Workshop on Geometry and Symmetry in Numerical Computation

数值计算中的几何与对称性研讨会

• 批准号: 0509873
• 负责人: Simon Tavener
• 金额: $ 1.67万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0509873&HistoricalAwards=false
• 关键词: Workshop; Geometry; Symmetry; Numerical; Computation

Some of the most exciting developments in modern computational science have resulted from exploiting ideas in areas of mathematics not traditionally associated with numerical computation. Conversely, numerical techniques have been applied to solve computational problems arising in "non-traditional" fields. A good example is the fruitful interaction between computational mathematics and algebraic geometry. Singularity theory, which builds on ideas of algebraic geometry, has been embraced by computational scientists to compute paths of critical points in multi-parameter systems of differential and partial differential equations. Symmetries and group actions are used to create numerical methods for specific types of problems with significantly improved accuracy and stability properties. Techniques in algebraic geometry are also very useful for finding solutions of differential equations on manifolds, and are currently being applied to develop algorithms to compute decompositions of higher order tensors. On the other hand, numerical techniques for continuation, homotopy and symmetry provide the basis for methods in numerical algebraic geometry that are used to compute solution components of systems of polynomial equations. The ability to carry out the numerical decomposition of polynomial systems has yielded applications in mechanical engineering including the understanding and design of mechanisms that transmit, control, or constrain relative motion, robotics, control theory (pole placement), integer programming, and statistics. The development of hybrid exact/approximate methods for finding solutions of polynomial equations gives rise to issues of errors and stability that confront numerical analysts in many other contexts. The Workshop on geometry and symmetry in numerical computation will bring together experts from computational mathematics and algebraic geometry in order to explore and develop the potential in this rich interdisciplinary area. The program has been planned specifically to introduce and attract students and young investigators to this area. Each session will begin with an introductory lecture followed by four talks by leading experts. The introductory speakers will prepare a short "guide" describing some basic language and results that the audience can use during the invited talks. The one hour lectures themselves will be aimed towards an audience of advanced graduate students and researchers from different areas of mathematics. We expect the Workshop to break down disciplinary barriers and encourage cross-fertilization between researchers from algebraic geometry and numerical analysis. The lectures and discussion sections will encourage students and young researchers to become involved in the intersection of algebraic geometry and numerical analysis, and will provide stimulation and support for those already engaged in this activity. Potential outcomes range from improved methods to compute large complex physical systems governed by systems of partial differential equations, to advances in computational methods for general relativity, to new geometric methods for the analysis of large data sets, and to more efficient numerical methods for robotics and control.

现代计算科学中一些最激动人心的发展是利用传统上与数值计算无关的数学领域的思想而产生的。相反，数值技术已被应用于解决“非传统”领域中出现的计算问题。计算数学和代数几何之间卓有成效的互动就是一个很好的例子。建立在代数几何思想基础上的奇点理论已经被计算科学家们所接受，用于计算多参数微分方程组和偏微分方程组中临界点的路径。对称性和群作用被用来为特定类型的问题创建具有显著改进的精度和稳定性的数值方法。代数几何中的技巧在求流形上的微分方程解时也非常有用，目前正被应用于开发计算高阶张量分解的算法。另一方面，延拓、同伦和对称的数值技巧为数值代数几何中用于计算多项式方程组解分量的方法提供了基础。对多项式系统进行数值分解的能力已经在机械工程中产生了应用，包括理解和设计传递、控制或约束相对运动的机构、机器人学、控制理论(极点配置)、整数规划和统计学。求解多项式方程的精确/近似混合方法的发展引起了数值分析者在许多其他情况下面临的误差和稳定性问题。关于数值计算中的几何和对称性的讲习班将汇集计算数学和代数几何的专家，以探索和开发这一丰富的跨学科领域的潜力。该项目是专门为引进和吸引学生和年轻研究人员到这一领域而设计的。每期会议将以一次介绍性讲座开始，然后由主要专家进行四次演讲。介绍性的演讲者将准备一份简短的“指南”，描述一些基本的语言和结果，听众可以在受邀的演讲中使用。一小时的讲座本身将面向来自不同数学领域的高级研究生和研究人员。我们期望研讨会打破学科壁垒，鼓励来自代数几何和数值分析的研究人员之间的交叉交流。讲座和讨论部分将鼓励学生和年轻的研究人员参与代数几何和数值分析的交叉，并将为那些已经从事这一活动的人提供激励和支持。潜在的成果范围从计算由偏微分方程组控制的大型复杂物理系统的改进方法，到广义相对论计算方法的进步，到用于分析大数据集的新几何方法，以及用于机器人和控制的更有效的数值方法。