# Synchronization, Control and Data Assimilation of the Lorenz System
## 4 Pruned-Enriching Approach
The final results are shown in Figure 7. In Figure 7, we initialize the ensemble parameters randomly in the interval \((0.5,30)\), and we measure the distance \(D\) in the parameters space (Equation (22)) for different \(\delta\). Notice that, starting from a large initial distance, larger values of \(\delta\) are more effective for the convergence. The opposite phenomenon is reported in Figure 7. Here, we assume that the true parameter values are approximately known with an error \(\epsilon\), and we test the dependence of the amplitude \(\delta\) in a fine-tuning regime. In this case, starting from a relatively small distance, smaller values of \(\delta\) are more effective. In Figure 7, we instead show the behavior of the variance of the parameter distance \(D\) during optimization for different amplitudes \(\delta\). With a small amplitude \(\delta\) for the parameter updating step, the elements of the ensemble rapidly converge to the local minimum before having explored the parameter space sufficiently, so small amplitude values of \(\delta\) can be useful only for a fine-tuning approach. Using large values of \(\delta\) instead is helpful for better exploration of the parameter landscape, and allows us to converge to the true values, but with noisy results. These considerations suggest the use of an adaptive value of the parameter \(\delta\). Inspired by these results, we modify the updating rule, introducing a variable amplitude \(\delta=\left(\delta^{(1)},\delta^{(2)},\ldots,\delta^{(h)}\right)\), where, for every ensemble member \(i\), \(\delta^{(i)}=\delta^{(i)}(d^{(i)})\)  
Figure 7: Parameter distance \(D\) after \(M\) repetitions for different amplitudes \(\delta\) with (**a**) \(\mathbf{Q}^{(i)}\) randomly initialized in the interval \((0.5,30)\) and (**b**) \(\mathbf{Q}^{(i)}\) initialized near the “true” values \(\mathbf{Q}\) by adding a random noise of amplitude \(\epsilon=0.5\). (**c**) Variance of the distance \(D\) for different amplitudes \(\delta\) for the last interval. The vertical lines indicate when the ensemble was restored to \(t=0\). The \(\mathbf{Q}^{(i)}\) are initialized as in (**a**). Then, for each replica, we define an amplitude that modulates the variation step in a self-consistent manner. To test this choice, we simply put  
\[\delta^{(i)}=d^{(i)}. \tag{24}\]  
In Figure 8, we plot the behavior of 50 randomly chosen ensemble members in the pruned-enriching optimization problem with the amplitude factor \(\delta\), defined as in Equation (24). We assume the situation to be the same as those of the last simulations, so we measure the real system only in the \(\mathbf{C}_{x}\) direction every \(\tau=0.2\). The ensemble dimension is equal to \(h=10{,}000\), and we choose the embedding dimension \(d_{e}=5\). In this simulation, we run the algorithm for \(M=8\) repetitions. The numerical results are reported in Table 2.  
We averaged these measurements over \(\Omega=50\) repetitions in order to estimate the influence of the stochastic elements of Algorithm 2. The results are reported in Table 3. It can be noticed that the values of the standard deviation over repetitions are essentially the same of those over ensemble, in Table 2, divided by \(\sqrt{\Omega}\), implying that the ensemble is self-averaging (i.e., averages over larger ensembles give the same results as averages over repetitions). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Real** & **ens. Mean** & \(\mathbf{\sigma}\) **ens** \\ \hline \(\sigma\) & 10.000000 & 9.997584 & 0.062233 \\ \(\beta\) & 2.666667 & 2.664959 & 0.031623 \\ \(\rho\) & 28.000000 & 28.017719 & 0.067698 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Estimated parameters obtained using the pruned-enriching algorithm with \(\delta\) variable. In the last column, we also show the standard deviation on the ensemble members.