OVERVIEW OF REFINEMENT AND LEAST-SQUARES METHODS. Dale E. Tronrud, Dept. of Chemistry, University of Oregon, Oregon, USA

The process of refinement is a large problem in function minimization. To reduce the amount of computation the methods chosen to minimize the function incorporate a number of assumptions. When these assumptions break down special procedures must be used.

The methods of minimization used in macromolecular refinement span the range from Simulated Annealing to Full-Matrix Least-Squares. The properties of Simulated Annealing, it being a stochastic method, are difficult to characterize and will be only touched upon. The other methods commonly used are classified as gradient descent and include Steepest Descent, Conjugate Gradient, and Preconditioned Conjugate Gradient (also known as Conjugate Direction). The Full-Matrix method can only be applied to small proteins whose crystals diffract to high resolution because of the huge amount of computer resources it requires.

Each of the gradient descent procedures are derived by making specific assumptions about the nature of the function being minimized. Because these assumptions usually are not valid for the crystallographic residual the methods will fail unless special precautions are taken by the crystallographer.

Many of these precautionary procedures are commonly known, such as rigid body refinement, but an understanding of the details of the methods themselves allows one to know when and what procedure to apply.

This talk will describe the various minimization methods used today and their relationships to one another. The assumptions and resulting limitations of each method will be discussed along with, where they exist, suggestions for diagnostics which should be monitored. Where there are no diagnostics for certain limitations, procedures will be given which must be applied blindly to prevent the refinement from "hanging up".