SI2-SSE: Collaborative Proposal: Symbolic-Numeric Approaches to Polynomials

SI2-SSE：协作提案：多项式的符号数值方法

• 批准号: 1440467
• 负责人: Daniel Bates
• 金额: $ 14.93万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1440467&HistoricalAwards=false
• 关键词: SI2; SSE; Collaborative; Proposal; Symbolic

Polynomial systems arise naturally in many areas of human endeavor. These include the modeling of tumor growth; the design of robotic devices; chemical systems arising in areas ranging from combustion to blood clotting; assorted problems in physics; plus many areas with mathematics. The solution of the polynomial systems answers questions critical to these endeavors. This research will be devoted to developing the next generation of Bertini, an open source software package, which has been used successfully by many researchers on many problems, which include all those mentioned above.Bertini will be rewritten in C++ to be scriptable and modular, which will allow it to be interfaced transparently with symbolic software. The new Bertini will include tools allowing the user to construct and manipulate homotopies based on the output of Bertini. A major focus of the research will be given to systems of polynomials arising from the discretization of systems of differential equations. The great challenge of these very large systems of polynomials is balanced by the great potential impact new efficient and robust methods of solution will have.

多项式系统自然出现在人类努力的许多领域。其中包括肿瘤生长的建模；机器人设备的设计；从燃烧到血液凝固等领域出现的化学系统；各种物理问题；加上许多数学领域。多项式系统的解决方案回答了对这些努力至关重要的问题。这项研究将致力于开发下一代 Bertini，这是一个开源软件包，许多研究人员已成功使用该软件包解决了许多问题，包括上述所有问题。Bertini 将用 C++ 重写，使其可脚本化和模块化，这将允许它与符号软件透明地连接。新的 Bertini 将包括允许用户根据 Bertini 的输出构建和操作同伦的工具。研究的一个主要焦点将是由微分方程组的离散化产生的多项式系统。这些非常大的多项式系统的巨大挑战与新的高效且稳健的解决方法所具有的巨大潜在影响相平衡。

项目成果

Homotopies for Connected Components of Algebraic Sets with Application to Computing Critical Sets

代数集连通分量的同伦及其在计算关键集中的应用

• DOI: 10.1007/978-3-319-72453-9_8
• 发表时间: 2017-11-15
• 作者: Daniel J. Bates;Daniel A. Brake;J. Hauenstein;A. Sommese;C. Wampler
• 通讯作者: C. Wampler