Regularization of Hypersensitive Problems for Numerical Computation with Empirical Data

这个项目的目的是发展正则化理论，稳健的数值算法，以及一个针对已知对数据扰动高度敏感的问题的软件包。代数计算中的一些基本问题仍然处于数值分析的前沿，需要可靠的算法和软件，这些问题就是这种性质。在以往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去策略，验证正则化理论，并生成软件NACLab。这项研究试图在数值分析、计算机代数、代数几何和微分拓扑学等科学领域架起桥梁。众所周知，超敏感问题在实际计算中是一个巨大的挑战，特别是在不可避免地使用经验数据的情况下。在解决这些问题方面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创新的几何分析，证明的正则化理论和有效的计算方法，以消除基础代数问题中可怕的超敏感性。这个项目本质上是多学科的，同时也是一个强大的、黑匣子类型的、公开可用的软件工具箱NACLab的主要成果，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础。该软件将为机器人、分子构象、化学平衡、纳什平衡、自动控制等应用领域以及代数几何等其他数学分支提供关键工具。

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.