# Bayesian Optimal Experimental Design for Race Tracking in Resin Transfer Moulding
## 6 Bayesian Optimal Experimental Design
### Results  
The optimal sensor locations for Model 1 are shown in Figure 7, calibrated using \(M=200\) different permeability scenarios. The choice of M was decided by investigating the robustness of the optimal locations trained with various amounts of data. This is omitted here for clarity but is contained in Appendix C, along with a brief discussion on the robustness of the design. The sensors tend to be distributed around the race-tracking areas. For the top and bottom race-tracking areas, it seems to be beneficial to have sensors at the start and end of these regions. There are some sensors (most notably sensor 2) that are positioned to gain pressure measurements at the inlet, where the large pressures make the influence of noise less confounding.  
Figure 8 illustrates how the objective function in Equation (42) changes with an increasing number of sensors, for Model 1. The results for (the more complex) Model 2 are given in Appendix B. The figure shows a monotonic decrease in objective function as more sensors are added, regardless of location, as expected. For the random design, there are points where adding a sensor gives no change in the objective. This is because the sensor is placed in a location with no information--for example, inside the central block. The optimal sensor locations lead to a significant reduction in uncertainty compared to  
Figure 7: The positions within the domain of the optimal sensor locations, as determined by the greedy algorithm for Model 1. Sensors are indicated in red and numbered in the order they appear in the greedy algorithm—i.e., the optimal locations of four sensors are locations 1, 2, 3, and 4. All candidate locations are shown in grey.  
using random sensor locations, with roughly five times lower variance in the permeability estimates. However, this difference would be expected to converge to zero as the number of sensors increases to the maximum [33; 34; 35; 36; 39; 40]. It is also clear in Figure 8 that the largest variances come from the race-tracking parameters \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) rather than the bulk parameters \(c_{1}\), \(c_{2}\), and \(c_{3}\). This is in agreement with the posterior variances shown in Section 5. The eighth sensor appears to have greatly reduced the uncertainty in \(c_{1},c_{2}\), and \(c_{3}\). However, the logarithmic scale of the graph emphasises this drop, and the sum of variances for all parameters still decreases, as seen in the upper graph (red line).  
The MAP estimate for each parameter is shown in Figure 9, along with 99% confidence intervals. These intervals are based on the Laplace approximation to the posterior about the MAP estimate (Equation (31)), rather than the approximation defined in Equation (43). The MAP estimate tends to converge to the truth as more sensors are added, as expected. The variance tends to decrease as more sensors are added, as in Figure 8, but not monotonically. When calculating the Laplace approximation about the MAP estimate, there is  
Figure 8: (**Top:**) Sum of the trace of the posterior covariance over all simulations versus the number of sensors used. The optimal solution is shown in red, and random solutions are shown in blue. (**Bottom:**) Individual variances of each posterior parameter (log scale), for the optimal design.  
no guarantee of strictly decreasing variances because the MAP estimate changes. There are some instances where the truth is not contained within the 99% confidence interval. This is due to a discrepancy between our Laplace approximation and the unapproximated posterior. One key reason for this discrepancy in this scenario is discussed in Section 7.1.