autocorrelation function and power-law density]{}

-   [power-law decaying autocorrelation function and uniform density]{}

Figs.(1-4) illustrate the corresponding results of our Monte-Carlo approach. The subfigures (a,c) within each figure show the resulting maps for model I and model II, respectively. Each of the subfigures (b,d) within each figure (1-4) shows both the prescribed data for the autocorrelation signal as well as the result of our optimization approach for model I and II, respectively. In comparing the given autocorrelation data and the results of our optimization scheme we deduce in the following a ’mean relative error’ per individual data point $\frac{K[h_f]}{\sqrt{m}}$ of the autocorrelation function.

Since it is not our goal to provide as precise data as possible for the optimized autocorrelation functions but to demonstrate the feasibility of our approach with a good accuracy for a few data points we restrict ourselves to auxiliary functions $h_f$ composed of a few elements $h_{f,i}$. For the case of model I we use (with one exception) four points, i.e. three elements $h_{f,i}$ for the decomposition of the auxiliary function $h_f$. For model II we use only two points, i.e. one element $h_{f,i}$. Typically a few ten thousand Monte-Carlo steps are performed. The most time 