Symbolic Computation Meets Computational Geometry and Data Approximation

符号计算满足计算几何和数据逼近

• 批准号: 2048906
• 负责人: Henry Schenck
• 金额: $ 15万
• 依托单位: Auburn University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-02-15 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2048906&HistoricalAwards=false
• 关键词: Symbolic; Computation; Meets; Computational; Geometry; Symbolic; Computation; Meets; Computational; Geometry

Modeling of functions and data approximation are two fundamental tools in computational geometry, computer graphics, and data analytics. For example, when an airplane wing is designed, rather than modeling the wing as one large object, the wing is conceptually broken into a number of small triangles, computations are performed on these small patches, and then the results are assembled back together into a coherent result. This research project studies this technique, and others like it, using tools from computational and symbolic algebra. Results of the research are expected to advance theoretical understanding of these approximations, as well as to provide potential speed-ups to computations used in computer graphics and industrial design.This research focuses on analysis and implementation of three central methods in approximation theory and geometric modeling: generalized barycentric coordinates, multidimensional splines, and polynomial interpolation. These methods involve algebraic techniques; from the computational standpoint this means that tools of symbolic algebra are applicable. The goal in all these projects is to compute the dimension of some space of functions, generally in terms of combinatorial and geometric data, and if possible to find a basis for that space. The theoretical tools to be employed are homological algebra and algebraic geometry; the investigations also use computation as a vehicle for experiment. The project will result in development of specialized software, which will be integrated into the Macaulay2 software package.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

功能和数据近似的建模是计算几何，计算机图形和数据分析的两个基本工具。例如，当设计飞机机翼，而不是将机翼建模为一个大物体时，在概念上将机翼分解为许多小三角形时，在这些小斑块上进行了计算，然后将结果组装回相干结果。该研究项目使用计算和符号代数的工具研究了该技术以及其他类似的技术。这项研究的结果有望提高对这些近似值的理论理解，并为计算机图形和工业设计中使用的计算提供潜在的速度。本研究重点是分析和实施近似理论和几何学建模中的三种核心方法：广义barycentric coordination，多维旋转spline spline spline spline spline spline spline spline spline和comletomial interpolation。这些方法涉及代数技术；从计算角度来看，这意味着符号代数的工具适用。所有这些项目中的目标是计算某些功能空间的维度，通常是在组合和几何数据方面，如果可能的话，可以找到该空间的基础。要使用的理论工具是同源代数和代数几何。调查还使用计算作为实验的工具。该项目将导致专门软件的开发，该软件将集成到Macaulay2软件包中。该奖项反映了NSF的法定任务，并认为使用基金会的知识分子优点和更广泛的影响审查标准，认为值得通过评估来获得支持。