** Next:** Methods to Find Global Minima

**A P Hammersley
ESRF, BP 220, 38043 Grenoble, France
E-mail: hammersley@esrf.fr**

The manner in which model fitting is implemented in **FIT2D** is presented
with emphasis on practical methods to overcome often stated problems.

Model fitting is the most powerful method of data analysis, although this statement should be balanced by also noting that it may also be the most biased. All depends on the suitability of the assumed model. If the model is not ``correct'' the results will be biased. As well as being used for 1-D and 2-D function fitting the same optimisation ``engine'' is used for less obvious model fitting problems, such as refining the experimental geometry from the shape of powder rings from a calibrant sample of accurately known D-spacings. An iterative non-linear Levenberg-Marquardt type algorithm with numerically calculated derivatives is used (Gill, Murray & Wright, 1981). This results in a robust and versatile system.

Note: All optimisation problems can be expressed mathematically as minimisation problems, and thus the terms ``minimisation'', ``minima'', and ``descent methods'' are often used.

There are three main concerns common to almost all optimisation problems:

- 1.
- Finding the global minimum and avoiding local minima.
- 2.
- Changes in the parameter values causing catastrophic changes in the sensitivity of the function being optimised (termed the ``objective function'').
- 3.
- Missing or noisy data preventing a unique solution, and requiring methods to add this uniqueness criterion.

Here practical solutions to these problems are discussed in detail.

- Methods to Find Global Minima
- Parameter Scaling
- Solution Uniqueness and Uniqueness Criteria
- Acknowledgements
- References
- About this document ...

6/11/1998