Preconditioning, analysis, and applications of numerical algebraic geometry methods

数值代数几何涉及使用数值方法从关于相应变种或方案的理想中提取数据。这一领域在过去20年中迅速发展，并在许多科学和工程领域得到了应用。这笔赠款资助了数值代数几何领域的四个项目。首先，PI和他的学生将研究同伦连续的几种形式的预条件，包括寻找(接近)最优的多重齐次和线性乘积开始系统的算法，以及使用对偶基来减少跟踪到“坏”终点的路径的数量。其次，PI和几个合作者将在三个应用领域开展工作：特殊机制(通过光纤产品)、用于代数几何相关应用的Macaulay2软件以及用于重复参数同伦的软件。第三，PI将致力于分析数值代数几何算法的复杂性。最后，PI和几位合作者将继续研究从标准延拓方法和Khovanskii-Rolle延拓方法中提取关于实代数集的信息的方法。多项式方程组出现在数学、科学和工程中的许多地方。一个完整的数学领域--代数几何--产生于寻找这类方程的解的需要。然而，直到20世纪60年代，还没有已知的求解此类方程组的通用技术。在那个时代开发的方法需要太多的内存才能有效，除了相对较小的问题。更新发展的方法-Sommese、Verschelde、Wanter、Li等人的数值方法，现在统称为数值代数几何-允许求解更大的多项式系统，从而使代数几何方法应用于更广泛的问题类别。然而，关于这些数值方法仍有许多需要理解的地方。该项目的目标包括解决这一方向上的四个未决问题。这项工作包括开发技术来简化其中一些计算，在流行和有用的软件包中实施有价值的算法，仔细分析与该领域的计算方法相关的计算成本，以及继续努力从作为这些方法的输出提供的数据中提取有用的真实世界数据。

Numerical algebraic geometry involves the use of numerical methods to extract data from ideals about the corresponding varieties or schemes. This area has grown rapidly over the last 20 years and has found applications in many areas of science and engineering. This grant is funding four projects in the area of numerical algebraic geometry. First, the PI and his students will investigate several forms of preconditioning for homotopy continuation, including algorithms for finding (near-)optimal multihomogeneous and linear product start systems, as well as the use of dual bases to reduce the number of paths tracked to "bad" endpoints. Second, the PI and several collaborators will work on three application areas: exceptional mechanisms (via fiber products), software in Macaulay2 for algebraic geometry-related applications, and software for repeated parameter homotopies. Third, the PI will work on analyzing the complexity of numerical algebraic geometry algorithms. Finally, the PI and several collaborators will continue to work on methods to extract information about real algebraic sets both from standard continuation methods and from Khovanskii-Rolle continuation. Polynomial systems of equations arise in many places throughout mathematics, science, and engineering. An entire mathematical field - algebraic geometry - grew out of the need to find solutions to these sorts of equations. Until the 1960s, though, there was no known general technique for solving such systems of equations. The methods developed in that era require too much memory to be effective except for relatively small problems. More recently developed methods - the numerical methods of Sommese, Verschelde, Wampler, Li, and others, now collectively known as numerical algebraic geometry - allow for the solution of much larger polynomial systems, opening the application of algebraic geometry methods to a wider class of problems. However, there is still much to understand about these numerical methods. The goals of this project include addressing four open problems in this direction. This work includes the development of techniques to streamline some of these computations, the implementation of valuable algorithms in popular and useful software packages, a careful analysis of the computational costs associated with the computational methods in this field, and the continued effort to extract useful real-world data from the data provided as output from these methods.