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

CONSTRAINED OPTIMIZATION

ENERGY FUNCTIONS ON GRAPHS

CONDITIONAL ELLIPTICITY

OVERVIEW

WAVELETS

MULTILEVEL ALGORITHMS

CONSTRAINED OPTIMIZATION

Minimize

wCv

Solution

Lv)v,v(a2

1)v(J

With

w

v

0C

CA *

Graph (nodes,edges) ),( 10

Nonzero Submatrices (n x n) of

Hilbert Space 0nRH

A

ENERGY FUNCTIONS ON GRAPHS

1

Minimal Seminorm Interpolation, Ref. 1

Molecular Biology, Ref. 3,4

0

ENERGY FUNCTIONS ON GRAPHS

J Potential Energy Change

Incremental Deformation

Atoms

Discretized Elliptic Boundary Value Problems, Ref. 2

1 Bonds

03RH

L External Force

C Traction

A Stiffness Matrix

1. Atom Positions

3. Nodal Parameters

21

WAVELETS

Large and Sparse

2. Torsion Angles

Boundary Element Method

Small and Dense

Parameterization Stiffness Matrix

Small and Sparse

32 Wavelet DiscretizationAnalogies

(N. P. describe position and orientation changes of protein atoms that separate 6 torsion angle bonds)

depends continuously on

The solution

CONDITIONAL ELLIPTICITY

and

of the constrained optimization problemv

w iff

is conditionally elliptic with respect to

C),(a

)C(v,||v||)v,v(a0 2

is elliptic, Ref. 5)C,C(),(a Then

TDTA *

CONDITIONAL ELLIPTICITY

2222

1111

AA

AAA

I0

BIT

22

11

D0

0AD

121

11 AAB 121

11212222 AAAAD

conditionally elliptic wrt 1i CC ),(a1

3C

2C

1 HHH 21

MULTILEVEL ALGORITHMS

}w)v(CC|)v,v(a{max)w,w(a 1i1i

iii Hw),w,w(a)w,w(a)w,w(aiii

restrictCi )ZM(H 21i2i

MULTILEVEL ALGORITHMS

1|nj||mi|

2|)n,m(w)j,i(w|)w,w(a1

),(ai

11

11M

has the same form, multigrid algorithms, Ref. 5

Can also construct multiresolution analysis on stratified nilpotent Lie groups, Ref. 6

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.

REFERENCES3. 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.