AF: Medium: Collaborative Research:Numerical Algebraic Differential Equations

AF：媒介：协作研究：数值代数微分方程

基本信息

批准号：
1564132
负责人：
Chee Yap
金额：
$ 59.15万
依托单位：
New York University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-07-01 至 2021-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1564132&HistoricalAwards=false
关键词：
AF Medium Collaborative Research Numerical

项目摘要

Many basic physical principles, like conservation of mass or momentum for a fluid, are captured mathematically as systems algebraic differential equations. Simplifying and solving these systems (which means reducing the number or complexity of the equations, and finding inputs that satisfy all equations) are fundamental to applications in many areas, including cellular biology, approximation for chemical reaction systems, combinatorics, and analysis. The theoretical and algorithmic study of such systems spans more than a century, using three methods: purely symbolic, numerical, and hybrid symbolic-numeric. Symbolic methods (the quadratic formula being the simplest example) give the strongest guarantees of reliability, at a high (even exorbitant) cost in computational time and memory, since the same algorithm solves both mathematically hard and easy instances. Numerical methods (the basis for computational simulation) allow small errors or approximations for speed; small intermediate errors produce corrupted outputs on singular and ill-conditioned (that is, nearly singular) input instances. In this project, a hybrid symbolic-numeric approach will be developed. Hybrid algorithms are more adaptive and have lower complexity than symbolic algorithms, and can avoid the errors of numerical algorithms.In more technical detail, the three investigators apply existing and develop new methods of symbolic-numeric computation and differential algebra, producing algorithms that run on all inputs. They bring together existing methods of numerical algebraic geometry and software packages, such as Bertini, with recent theoretical results in differential algebra that provide upper bounds needed for guaranteed results. New near-optimal root isolation techniques are developed, implemented, and applied to solve systems of differential equations with finitely many solutions. The work spans from theory to producing practical tools.As part of this project the three investigators mentor and train students in symbolic and numeric computation at CUNY (noted for serving minority and low-income students) and NYU, and more broadly in New York City and Long Island, by activities ranging from developing a Symbolic-Numeric Computing course for graduate students at the Computer Science program of the CUNY Graduate Center and NYU, to advising high school students in projects.

许多基本物理原理，例如流体的质量守恒或动量守恒，在数学上都被捕获为系统代数微分方程。简化和求解这些系统（这意味着减少方程的数量或复杂性，并找到满足所有方程的输入）是许多领域应用的基础，包括细胞生物学、化学反应系统的近似、组合学和分析。对此类系统的理论和算法研究跨越了一个多世纪，使用了三种方法：纯符号、数值和符号-数值混合。符号方法（二次公式是最简单的例子）提供了最强有力的可靠性保证，但计算时间和内存成本很高（甚至过高），因为相同的算法可以解决数学上困难和简单的实例。数值方法（计算模拟的基础）允许较小的速度误差或近似值；小的中间错误会在奇异和病态（即接近奇异）输入实例上产生损坏的输出。在该项目中，将开发一种混合符号数字方法。混合算法比符号算法更具适应性，复杂度更低，并且可以避免数值算法的错误。在更多技术细节上，三位研究人员应用现有的符号数值计算和微分代数方法并开发新方法，产生了运行在所有输入。它们将现有的数值代数几何方法和软件包（例如 Bertini）与微分代数的最新理论结果结合在一起，提供了保证结果所需的上限。新的近最优根隔离技术被开发、实现并应用于求解具有有限多个解的微分方程组。这项工作涵盖从理论到生产实用工具。作为该项目的一部分，三位研究人员在纽约市立大学（以服务少数族裔和低收入学生而闻名）和纽约大学以及更广泛的纽约市指导和培训学生进行符号和数值计算和长岛，开展的活动包括为纽约市立大学研究生中心和纽约大学计算机科学项目的研究生开发符号数字计算课程，以及为高中生提供项目咨询。