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Symbolic Computation Meets Computational Geometry and Data Approximation

符号计算满足计算几何和数据逼近

基本信息

批准号：
1818646
负责人：
Henry Schenck
金额：
$ 15万
依托单位：
Iowa State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-06-15 至 2021-03-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1818646&HistoricalAwards=false
关键词：
Symbolic Computation Meets Computational Geometry

项目摘要

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.

函数建模和数据近似是计算几何、计算机图形学和数据分析中的两个基本工具。例如，当设计飞机机翼时，不是将机翼建模为一个大物体，而是将机翼在概念上分成许多小三角形，对这些小块进行计算，然后将结果组合在一起形成一个连贯的结果。本研究项目使用计算代数和符号代数的工具来研究这种技术，以及其他类似的技术。研究结果有望促进对这些近似的理论理解，并为计算机图形学和工业设计中使用的计算提供潜在的加速。本研究著重在分析和实现近似理论和几何建模中的三种中心方法：广义质心坐标、多维样条和多项式插值。这些方法涉及代数技巧；从计算的角度来看，这意味着符号代数的工具是适用的。所有这些项目的目标都是计算一些函数空间的维数，通常是根据组合和几何数据，如果可能的话，找到该空间的基。使用的理论工具是同调代数和代数几何；研究也使用计算作为实验的载体。该项目将开发专门的软件，这些软件将集成到Macaulay2软件包中。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。