Optimal Polyhedral Homotopies on Supercomputers for Algebraic Sets

超级计算机上代数集的最优多面体同伦

• 批准号: 0713018
• 负责人: Jan Verschelde
• 金额: $ 25.5万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0713018&HistoricalAwards=false
• 关键词: Optimal; Polyhedral; Homotopies; Supercomputers; Algebraic

Algebraic sets are solutions of polynomial systems. Exploiting the sparse structure of a polynomial system, polyhedral homotopies are optimal for computing all isolated solutions of general systems in the sense that every solution path converges to a solution. The goal of this proposal is to develop new homotopy algorithms that will be generically optimal for computing numerical representations of positive dimensional solution sets. To solve large polynomial systems, the new homotopy algorithms will be made suitable to run on supercomputers.Polynomial systems arise frequently in many problems in science and engineering. The algorithms to solve polynomial systems are implemented in an open source software package PHCpack, available for free on the web.User-friendly interfaces to PHCpack (in computer algebra systems such as Maple (commercial), SAGE (open source), and in scientific software systems such as MATLAB (commercial), Octave (open source)) will continue to benefit many scientists and engineers who solve polynomial systems in their research.The principal investigator teaches his students to transfer mathematical technology through software with an eye towards high performance computing.

代数集是多项式系统的解。 利用多项式系统的稀疏结构，多面体同伦是计算一般系统的所有孤立解的最佳选择，因为每个解路径都收敛到一个解。 该提案的目标是开发新的同伦算法，该算法对于计算正维解集的数值表示通常是最佳的。 为了解决大型多项式系统，新的同伦算法将适合在超级计算机上运行。多项式系统在科学和工程中的许多问题中经常出现。 求解多项式系统的算法在开源软件包 PHCpack 中实现，可在网络上免费获得。PHCpack 的用户友好界面（在计算机代数系统中，例如 Maple（商业）、SAGE（开源），以及在科学中） MATLAB（商业）、Octave（开源）等软件系统将继续使许多在研究中求解多项式系统的科学家和工程师受益。首席研究员教他的学生通过软件转移数学技术，着眼于高性能计算。