# Using Feedback Strategies in Simulated Annealing with Crystallization Heuristic and Applications
## 4 SA with Incomplete Cost Evaluation Applied to EIT
### Results and Discussion  
The electrical potential measurements were taken from two different impedance distributions. Figure 5 displays the two distributions, which consisted of placing three phantoms made of cucumber slices in random positions inside of a saline solution. The reconstructed images are shown in Figure 6 for the two configurations. The saline water has almost constant conductivity. The cucumbers are visible according to the established arrangement. These results have a good contrast, determining the boundaries of the internal elements. When compared with the results of other approaches, these results show an improvement in the internal element boundaries determination [30; 42]. Figure 7 shows the convergence of SA with the crystallization factor for both arrangements; the two graphs are shown in blue. Averagely, larger values of the cost function show that the exploration phase is happening at higher temperatures. The graph shows that the exploration phase is very short for both arrangements. At lower temperatures, lower values of the cost function on average show that the refinement phase happens. [PERSON] and [PERSON] [43] showed that the application of interval SA with the crystallization factor to EIT created an emergent behavior: external layers converged first and internal layers converged later. The possibility of convergence layer by layer was first proposed  
Figure 5: Electrical potential measurements were taken for two conductivity distributions. (**a**) Physical linear arrangement. (**b**) Physical triangular arrangement. Figure 6: The reconstructed for the two conductivity distributions in \((\Omega\cdot\text{m})^{-1}\). (**a**) Linear arrangement reconstructed. (**b**) Triangular arrangement reconstructed. by [PERSON] et al. [44]. However, it was shown to be unstable in the presence of noise. The algorithm proposed by [PERSON] and [PERSON] [43] is the first stable implementation with this property [45]. [PERSON] et al. [46] developed a new heuristic, called outside-in, using this emergent behavior. In the two graphs in red shown in Figure 7, the number of conjugated gradient iterations is represented on average for each temperature. The mesh used in the reconstruction has more than 1000 nodes. Consequently, the linear system (6) has more than 1000 equations, and the number of iterations that the conjugated gradient must perform is the same as the number of equations. Considering Figure 7, the maximum number of conjugated gradient iterations is 35. As the cost function is partially evaluated, the number of conjugated iterations is kept at a small number. The number of conjugated gradient iterations at higher temperatures is smaller when compared to the iterations at lower temperatures. As previously pointed, the cost function is partially evaluated and, simultaneously, the interval \(\Delta E_{i}=\left[E_{i}^{\min},E_{i}^{\max}\right]\) in which the exact value is contained is also determined. At higher temperatures, SA performs exploration and the determined \(\Delta E_{i}\) is larger. Usually, the candidates are far from the final solution and larger errors are present. At lower temperatures, SA performs refinement, and the determined \(\Delta E_{i}\) must be small. The number of conjugated gradient iterations is kept at a reasonable level (smaller than 35 iterations) because the initial linear system solution for the conjugated gradient current is the final linear system solution from the previous conjugated gradient evaluation. Considering that only a small modification happens, remember that only one parameter is modified at a time, as shown in Algorithm 2; the final linear system solution from the previous conjugated gradient evaluation is a good initial linear system solution for the current conjugated gradient evaluation. Table 5 shows the final cost and the number of cost function evaluations for the two experimental arrangements. Occasionally, the linear arrangement converges first with a smaller number of cost function evaluations.