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发布日期 : 2019-07-12 浏览次数 :

报告1: Recent Development on Some Numerical Methods for Stochastic Partial Differential Equations


时间地点:2019898:50-9:30 孔子会堂

报告摘要:In this talk we review our recent results on some important numerical issues for stochastic partial differential equations, inclusive of numerical stability, exponential integrability, invariant measure, numerical ergodicity, strong convergence and weak convergence, etc. We present stochastic symplectic methods, stochastic multi-symplectic methods, conservative methods, splitting methods for some specific stochastic partial differential equations, such as stochastic Schroedinger equations, stochastic Maxwell equation, etc. And we study their convergence analysis, stability, exponential integrability, ergodicity and other dynamical behaviors. Convergence rates of the considered numerical methods are given. Based on Malliavin calculus, we show some results on density functions of numerical solutions for some stochastic partial differential equations. Both theoretical and numerical results are presented.

洪佳林简介:洪佳林, 研究员、博士生导师、中国科学院数学与系统科学研究院副院长。1994年在吉林大学获得博士学位,1995年至1996年在应用数学研究所作博士后,1996年11月在计算数学与科学工程计算研究所任副研究员,1997年3月至1999年3月受西班牙科学教育部资助在西班牙Valladolid大学做研究工作,1999年1月至今,历任数学与系统科学研究院副研究员、研究员、博士生导师。主要研究方向: 动力系统保结构算法理论与应用,包括确定与随机哈密尔顿系统辛几何算法、确定与随机哈密尔顿偏微分方程的多辛几何算法、李群算法以及确定与随机微分系统的守恒型算法等。在SIAM J. Numer. Anal、J. Comput. Phys.、Math. Comput.、 Stud. Math.、中国科学等国际学术刊物上发表研究论文70余篇。


报告2: Adaptive FEM for Helmholtz equation with high wave number


时间地点:2019899:30-10:10 孔子会堂



报告3: Higher order structure-preserving numerical schemes for nonlinear time-dependent problems


时间地点:20198910:40-11:20 孔子会堂

报告摘要:In this talk, we discuss local discontinuous Galerkin method for solving the nonlinear wave equations which contain nonlinear high order derivatives. The discretization results in an extremely local, element based discretization, which is beneficial for parallel computing and maintaining high order accuracy on unstructured meshes. In particular, the methods are well suited for hp-adaptation, which consists of local mesh refinement and/or the adjustment of the polynomial order in individual elements. We will present a high order semi-implicit time discretization method for highly nonlinear PDEs, which consist of the surface diffusion and Willmore flow of graphs, the phase field models, etc. These PDEs are high order in spatial derivatives, which motivates us to develop implicit or semi-implicit time marching methods to relax the severe time step restriction for stability of explicit methods. In addition, these PDEs are also highly nonlinear, fully implicit methods will incredibly increase the difficulty of implementation. In particular, we can not well separate the stiff and non-stiff components for these problems, which leads to traditional implicit-explicit methods nearly meaningless. In this paper, a high order semi-implicit time marching method and the local discontinuous Galerkin (LDG) spatial method are coupled together to achieve high order accuracy in both space and time, and to enhance the efficiency of the proposed approaches, the resulting linear or nonlinear algebraic systems are solved by multigrid solver.  We also present a new approach to obtain positivity preserving DG discretization. The positivity preserving DG discretization is then reformulated as a Karush-Kuhn-Tucker (KKT) problem, which is frequently encountered in constrained optimization. We develop an efficient active set semi-smooth Newton method that is suitable for the KKT formulation of time-implicit positivity preserving DG discretizations. Numerical simulation results in one and two dimensions are presented to illustrate that the combination of the LDG method for spatial approximation, semi-implicit temporal integration with the multigrid solver provides a practical and efficient approach when solving this family of problems.



报告4: Long-time accurate symmetrized implicit-explicit BDF methods for a class of parabolic equations with non-selfadjoint operators


时间地点:20198911:20-12:00 孔子会堂

报告摘要:An implicit-explicit multistep method based on the backward difference formulae (BDF) is proposed for time discretization of parabolic equations with a non-selfadjoint operator. Implicit and explicit schemes are used for the self-adjoint and anti-selfadjoint parts of the operator, respectively. For a k-step method, some correction terms are added to the starting k-1 steps to maintain kth-order convergence without imposing further compatibility conditions at the initial time. Long-time kth-order convergence for the numerical method is proved under the assumptions that the operator is coercive and the non-selfadjoint part is low order. Such an operator often appears in practical computation (such as the Stokes-Darcy sys- tem), but may violate the standard sectorial angle condition used in the literature for analysis of BDF. In particular, the proposed method and analysis in this paper extend the long-time energy error analysis of the Stokes-Darcy system in Chen, Gunzburger, Sun & Wang [14,15] to general symmetrized and decoupled BDF methods up to order 6 by using the generating function technique.

李步扬简介:李步扬博士于2005年在邵阳大学取得数学学士学位,并分别于2007、2009 及2012年在香港城市大学取得应用数学硕士、哲学硕士及博士学位。李博士于2012年12月开始任职于南京大学,并于2015年7月晋升为副教授。在2015年 6月至2016年5月期间,李博士在德国图宾根大学兼任洪堡学者的工作。2016年6月加入香港理工大学应用数学系担任助理教授一职。主要研究方向是偏微分方程的数值解法和数值分析,在SIAM J. Numer. Anal., SIAM J. Sci. Comput., Math. Comput., Numer. Math. 等计算数学顶级期刊上发表论文40多篇。


报告5: A weak Galerkin finite element scheme for the Cahn-Hilliard equation


时间地点:20198108:10-8:50 孔子会堂

报告摘要:This talk presents a weak Galerkin (WG) finite element method for the Cahn- Hilliard equation. The WG method makes use of piecewise polynomials as approximating functions, with weakly partial derivatives (first and second order) computed locally by using the information in the interior and on the boundary of each element. A stabilizer is constructed and added to the numerical scheme for the purpose of providing certain weak continuities for the approximating function. A mathematical convergence theory is developed for the corresponding numerical solutions, and optimal order of error estimates are derived. Some numerical results are presented to illustrate the efficiency and accuracy of the method.

张然简介:张然,教授,博士生导师。现任吉林大学数学学院副院长。2013年,入选教育部新世纪人才奖励计划。2016年,入选教育部“长江学者奖励计划”青年学者。从事随机微分、积分方程数值解、多尺度分析及应用、金融衍生产品的数值计算等课题研究。曾多次访问香港中文大学、香港浸会大学、美国密西根州立大学、美国奥本大学、新加坡国立大学等。目前主持包括国家自然科学基金委重大研究计划重点项目1项,联合项目1项,以及面上项目1项。此外,主持完成自然科学基金项目2项。在包括计算数学领域的顶级期刊《SIAM J Numerical Analysis》、《SIAM J Scientific Computing》、《IMA J Numerical Analysis》等上发表学术论文50余篇。2016年,获得吉林省自然科学奖三等奖(第一完成人)。


报告6: Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness


时间地点:20198108:50-9:30 孔子会堂

报告摘要:The accuracy of deep learning, i.e., deep neural networks, can be characterized by dividing the total error into three main types: approximation error, optimization error, and generalization error. Whereas there are some satisfactory answers to the problems of approximation and optimization, much less is known about the theory of generalization. Most existing theoretical works for generalization fail to explain the performance of neural networks in practice. To derive a meaningful bound, we study the generalization error of neural networks for classification problems in terms of data distribution and neural network smoothness. We introduce the cover complexity (CC) to measure the difficulty of learning a data set and the inverse of modules of continuity to quantify neural network smoothness. A quantitative bound for expected accuracy/error is derived by considering both the CC and neural network smoothness. We validate our theoretical results by several data sets of images. The numerical results verify that the expected error of trained networks scaled with the square root of the number of classes has a linear relationship with respect to the CC. In addition, we observe a clear consistency between test loss and neural network smoothness during the training process.

唐贻发简介:唐贻发,中国科学院数学与系统科学研究院二级研究员、博士生导师。1966年9月生,1987年毕业于复旦大学数学系,同年进入中国科学院计算中心,师从冯康教授学习辛算法,先后获硕士、博士学位. 研究方向:动力系统的几何算法、分数阶微分方程数值分析. 在国际SCI刊物上发表论文80余篇,主要在“多步法的辛性”、“辛算法形式能量及其收敛性分析”、“非线性Schrödinger方程、等离子体导心系统的正则化与辛模拟”、“含时Maxwell方程的辛谱元离散方法”、“二维时空分数阶Bloch-Torrey方程有限元方法”等方面做出有影响的工作,是1997年国家自然科学一等奖获奖项目“哈密尔顿系统的辛几何算法”的五位主要参加者之一。


报告7: A coarse-graining framework for spiking neuronal networks: from local, low-order moments to large-scale spatiotemporal activities


时间地点:20198109:30-10:10 孔子会堂

报告摘要:In this talk we provide a general methodology for systematically reducing the dynamics of a class of integrate-and-fire networks down to an augmented 4-dimensional system of ordinary-differential-equations. The class of integrate-and-fire networks we focus on are homogeneously-structured, strongly coupled, and fluctuation-driven. Our reduction succeeds where most current firing-rate and population-dynamics models fail because we account for the emergence of ‘multiple-firing- events’ involving the semi-synchronous firing of many neurons. These multiple-firing-events are largely responsible for the fluctuations generated by the network and, as a result, our reduction faithfully describes many dynamic regimes ranging from homogeneous to synchronous. Our reduction is based on first principles, and provides an analyzable link between the integrate-and-fire network parameters and the relatively low-dimensional dynamics underlying the 4-dimensional augmented ODE.

张继伟简介:张继伟,武汉大学数学与统计学院教授。2009年在香港浸会大学获得博士学位,随后在南洋理工大学和纽约大学克朗所从事博士后研究,2014年5月在北京计算科学研究中心工作,2018年11月到武汉大学工作。现主持1项国家自然科学基金面上项目,并参与1项重点项目,2015年入选“第十一批青年千人计划”。主要研究领域包括偏微分方程和非局部模型的数值解法,以及神经科学的建模与计算。其成果发表在SIAM Journal on Scientific Computing, SIAM Journal on Numerical Analysis, Mathematic of Computation, Journal of Computational Neuroscience等业内期刊上。


报告8: Hamilton系统的小波保结构算法及其应用


时间地点:201981010:40-11:20 孔子会堂

报告摘要:许多重要的数学物理方程都可以表示成Hamilton 系统的形式,Hamilton 系统内在具有守恒特性和辛几何结构。现代计算方法的基本原则是尽可能保持原问题的本质特征。因此,研究Hamilton 系统框架下的偏微分方程,以及能够保持其物理守恒律及辛几何结构的数值方法是非常有意义的。本报告基于小波配点方法,对一些重要的非线性偏微分方程进行数值研究,并构造了一系列的小波保结构算法,同时给出了这些算法的离散守恒性质、收敛特性以及数值稳定性等必要的理论分析,保证了算法在长时间数值模拟中的可靠性。

宋松和简介:宋松和,男,1965年出生,湖南湘乡人,现任国防科学技术大学理学院教授、博士生导师、计算数学方向学术带头人,担任学校科技委委员、校理学学部学部委员. 1986年7月毕业于湘潭大学数学专业,1989年7月获得中国科学院计算中心计算数学专业硕士学位,1996年7月获得中国科学院计算数学研究所计算数学专业博士学位,1998年在北京航空航天大学动力学流体机械专业博士后出站。长期从事偏微分方程数值解及其应用、计算流体力学的研究,主要研究内容包括:双曲型守恒律方程高分辨方法、非结构网格有限体积方法、非结构网格生成技术、辛几何算法、图像处理等研究。主持国家自然科学基金9项(2项重大研究计划培养项目、4项面上基金、3项专项基金),发表高水平科研论文80多篇,已经培养博士和硕士研究生40多名。享受军队优秀专业技术人才一类岗位津贴、获得军队育才奖,湖南省自然科学二等奖。现任中国计算数学学会常务理事、湖南省数学会常务理事、“计算数学”、“高校计算数学学报”等期刊编委。


报告9: 时间分数阶相场模型的能量稳定性


时间地点:201981011:20-12:00 孔子会堂

周涛简介:周涛,中科院数学与系统科学研究院副研究员,曾在瑞士洛桑联邦工学院从事博士后研究,长期从事不确定性量化高精度算法研究。2016年曾获中国工业与应用数学学会优秀青年学者奖,2017年入选中科院数学与系统科学研究院“陈景润未来之星”称号。2018年获国家自然科学基金委“优秀青年科学基金”。现担任国际不确定性量化期刊Inter. J. for Uncertainty Quantification副主编(Associate Editor in Chief),东亚应用数学杂志EastAsian Journal on Applied Mathematics执行主编(Managing Editor)。同时担任期刊Communications in Computational Physics和Numer. Math.: Theory, Methods and Applications的编委。


报告10: Volterra积分方程的连续配置方法的收敛性


时间地点:20198118:10-8:50 孔子会堂


梁慧简介:梁慧,哈尔滨工业大学(深圳)理学院教授。任SCI期刊Computational & Applied Mathematics编委、中国仿真学会仿真算法专委会委员、黑龙江省数学会理事。主要的研究方向为:延迟微分方程、Volterra积分方程、分数阶微分方程的数值分析。主持国家自然科学基金、青年基金、黑龙江省普通本科高等学校青年创新人才培养计划等9项科研项目,获中国系统仿真学会“2015年优秀论文”奖、2018第二届黑龙江省数学会优秀青年学术奖。目前共被SCI收录文章26篇,其中JCR一区15篇,发表在SIAM Journal on Numerical Analysis、IMA Journal of Numerical Analysis、Journal of Scientific Computing、BIT Numerical Mathematics、Advances in Computational Mathematics、Applied Numerical Mathematics 等15种不同的国际杂志上。


报告11: Some high-order numerical schemes for the nonlinear time fractional parabolic problems


时间地点:20198118:50-9:30 孔子会堂

报告摘要:Several high-order numerical methods are proposed to solve nonlinear time fractional parabolic problems with non-smooth solutions. The optimal error estimate in the $L^2$-norm is obtained without any time step restrictions dependent on the spatial mesh size. Such unconditional convergence results are proved by using the recent fractional discrete gronwall’s inequalities and the temporal spatial error splitting argument. Numerical experiments are presented to confirm the theoretical results.

李东方简介:李东方,华中科技大学数学与统计学院教授,中国系统仿真学会仿真算法专业委员会委员。主要从事微分方程数值解、系统仿真和信号处理等方面的研究。曾先后赴加拿大McGill大学,香港城市大学从事博士后研究。截至目前在SIAM. J. Numer. Anal. SIAM. J. Sci. Comput.,J. Comp. Phys., Appl. Comp. Harm. Appl.等多个国际著名计算学科SCI期刊上发表第一或者通讯作者论文30余篇。主持国家自然科学基金面上项目、青年基金各一项,博士后基金一项,参与多项国家自然科学基金。先后获得华中科技大学学术新人奖、香江学者奖等。


报告12: 非全局Lipschitz条件下随机微分方程的数值方法收敛性


时间地点:201981110:00-10:40 孔子会堂


王小捷简介:王小捷,中南大学数学与统计学院副教授,硕士生导师。2012年6月毕业于中南大学计算数学专业,获理学博士学位,2012年9月进入中南大学工作,2014年10月评上副教授职称。研究方向为随机偏微分方程以及随机常微分方程数值方法以及计算金融,发表SCI收录的学术论文21篇,一部分研究成果发表在 “SIAM Journal on Numerical Analysis”、“Mathematics of Computation”、“SIAM Journal on Scientific Computing”、“IMA Journal of Numerical Analysis”等计算数学国际顶尖刊物上。现主持一项国家自然科学基金面上项目、一项湖南省自然科学基金青年项目、中南大学第五批创新驱动计划项目和 2015年第二批“中南大学升华育英”人才计划项目。已主持完成一项国家自然科学基金青年项目、一项中国博士后科学基金特别资助项目和一项中国博士后科学基金面上项目。


报告13: Efficient gauge-invariant method for the time-dependent Ginzburg-Landau equations


时间地点:201981110:40-11:20 孔子会堂

报告摘要:We propose and analyze an efficient gauge-invariant method for numerical solution of the time-dependent Ginzburg-Landau (TDGL) equations in the two-dimensional space. The proposed method uses the well-known gauge-invariant finite difference approximations with staggered variables in a rectangular mesh, and a stabilized semi-implicit Euler discretization for time integration. The resulted fully discrete system leads to two decoupled linear systems at each time step, thus can be efficiently solved. We prove that the proposed method unconditionally preserves the point-wise boundedness of the solution and is also energy-stable. Moreover, the proposed method under the zero-electric potential gauge is shown to be equivalent to a mass-lumped version of the lowest order rectangular Nedelec edge element approximation and the Lorentz gauge scheme to a mass-lumped mixed finite element method. These indicate the method is also effective in solving the TDGL problems in non-convex domains although the solutions are often of low-regularity in such situation. Various numerical experiments are also presented to demonstrate effectiveness and robustness of the proposed method.

高华东简介:高华东,华中科技大学数学与统计学院副教授。分别在香港城市大学数学系(2014年),南开大学数学科学学院(2011年)和大连理工大学应用数学系(2008年)获得博士,硕士,学士学位。研究方向包括数值分析: 微分方程数值解, 有限元方法与差分方法, 尤其是对非线性抛物问题的数值求解与分析; 数学建模与计算物理: 多孔介质中热和水汽的传导流动, 计算超导现象, 计算微磁学, 计算电热学。目前主持面上基金一项,已正式发表论文十余篇。