# Active Optics and Aberration Correction Technology for Sparse Aperture Segmented Mirrors
## 3 Correction Quantity-Solving Method
### Improved Differential Evolution Algorithm  
Equation (34) requires recalculation of the weight matrix \(P_{L}\) and the ridge parameter \(\eta\) in each iteration step, which is done by the differential evolution algorithm, also known as the DE algorithm in this work.  
The basic idea of the DE algorithm is to use 2 randomly selected different vectors in the population to interfere with the existing vectors. Each vector in the population is disturbed, and the new vectors replace the previous generation if the corresponding fitness function is smaller than their previous generation [17; 18]. The DE algorithm reduces the probability of falling into a local solution by using real-number encoding. It is to find the optimal solution efficiently. The implementation DE algorithm is expressed in Equations (35)-(39). Each individual in the population represents a set of solutions in the solution space. A single individual can be expressed as follows.  
\[X_{i,G}=(x_{1,i,G},x_{2,i,G},...,x_{M,i,G}) \tag{35}\]  
where \(i\) represents the \(i^{th}\) individual in the solution space. \(M=n+1\) represents the dimension of the solution vector, such that \(x_{1,i,G}=\eta\), and \(x_{j,i,G}=p_{j-1},j=2,3,\cdots,M\). \(G\) represents the current evolutionary generation. To ensure that the initial population is spread throughout the solution space, the slant tent chaotic mapping function is chosen to initialize the population. The implementation can be seen in the following equation.  
\[\begin{array}{l}x_{j,1,0}=rand(0,1)\times(x_{j,\max}-x_{j,\min})+x_{j,\min}\\ X_{j,j,0}=\begin{cases}\frac{x_{j,(i-1),0}}{rand(0,1)},x_{j,(i-1),0}\in[0, \text{rand}(0,1))\\ \frac{1-x_{j,(i-1),0}}{(1-rand(0,1))},x_{j,(i-1),0}\in[\text{rand}(0,1),1]\end{cases}\end{array} \tag{36}\]  
where \(x_{j,max}\) and \(x_{j,min}\) represents the upper and lower bounds of the optimization parameters given by \(j\). Traversing the entire solution space of the function avoids getting into local solutions and speeds up the convergence of the algorithm. The premature problem is avoided, and the performance of the algorithm is improved by the variational mixture approach of Equation (37).  
\[V_{i,G}=X_{i,G}+0.5\times F_{s}\times(R_{B}\times(X_{\text{best},G}-R_{B} \times X_{i,G})) \tag{37}\]  
In the above equation, \(R_{B}\) is a random constant obeying normal distribution and \(X_{best,G}\  
\(CR\) of the system is designed to be larger, the population changes faster. Moreover, the damage to the best-adapted individuals is also larger. On the contrary, the population evolves slowly, and the search speed decreases [19]. Thus, the crossover probability \(CR\) can reflect the global search ability and local search ability of the algorithm. The designs of \(F_{s}\) and \(CR\) in the DE algorithm were set as an adaptive function of the fitness function. The idea of dominant poles is adopted in selecting the fitness function. Firstly, the system indicators are calculated as a second-order system with 2 poles \(\lambda_{1,2}\) as the dominant poles of the system. The definition of the fitness function is given in Equation (40).  
\[\begin{array}{c}E_{E}=1+G(\lambda_{1,2})\\ J=\text{Re}^{2}\{E_{E}\}+\text{Im}^{2}\{E_{E}\}\end{array} \tag{40}\]  
where \(E_{E}=0\) is the characteristic equation of the system. Re and Im are the real and imaginary parts, respectively. At this time, the DE algorithm selects the individual with a small adaptation as the next generation in the selection operation by trying to make \(E_{E}(\lambda_{1,2})=0\). The evaluation function of the DE algorithm is set for the accuracy of parameter solving in the GL estimation process, as shown in the following equation.  
\[E_{DE}=\frac{1}{1+MSE(\bar{X}_{GL})} \tag{41}\]  
If the solution of Equation (41) is close to 1, the solution is more accurate.