AF: Medium: Collaborative Research: Numerical Algebraic Differential Equations

许多基本的物理原理，如流体的质量守恒或动量守恒，在数学上都是代数微分方程组。 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，组合数学和分析。 这种系统的理论和算法研究跨越了一个多世纪，使用三种方法：纯符号，数值和混合符号-数值。 符号方法（二次公式是最简单的例子）给出了可靠性的最强保证，但在计算时间和内存方面的成本很高（甚至过高），因为同一算法可以解决数学上困难和容易的实例。 数值方法（计算模拟的基础）允许小的误差或近似的速度;小的中间误差会在奇异和病态（即近似奇异）输入实例上产生损坏的输出。在这个项目中，将开发一种混合的符号-数字方法。混合算法比符号算法具有更好的适应性和更低的复杂性，并且可以避免数值算法的错误。在更多的技术细节中，三位研究人员应用现有的符号-数值计算和微分代数的新方法，产生在所有输入上运行的算法。他们汇集了现有的方法，数值代数几何和软件包，如贝尔蒂尼，最近的理论成果，在微分代数，提供上限所需的保证结果。新的近最佳根隔离技术的开发，实施，并应用于解决系统的微分方程与众多的解决方案。 工作范围从理论到生产实用工具。作为这个项目的一部分，三名研究人员在纽约市立大学指导和培训学生进行符号和数值计算（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和长岛，通过各种活动，从发展一个象征性的，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数值计算课程，为高中生提供项目建议。

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.