1. ENERGY FUNCTIONS ON GRAPHS, WAVELETS, AND MULTILEVEL ALGORITHMS Wayne M. Lawton Department of Computational Science National University of Singapore Block S17, Level 7 10 Kent Ridge Crescent Singapore 119260 wlawton@cz3.nus.edu.sg

2. CONSTRAINED OPTIMIZATION ENERGY FUNCTIONS ON GRAPHS CONDITIONAL ELLIPTICITY OVERVIEW WAVELETS MULTILEVEL ALGORITHMS

3. CONSTRAINED OPTIMIZATION Minimize Solution With

4. Graph (nodes,edges) Nonzero Submatrices (n x n) of Hilbert Space ENERGY FUNCTIONS ON GRAPHS

5. Minimal Seminorm Interpolation, Ref. 1 Molecular Biology, Ref. 3,4 ENERGY FUNCTIONS ON GRAPHS Potential Energy Change Incremental Deformation Atoms Discretized Elliptic Boundary Value Problems, Ref. 2 Bonds External Force Traction Stiffness Matrix

6. 1. Atom Positions 3. Nodal Parameters WAVELETS Large and Sparse 2. Torsion Angles Boundary Element Method Small and Dense ParameterizationStiffness Matrix Small and Sparse Wavelet Discretization Analogies (N. P. describe position and orientation changes of protein atoms that separate 6 torsion angle bonds)

7. depends continuously on The solution CONDITIONAL ELLIPTICITY and of the constrained optimization problem iff is conditionally elliptic with respect to is elliptic, Ref. 5 Then

8. CONDITIONAL ELLIPTICITY

9. conditionally elliptic wrt MULTILEVEL ALGORITHMS

10. has the same form, multigrid algorithms, Ref. 5 Can also construct multiresolution analysis on stratified nilpotent Lie groups, Ref. 6

11. 1. Y. Yu, W. Lawton, S. L. Lee and S. Tan, “Wavelet based modeling of nonlinear systems”, pages 119-148 in Nonlinear Modelling: Black-Box Techniques, edited by Johanes A. K. Suykens and Joos Vandewalle, Kluwer, Boston, 1998. REFERENCES 2. W. Lawton, “Mathematical methods for active geometry”, Annals of Numerical Mathematics, Vol. 3, pages 163-180, 1996.

12. REFERENCES 3. W. Lawton, L. Ngee, T. Poston, R. Raghavan, S. R. Ranjan, R. Viswanathan, Y. P. Wang and Y. Yu, “Variational methods in biomedical computing”, pages 447-456 in Computational Science for the 21st Century, John Wiley, 1997. 4. W. Lawton, S. Meiyappan, R. Raghavan, R. Viswanathan, and Y. Yu, “Proteinmorphosis: a mechanical model for protein conformational changes”, submitted. 5. W. Lawton, “Conditional ellipticity and constrained optimization”, Computational Mathematics, Guangzhou,1997. 6. W. Lawton, “Infinite convolution products and refinable distributions on Lie groups”, to appear in Transactions AMS.