Regularization of Hypersensitive Problems for Numerical Computation with Empirical Data

Northeastern Illinois University, Chicago IL

Investigators

Abstract

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.