Analyzing & Improving Numerical Algorithms for Eigenproblems Using Singularity Theory and Riemannian Geometry

这项工作得到了国家科学基金会教师早期职业发展奖的支持。研究部分将侧重于应用新的几何思想来分析和改进数值算法。该计划的一部分将涉及推导和研究格拉斯曼流形的优化技术，该技术适用于求解凝聚态物理和材料科学中使用的薛定谔方程的局部密度近似的Kohn-Sham方程。该计划的第二部分涉及应用奇点理论来分析和解释在特征系统分析和系统与控制中广泛使用的某些数值算法的行为。教育部分将包括麻省理工学院科学计算新课程的进一步发展。带有投影系统的计算机将用于数值分析和应用数学的本科和研究生课程，以便为材料带来新的见解。还将对新的研究人员进行指导和培训，并将把研究和教育活动结合起来。美国国家科学基金会强烈鼓励教师作为教育工作者和研究人员的早期发展。教师早期职业发展（Career）计划是一个基金会范围内的计划，为初级教师的整体职业发展提供支持。它在一个项目中结合了对最广泛意义上的高质量研究和教育的支持，以及那些传统上在科学和工程领域代表性不足的人的充分参与。该计划加强并强调了基金会对包括研究和教育在内的全面，平衡的学术生涯发展的重要性。

9501278 Edelman This work is supported by a National Science Foundation Faculty Early Career Development Award. The research component will focus on the application of new geometrical ideas to analyze and improve numerical algorithms. Part of the program will involve the derivation and study of an optimization technique on the Grassman manifold suitable for solving the Kohn-Sham equations of the local density approximation to Schroedinger's equation used in condensed matter physics and in materials science. The second part of the program involves the application of singularity theory to analyze and explain the behavior of certain numerical algorithms widely used in eigensystem analysis and systems and control. The education component will include further development of a new course in scientific computing at the Massachusetts Institute of Technology. Computers with a projection system will be used in undergraduate and graduate courses in numerical analysis and applied mathematics in order to bring new insights into the material. Mentoring and training of new researchers will also take place and will integrate the research and educational activities. The National Science Foundation strongly encourages the early development of academic faculty as both educators and researchers. The Faculty Early Career Development (CAREER) Program is a Foundation- wide program that provides for the support of junior faculty within the context of their overall career development. It combines in a single program the support of quality research and education in the broadest sense and the full participation of those traditionally underrepresented in science and engineering. This program enhances and emphasizes the importance the Foundation places on the development of full, balanced academic careers which include both research and education.