The mock data for the absorbance are generated by an Abel transformation of the density function:

(15) |

which is evaluated at equally spaced points and multiplied by the model matrix defined by (7). Then, Gaussian noise with zero mean and standard deviation

Before starting the reconstruction by pressing the button `Compute`,
the standard deviation of the data points corresponding to in
(11), the value of the default model
which is
chosen flat and the initial value of the hyperparameter have to be
specified. Moreover, the number of data points `N_data` and the number
of control points `N_knots` for spline interpolation may be set.

- Increase the initial value for
`alpha`to`1.e6`and watch how the reconstruction converges, starting from the default model. - Set the default model to
`1.e-1`: The reconstructed density no longer vanished at . A look at Fig. 1 reveals that there is only one measurement taking into account the density at the center, while all measurement include information on . Therefore, the density near the center is mainly determined by our choice for the default model -- in case of insufficient data we will get what we used as input! - Increase both
`Noise`and`Standard deviation`to`0.03`and`0.003`; in the latter case you may need to increase the number of knot points for the spline interpolation, for otherwise there are not enough degrees of freedom to reduce the misfit .Note that you will probably not achive convergence if you choose the noise level greater that the standard deviation for the reconstruction.

- Restore default values (press
`Reset`) and try different numbers of knot points`N_knot`. For too few points the criterion cannot be satisfied, if too many points are used, ripples appear in the reconstructed density -- the data are overfitted.

Danilo Neuber 2003-10-03