AF: Medium: Collaborative Research: Numerical Algebraic Differential Equations

AF：媒介：协作研究：数值代数微分方程

• 批准号: 1563942
• 负责人: Alexey Ovchinnikov
• 金额: $ 60.82万
• 依托单位: CUNY Queens College
• 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 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1563942&HistoricalAwards=false
• 关键词: AF; Medium; Collaborative; Research; Numerical

Many basic physical principles, like conservation of mass or momentum for a fluid, are captured mathematically as systems of 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.

许多基本的物理原理，例如对流体的质量保护或动量，被数学捕获为代数微分方程的系统。 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式的投入）是许多领域的应用，包括蜂窝生物学，化学反应系统，组合剂和分析的近似值。 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：纯粹的符号，数值和混合符号数字。 符号方法（二次公式是最简单的示例）给出了可靠性的最强保证，因为相同的算法在数学上硬化和轻松实例上都解决了计算时间和内存的高（甚至过高）成本。 数值方法（计算模拟的基础）允许速度较小的错误或近似值；小的中间错误会在单数和不良条件（即几乎是单数）的输入实例上产生损坏的输出。在这个项目中，将开发一种混合符号数字方法。与符号算法相比，混合算法更具适应性，并且具有较低的复杂性，并且可以避免数值算法的错误。在更技术细节中，这三名研究人员应用了现有的符号数字计算和差异代数的新方法，并产生了所有输入的算法。他们将现有的数值代数几何形状和软件包（例如Bertini）的方法汇总在一起，以及差异代数的最新理论结果，这些差异代数提供了保证结果所需的上限。开发，实施并应用了新的近乎最佳的根隔离技术，以求解具有许多解决方案的微分方程系统。 这项工作跨越从理论到生产实用工具。作为该项目的一部分，三位调查人员的导师和培训学生在CUNY（以少数群体和低收入学生的服务）和纽约大学的象征和数字计算，以及在纽约市和长岛上，通过开发一项象征性的计算学生的毕业生，包括Cun compersy Science for Cun cunsy compersy for Cun compersy comply for Cun compersy comply the Cun cunsy for cun，项目。