# Synchronization, Control and Data Assimilation of the Lorenz System
## 4 Pruned-Enriching Approach
Simulation data as in Figure 8.  
Figure 8: The distance between variables (black points, right axis) and parameters (color points, left axis) as a function of iterations (repetitions times number of samples of a trajectory) with the pruned-enriching method for 50 randomly selected replicas of the \(h=10{,}000\) used to estimate the parameters. We used \(\mathrm{d}t=10^{-2}\), \(\tau=0.2\) and \(T=100\), so the number of samples (time series) of a trajectory is 500, and we show \(M=8\) iterations. We consider the situation where only measurements in the coupling direction \(C=[1,0,0]\) are available at every \(k=20\) temporal steps \(\mathrm{d}t\), and we used an embedding space of dimension \(d_{e}=5\). The pruned-enriching procedure is similar to a genetic algorithm without cross-over. In general, a cross-over operation aims to combine locally-optimized solutions, and depends crucially on the coding of the solution in the \"genome\". When the cost function is essentially a sum of the functions of variables coded in the solution in nearby zones, the cross-over can indeed allow a jump to an almost-optimal instance. In this case, we encode the parameters simply using their current values (a single genome is a vector of 3 real numbers), so there is no indication for this option to be present. It is, however, possible to pick parameters from \"parents\" instead of randomly altering them, i.e., performing \"gene exchanging\". Because the pool of tentative solutions is obtained by cloning those corresponding to the lowest distances from the master, we expect little improvement using parameter exchange. To add gene exchanging to our procedure, we modify the algorithms such that, for every element of the second half of the ensemble, we choose randomly whether to update the parameters as in Algorithm 2 or perform the gene exchanging step, generating the new replica from two parents randomly chosen from the first half of the ensemble. Children randomly inherit two of the parameters from one parent and one from the other. With no prior information about the true parameter values, we randomly initialize the initial states and the parameters. Therefore, using the gene exchanging operation can introduce a bias in the early stages of the optimization problem, as can be seen in Figure 9, where we compare the estimated parameters for different repetitions using an amplitude \(\delta=0.8\) with (Figure 9a) and without (Figure 9b) gene exchanging. As in the other simulations, we randomly initialize the initial states and parameter values of the ensemble, we evolve the system, after the transient time \(T_{Trans}\), up to \(T=100\) with \(dt=0.01\), and we assume that it is possible to measure the \(x\) direction with \(\tau=0.2\). The gene exchanging operation allows jumps to be made in the parameters space, but, in the early stage of the optimization process, these jumps can cause the ensemble to converge on the wrong values. On the other hand, gene exchanging can reduce the variance of the ensemble estimation, so it can help in the final steps, or for fine tuning. Future work is needed to explore this option in more detail. \begin{table}
\begin{tabular}{c c c c} \hline  & **Real** & **Mean** & \(\boldsymbol{\sigma}\) \\ \hline \(\sigma\) & 10.000000 & 9.999346 & 0.013472 \\ \(\beta\) & 2.666667 & 2.666094 & 0.005040 \\ \(\rho\) & 28.000000 & 28.003983 & 0.042175 \\ \hline \end{tabular}
\end{table} 
Table 3: Estimated parameters obtained using the pruned-enriching algorithm with \(\delta\) variable, averaged over \(\Omega=50\) realizations. Figure 9: Estimation (x) and standard deviation (blue area) of the parameters computed using the first half of the ensemble (**a**) with and (**b**) without cross-over. The true values are also shown (dotted orange lines).