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CDS&E-MSS: Computational Developments of Power Series Methods to Solve Polynomial Systems

CDS

基本信息

批准号：
1854513
负责人：
Jan Verschelde
金额：
$ 23.6万
依托单位：
University of Illinois at Chicago
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1854513&HistoricalAwards=false
关键词：
CDS&E MSS Computational Developments Power

项目摘要

Polynomial systems occur in many mathematical models. For example, the design of a robot arm requires the solutions of a system of polynomial equations, as those solutions determine the design parameters of the robot arm, for it to reach desired positions. The proposed research project develops algorithms and software to solve polynomial systems more efficiently and accurately. Massively parallel algorithms on graphics processing units (GPUs) will be capable of solving large polynomial systems. A web server at www.phcpack.org will give anyone via the internet access to the developed software, the free and open source package PHCpack and its scripting interface phcpy. The award will provide support graduate student training through research.The proposed research addresses several computational challenges occurring in the exploitation of the sparse structure when solving a polynomial system. The tropical prevariety provides a collection of polyhedral cones of pretropisms, which are candidates leading forms of Puiseux series of positive dimensional solution sets of the polynomial system. As faces of cones may correspond to higher dimensional solution sets, the exploration of the tropical prevariety will proceed by generalized cascade homotopies which move from higher to lower dimensional solutions. The second computational challenge concerns the robustness of numerical algorithms to track solution paths defined by polynomial homotopies. Using rational approximations based on power series developments, singularities near a solution path are detected which allows to reduce the step size to avoid divergence from the solution path. Following the theory of analytic continuation, rational approximations are very effective tools to deduce information about singularities of the solution paths as function of the continuation parameter in the homotopy. The third challenge concerns the integration of pipelining, multithreading, and GPU acceleration into the solver, as needed to solve large problems.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.

多项式系统出现在许多数学模型中。例如，机器人臂的设计需要多项式方程组的解，因为这些解确定机器人臂的设计参数，以使其到达期望的位置。拟议的研究项目开发的算法和软件，以解决多项式系统更有效，更准确。图形处理器（GPU）上的大规模并行算法将能够解决大型多项式系统。www.phcpack.org上的一个网络服务器将使任何人都可以通过互联网访问所开发的软件、免费的开源软件包PHCPack及其脚本接口phcpy。该奖项将通过研究提供支持研究生培训。拟议的研究解决了在解决多项式系统时利用稀疏结构时出现的几个计算挑战。热带预变种提供了一个多面锥的预变种集合，它们是多项式系统的正维解集的Puiiiix级数的候选领导形式。由于锥面可能对应于更高维的解集，热带前变的探索将通过从高维向低维解移动的广义级联同伦进行。第二个计算挑战涉及跟踪由多项式同伦定义的解路径的数值算法的鲁棒性。使用有理逼近的基础上的幂级数的发展，奇异性附近的解决方案的路径检测，这允许减少步长，以避免从解决方案的路径divergence.Follow解析延拓理论，有理逼近是非常有效的工具，推导信息的奇异性的解决方案的路径作为连续参数的同伦函数。第三个挑战是将流水线、多线程和GPU加速集成到求解器中，以解决大型问题。该奖项反映了NSF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。

项目成果

期刊论文数量（8）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Exporting Ada Software to Python and Julia

将 Ada 软件导出到 Python 和 Julia

DOI：
10.1145/3577949.3577961
发表时间：
2022
期刊：
ACM SIGAda Ada Letters
影响因子：
0
作者：
Verschelde, Jan
通讯作者：
Verschelde, Jan

Locating the Closest Singularity in a Polynomial Homotopy

DOI：
10.48550/arxiv.2205.07380
发表时间：
2022-05
期刊：
影响因子：
0
作者：
J. Verschelde;Kylash Viswanathan
通讯作者：
J. Verschelde;Kylash Viswanathan

Robust Numerical Tracking of One Path of a Polynomial Homotopy on Parallel Shared Memory Computers

并行共享内存计算机上多项式同伦一条路径的鲁棒数值跟踪

DOI：
10.1007/978-3-030-60026-6_33
发表时间：
2021
期刊：
vol 12291
影响因子：
0
作者：
Telen, Simon;Van Barel, Marc;Verschelde, Jan
通讯作者：
Verschelde, Jan

Accelerated Polynomial Evaluation and Differentiation at Power Series in Multiple Double Precision

多重双精度幂级数的加速多项式计算和微分

DOI：
10.1109/ipdpsw52791.2021.00111
发表时间：
2021
期刊：
2021 IEEE International Parallel and Disributed Processing Symposium Workshops (IPDPSW
影响因子：
0
作者：
Verschelde, Jan
通讯作者：
Verschelde, Jan

Parallel Software to Offset the Cost of Higher Precision

并行软件可抵消更高精度的成本