of virus into cancer cells is related to the presence of the coxsackie-adenovirus receptor (CAR). Mitogen-activated protein kinase kinase (known as MEK) inhibitors can promote CAR expression, resulting in enhanced adenovirus entry into cancer cells. However, MEK inhibitors can also cause G1 cell-cycle arrest, inhibiting reproduction of the virus. To design an effective synergistic therapy, the promotion of virus infection must be optimally balanced with inhibition of virus production. We introduce a mathematical model to describe the effects of MEK inhibitors and viruses on tumor cells, and use it to explore the reduction of the tumor size that can be achieved by the combined therapies. Furthermore, we find an optimal dose of inhibitor: Poptimal = 1 − μ/δ for a certain initial density of cells (where μ is the removal rate of the dead cells and δ is the death rate of the infected cells). The optimal timing of MEK inhibitors is also numerically studied. Asymptotic stability of numerical methods for linear delay parabolic differential equations Hongjiong Tian Comp. Math. Appl., 56 (2008) 1758–1765 Abstract This paper is concerned with the asymptotic stability property of some numerical processes by discretization of parabolic differential equations with a constant delay. These numerical processes include forward and backward Euler difference schemes and Crank–Nicolson difference scheme which are obtained by applying step-by-step methods to the resulting systems of delay differential equations. Sufficient and necessary conditions for these difference schemes to be delay-independently asymptotically stable are established. It reveals that an additional restriction on time and spatial stepsizes of the forward Euler difference scheme is required to preserve the delay-independent asymptotic stability due to the existence of the delay term. Numerical experiments have been implemented to confirm the asymptotic stability of these numerical methods. Asymptotic stability, contractivity and dissipativity of one-leg θ-method for non-autonomous delay functional differential equations Hongjiong Tian and Ni Guo Appl. Math. Comp., 203(2008), 333-342. Abstract This paper focuses on asymptotic stability, contractivity and dissipativity of non-autonomous nonlinear delay functional differential equations with bounded lag, and the corresponding dynamical properties of one-leg -method. Sufficient conditions for these delay functional differential equations to be dissipative, asymptotically stable and contractive are established. One-leg -method is constructed to solve such equations numerically. An important result on the growth of solution of a class of difference inequalities with variable coefficients is obtained. Finally, it is proved that the one-leg -method is asymptotically stable, contractive and dissipative if and only if = 1. Numerical examples are given to confirm our theoretical results. Composite generalized Laguerre-Legendre pseudospectral method for Fokker--Planck equation in an infinite channel Tianjun Wang and Benyu Guo Applied Numerical Mathematics, 58(2008), 1448-1466 Abstract In this paper, we propose a composite generalized Laguerre-Legendre pseudospectral method for the Fokker-Planck equation in an infinite channel, which behaves like a parabolic equation in one direction, and behaves like a hyperbolic equation in other direction. We establish some approximation results on the composite generalized Laguerre-Legendre-Gauss-Radau interpolation, with which the convergence of proposed composite scheme follows. An efficient implementation is provided. Numerical results show the spectral accuracy in space of this approach and coincide well with theoretical analysis. The approximation results and techniques developed in this paper are also very appropriate for many other problems on multiple-dimensional unbounded domains, which are not of standard types. Fourth-order compact finite difference method for fourth-order nonlinear elliptic boundary value problems Yuanming Wang and Benyu Guo J. Comp. Appl. Math., 221 (2008), 76–97. Abstract A compact finite difference method with non-isotropic mesh is proposed for a two-dimensional fourth-order nonlinear elliptic boundary value problem. The existence and uniqueness of its solutions are investigated by the method of upper and lower solutions, without any requirement of the monotonicity of the nonlinear term. Three monotone and convergent iterations are provided for resolving the resulting discrete systems efficiently. The convergence and the fourth-order accuracy of the proposed method are proved. Numerical results demonstrate the high efficiency and advantages of this new approach. A fourth-order compact finite difference method for higher-order Lidstone boundary value problems Yuanming Wang, Haiyun Jiang and Ravi P. Agarwal Comp. Math. Appl., 56 (2008), 499–521. Abstract A compact finite difference method is proposed for a general class of 2nth-order Lidstone boundary value problems. The existence and uniqueness of the finite difference solution is investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. A monotone iteration process is provided for solving the resulting discrete system efficiently, and a simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations. The convergence of the finite difference solution and the fourth-order accuracy of the proposed method are proved. Numerical results demonstrate the high efficiency and advantages of this new approach. A monotone compact implicit scheme for nonlinear reaction-diffusion equations Yuanming Wang and Benyu Guo J. Comp. Math., 26(2008), 123–148. Abstract A monotone compact implicit finite difference scheme with fourth-order accuracy in space and second-order in time is proposed for solving nonlinear reaction-diffusion equations. An accelerated monotone iterative method for the resulting discrete problem ispresented. The sequence of iteration converges monotonically to the unique solution of the discrete problem, and the convergence rate is either quadratic or nearly quadratic, depending on the property of the nonlinear reaction. The numerical results illustrate the high accuracy of the proposed scheme and the rapid convergence rate of the iteration. Global asymptotic stability of 3-species Lotka–Volterra models with diffusion and time delays Yuanming Wang Appl. Math. Comp., 195 (2008), 34–48. Abstract This paper is concerned with three 3-species time-delayed Lotka–Volterra reaction–diffusion models with homogeneous Neumann boundary condition. Some simple conditions are obtained for the global asymptotic stability of the nonnegative semitrivial constant steady-state solutions. These conditions are explicit and easily verifiable, and they involve only the reaction rate constants and are independent of the diffusion and time delays. The result of global asymptotic stability not only implies the nonexistence of positive steady-state solution but also gives some extinction results of the models in the ecological sense. The instability of some nonnegative semitrivial constant steady-state solutions is also shown. The conclusions for the reaction–diffusion systems are directly applicable to the corresponding ordinary differential systems. Jacobi rational approximation and spectral method for differential equations of degenerate type Zhongqing Wang and Benyu Guo Math. Comp., 77(2008), 883-907. Abstract We introduce an orthogonal system on the half line, induced by Jacobi polynomials. Some results on the Jacobi rational approximation are established, which play important roles in designing and analyzing the Jacobi rational spectral method for various differential equations, with the coefficients degenerating at certain points and growing up at infinity. The Jacobi rational spectral method is proposed for a model problem appearing frequently in finance. Its convergence is proved. Numerical results demonstrate the efficiency of this new approach. Mixed spectral method for three-dimensional exterior problems using spherical harmonic and generalized Laguerre functions Zhongqing Wang, Benyu Guo and Wei Zhang J. Comp. Appl. Math., 217(2008), 277-298. Abstract In this paper, we develop the mixed spectral method for three-dimensional exterior problems, using spherical harmonic and generalized Laguerre functions. Some basic approximation results are established. The mixed spectral schemes are proposed for two model problems. Their convergences are proved. Numerical results demonstrate the efficiency of this new approach. Modified Laguerre spectral and pseudospectral methods for nonlinear partial differential equations in multiple dimensions Chenglong Xu and Benyu Guo Appl. Math. Mech., 29(2008), 311-331. Abstract The Laguerre spectral and pseudospectral methods are investigated for multidimensional nonlinear partial difierential equations.Some results on the modified Laguerre orthogonal approximation and interpolation are established,which play important roles in the related numerical methods for unbounded domains.As an example,the modified Laguerre spectral and pseudospectral methods are proposed for two-dimensional Logistic equation.The stability and convergence of the suggested schemes are proved.Numerical results demonstrate the high accuracy of these approaches. Bayesian Design Comparison for Accelerated Life Tests Haiyan Xu, Yincai Tang and Heliang Fei 14th ISSAT International Conference on Reliability and Quality in Design, Florida, USA, 29-33, 2008. Abstract With Bayesian methods, there are two popular criteria on which accelerated life tests (ALT) design is based. In this article, the theoretical basis of the two criteria is unmasked, and relationships between the two criteria are derived. An example is also presented to illustrate the application of the two criteria in constant-stress ALT situation with log-normal distribution and Type-I censoring. Heat transfer in composite materials with Stefan-Boltzmann interface conditions Gufan Yang, Masahiro Yamamoto and Jin Cheng Math. Methods Appl. Sci. 31 (2008), 1297-1314. Abstract In this paper, we discuss nonstationary heat transfer problems in composite materials. This problem can be formulated as the parabolic equation with Stefan-Boltzmann interface conditions. It is proved that there exists a unique global classical solution to one-dimensional problems. Moreover, we propose a numerical algorithm by the finite difference method for this nonlinear transmission problem. Bayesian robust designs for linear models with possible bias and correlated errors Rongxian Yue and Xiaodong Zhou Metrika, (2008) Abstract Consider the design problem for the approximately linear model with serially correlated errors. The correlated structure is the qth degree moving average process, MA(q), especially for q = 1, 2. The optimal design is derived by using Bayesian approach. The Bayesian designs derived with various priors are compared with the classical designs with respect to some specific correlated structures. The results show that any prior knowledge about the sign of the MA(q) process parameters leads to designs that are considerately more efficient than the classical ones based on homoscedastic assumptions. Minimax Designs for Approximately Linear Multiresponse Models Rongxian Yue and Xin Liu Proceedings of Fifth International Conference on Fuzzy Systems and Knowledge Discovery, 2008, 1 (2008), 324-328. Abstract This paper considers the minimax design problem in approximately linear multiresponse regression models. The difference between the fitted linear models and the true responses is a vector of non-linear contaminations restricted by a bound on its L_2 norm in the design region. We derive a theory to guide in the construction of designs which minimize the maximum loss. The minimax design is found for a bivariate regression model with two responses. Mixed Fourier-Laguerre spectral and pseudospectral methods for exterior problems using generalized Laguerre functions Rong Zhang, Zhongqing Wang and Benyu Guo J. Scientific Computing, 36(2008), 263-283. Abstract In this paper, we develop the mixed spectral and pseudospectral methods for two-dimensional exterior problems, by using the scaled generalized Laguerre functions. Some basic results on the mixed Fourier-Laguerre orthogonal approximation and Gauss-type interpolation are established, which play important roles in the related spectral and pseudospectral methods. As an example, we propose the mixed spectral and pseudospectral schemes for a model problem. The convergence of proposed schemes are proved. Numerical results demonstrate the spectral accuracy efficiency of this new approach. Return |