Exercises

Using the Program

The mock data for the absorbance $\boldsymbol{A}$ are generated by an Abel transformation of the density function:

$$\rho(r) = r^2 (1-r^2) + \frac{1}{5} \exp \left\{ -250\left(r-\frac{1}{4}\right)^2 \right\},$$ (15)

which is evaluated at $N_0 = \mbox{\texttt{N\_grid}}$ equally spaced points and multiplied by the model matrix $\tilde{M}$ defined by (7). Then, Gaussian noise with zero mean and standard deviation Sigma noise is added; seed may be changed to obtain different sequencies of normally distributed random numbers.

Before starting the reconstruction by pressing the button Compute, the standard deviation of the data points corresponding to $\sigma_i$ in (11), the value of the default model $\boldsymbol{m}$ which is chosen flat and the initial value of the hyperparameter $\alpha$ 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.

Suggested exercises

• 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 $r=0$. A look at Fig. 1 reveals that there is only one measurement taking into account the density $\rho(0)$ at the center, while all measurement include information on $\rho(R)$. 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 $\chi^2 \leq N_d$.

  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 $\chi^2 = N_d$ cannot be satisfied, if too many points are used, ripples appear in the reconstructed density -- the data are overfitted.