Regularization of Hypersensitive Problems for Numerical Computation with Empirical Data

该项目的目的是发展正则化理论、鲁棒的数值算法和一个软件包，用于解决已知对数据扰动高度敏感的问题。代数计算中的一些基本问题仍然处于数值分析的前沿，并且需要可靠的算法和软件，就是这种性质。在先前NSF支持下开发的新理论和算法/软件的基础上，PI建议设计缺陷特征值问题的算法，开发多项式系统的数值消除策略，验证正则化理论，并制作软件NAClab。本研究试图将数值分析、计算机代数、代数几何和微分拓扑等科学领域连接起来。众所周知，在实际计算中，特别是当不可避免地使用经验数据时，超敏感问题是一个巨大的挑战。在解决这些问题方面取得的进展将使广泛的应用成为可能。这个项目的智力价值在于一个创新的几何分析，证明了正则化理论和一个有效的计算方法，以消除在基本代数问题中可怕的超敏感性。该项目本质上是多学科的，其主要成果是一个强大的、黑盒型的、公开可用的软件工具箱NAClab，以解决科学/工程中出现的高度敏感的代数问题，并作为未来算法开发的基石。该软件将为机器人、分子构象、化学平衡、纳什平衡、自动控制以及代数几何等其他数学分支等应用领域提供关键工具。

The aim of this project is the development of regularization theories, robust numerical algorithms, and a software package for problems that are known to be highly sensitive to data perturbations. Some of the fundamental problems in algebraic computation that remain at the frontier in numerical analysis, and where reliable algorithms and software are in demand, are of this nature. Extending on novel theories and algorithms/software developed under previous NSF support, the PI proposes to design algorithms for defective eigenvalue problems, to develop a numerical elimination strategy for polynomial systems, to validate the regularization theories, and to produce software, NAClab. This research attempts to bridge scientific fields of numerical analysis, computer algebra, algebraic geometry, and differential topology. Hypersensitive problems are known to be formidable challenges in practical computation particularly when empirical data are inevitably used. Advances in attacking those problems will enable wide range of applications. The intellectual merit of this project lies in an innovative geometric analysis, proven regularization theory and an effective computational methodology for striking out the dreaded hypersensitivity in fundamental algebraic problems. This project is multidisciplinary in nature along with a major outcome in a robust, blackbox-type, and publicly available software toolbox NAClab to solve highly sensitive algebraic problems arising in sciences/engineering and to serve as building blocks for future algorithmic development. The software will supply critical tools for application areas such as robotics, molecular conformation, chemical equilibrium, Nash equilibria, automatic control, as well as other branches of mathematics such as algebraic geometry.