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，都可以用代数微分方程系统的数学形式表达出来。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。这种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、数值和混合符号-数字。符号方法（二次公式是最简单的例子）提供了最强的可靠性保证，但在计算时间和内存方面的成本很高（甚至过高），因为相同的算法可以解决数学上困难和容易的实例。数值方法（计算模拟的基础）允许速度的小误差或近似值；小的中间错误会在奇异和病态（即几乎奇异）输入实例上产生损坏的输出。在这个项目中，将开发一种混合符号-数字方法。混合算法比符号算法具有更强的适应性和更低的复杂度，并且可以避免数值算法的误差。在更多的技术细节上，三位研究者应用了现有的和开发的符号数值计算和微分代数的新方法，产生了在所有输入上运行的算法。他们将现有的数值代数几何方法和软件包（如Bertini）与最近的微分代数理论结果结合在一起，这些理论结果提供了保证结果所需的上界。新的近最优根隔离技术被开发、实施并应用于求解具有有限多解的微分方程系统。这项工作的范围从理论到生产实用工具。作为该项目的一部分，三位研究者在纽约市立大学（以服务少数族裔和低收入学生而闻名）和纽约大学，以及纽约市和长岛更广泛地指导和培训学生的符号和数值计算，其活动范围从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-数值计算课程，到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.