Collaborative Research: AF: Small: Real Solutions of Polynomial Systems

合作研究：AF：小：多项式系统的实数解

• 批准号: 2331400
• 负责人: Jonathan Hauenstein
• 金额: $ 30万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-01-01 至 2026-12-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2331400&HistoricalAwards=false
• 关键词: Collaborative; Research; AF; Small; Real

Many problems in science, engineering, and industry can be formulated as computing real solutions to systems of nonlinear polynomial equations. Some examples include synthesizing a mechanical linkage for a robot to perform given tasks, analyzing chemical reaction networks, and computing bifurcations in a biological process. Due to this ubiquity, there is a strong demand to tackle many large systems that naturally arise in real world settings. This project aims to develop new algorithmic approaches for efficiently and rigorously computing real solutions and test these new approaches on a variety of problems in science and engineering. Furthermore, this project will directly support the training and mentoring of graduate and undergraduate students, and will impact many other students through curriculum development in computational real algebraic geometry related to this research project. The technical aims of this project are divided into two thrusts. The first thrust develops efficient and rigorous algorithms for computing real solutions to systems of polynomial equations that have only finitely many solutions. This project focuses on developing new mathematical theories and algorithms using homotopy continuation for computing either at least one or all real solutions without needing to compute all complex solutions. For many real world problems of interest, the number of complex solutions is many orders of magnitude larger than the number of real solutions so avoiding non-real solutions is beneficial for algorithmic efficiency. The second thrust develops an algorithm for sampling smooth points in each connected component of the set of real solutions for polynomial systems with infinitely many solutions. By reducing to sample points, the algorithms from the first thrust can be applied to such positive dimensional systems. Moreover, sampling real smooth points is useful for determining the dimension of the real solution set and deciding connectedness.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.

科学、工程和工业中的许多问题都可以表述为计算非线性多项式方程组的实数解。 一些示例包括合成机器人执行给定任务的机械连杆、分析化学反应网络以及计算生物过程中的分叉。 由于这种普遍存在，因此迫切需要解决现实世界环境中自然出现的许多大型系统。 该项目旨在开发新的算法方法，以高效、严格地计算实际解决方案，并在科学和工程中的各种问题上测试这些新方法。 此外，该项目将直接支持研究生和本科生的培训和指导，并将通过与该研究项目相关的计算实代数几何课程开发来影响许多其他学生。 该项目的技术目标分为两个主旨。 第一个推动力开发高效且严格的算法，用于计算只有有限多个解的多项式方程组的实数解。 该项目专注于开发新的数学理论和算法，使用同伦延拓来计算至少一个或所有实数解，而不需要计算所有复杂的解。 对于许多感兴趣的现实世界问题，复杂解决方案的数量比实际解决方案的数量大许多数量级，因此避免非实际解决方案有利于算法效率。 第二个推动力开发了一种算法，用于对具有无限多个解的多项式系统的实解集的每个连通分量中的平滑点进行采样。 通过减少样本点，第一推力的算法可以应用于此类正维系统。 此外，对真实平滑点进行采样对于确定真实解决方案集的维度和决定连通性很有用。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。