Next: Methods to Find Global Minima
A P Hammersley
ESRF, BP 220, 38043 Grenoble, France
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:
Here practical solutions to these problems are discussed in detail.