Optimisation

The fitting of the model to polarization data involves adjusting the brightness of emission points on the surface of the white dwarf. An array of 1740 elements is used to represent points on the white dwarf surface.As the fitting proceeds, the brightness along the surface of the white dwarf is adjusted in order to minimise the terms in the following equation:

where x is any one solution. The first term is the chi squared of the model fit to the data, where k is the number of data points. The second term is the regularisation term which is a function that leads to the smoothest possible solution. The regularisation term (similar to the Tikhonov regularisation, see Piskunov et. al. 1990 and references therein) is simply the difference in brightness between an emission point ( an array element ) and its four nearest companions (l), squared and summed for all emission points over the white dwarf surface. Lambda is the Lagrangian multiplier and it defines the strength of the regularisation term. This simple but effective regularisation scheme finds the locally smoothest possible solution of the above equation that still fits the data.

The first stage in optimising the data then proceeds with the use of a Genetic Algorithm and then refined with the use of the Powells method.