A transformed function is a coarse approximation to the original function, with small and narrow minimizers being removed while the overall structure of the function is maintained. This property allows a local minimization procedure to skip less interesting local minimizers and to concentrate on regions with low function values, where a global minimizer is most likely to be located.

The smoothing transform, called the Gaussian transform, depends on a parameter that controls the degree of smoothing. The original function is obtained if the parameter is zero, while smoother functions are obtained as the parameter increases.

Further information on Gaussian smoothing can be found in the following papers:

- Jorge Moré and Zhijun Wu,
**Distance geometry optimization for protein structures,**Journal on Global Optimization, 15 (1999), pp. 219-234. - Jorge Moré and Zhijun Wu,
**Epsilon-optimal solutions to distance geometry problems via global continuation,**in Global Minimization of Nonconvex Energy Functions: Molecular Conformation and Protein Folding, P. M. Pardalos, D. Shalloway, and G. Xue, eds., pages 151-168, American Mathematical Society, 1996. - Jorge Moré and Zhijun Wu,
**Issues in large-scale global molecular optimization,**in Large Scale Optimization with Applications: Molecular Structure and Optimization, Lorenz T. Biegler, Thomas F. Coleman, Andrew R. Conn and Fadil N. Santosa, eds., IMA Volumes in Applied Mathematics and Applications 94, pages 99-122, Springer-Verlag, 1997. - Jorge Moré and Zhijun Wu,
**Smoothing techniques for macromolecular global optimization,**in Nonlinear Optimization and Applications, G. Di Pillo and F. Giannessi, eds., pages 297-312, Plenum Press, 1996. - Jorge Moré and Zhijun Wu,
**Global continuation for distance geometry problems,**SIAM J. Optim. 7, no. 3 (August 1997), pp. 814-836.

Solutions have been obtained
for a set of model problems with up to 1500 variables, with
excellent performance on the IBM SP.
We have also used the algorithms
to determine the structure of protein fragments,
such as the the active fragment of E. Coli STh enterotoxin shown here,
or the DNA-binding protein shown above.

Stable configurations for a set of small systems have been obtained by using Gaussian smoothing algorithms on the IBM Quad. We are now working on larger ionic systems. An important goal of this work is to find the stable configurations for very large systems, say, systems of 200,000 ions. The optimal structure with 200 ions is shown here.

This work is joint with John Schiffer and Steve Pieper
of Argonne's Physics division.