Symbolic Computation Meets Computational Geometry and Data Approximation

函数建模和数据逼近是计算几何、计算机图形学和数据分析中的两个基本工具。例如，在设计飞机机翼时，不是将机翼建模为一个大对象，而是在概念上将机翼分解为多个小三角形，在这些小面片上执行计算，然后将结果重新组装成一致的结果。这个研究项目使用计算和符号代数中的工具来研究这种技术以及其他类似的技术。本研究重点分析和实现了逼近理论和几何建模中的三种核心方法：广义重心坐标、多维样条和多项式插值。这些方法涉及代数技术；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某些函数空间的维度，通常是根据组合和几何数据，如果可能的话，为该空间找到一个基础。要使用的理论工具是同调代数和代数几何；研究还使用计算作为实验的工具。该项目将导致开发专门的软件，并将其整合到Macaulay2软件包中。该奖项反映了NSF的法定使命，并已通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.