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

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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.