# Bayesian Optimal Experimental Design for Race Tracking in Resin Transfer Moulding
## 7 Discussion and Conclusions
### Non-Differentiability  
The Laplace approximation of the posterior, along with all gradient-based optimisation techniques that are used to find the MAP estimate, relies on the RTM process being differentiable. In this section, we discuss how this assumption only holds under specific experiments. When pressure measurements are extracted at fixed, equal time intervals within a predefined period, the pressure data from a given sensor can be split into three distinct phases:  
1. Before the resin reaches the sensor, the sensor gives 0 pressure (with noise). 2. Once the resin reaches the sensor, the pressure increases from 0 in a decaying, quasi-logarithmic fashion. 3. When the resin has almost filled the mould, the limited size of the exit vent means that the boundary condition on the resin changes from a Dirichlet boundary to a zero-flow Neumann boundary for the large majority of the boundary. This rapid change in the boundary condition causes a large and sudden increase in pressure throughout the mould. This change is sometimes so quick that it is 'almost discontinuous'. This is illustrated in Figure 10. The three phases are clearly visible here. The 'almost discontinuity' that appears in the posterior comes from the following: if, in the example shown in Figure 10, there was a measurement taken at \(t=23\),000 s, then a slight increase in any of the components in the permeability vector \(\mathbf{k}\) from either the bulk permeability or from the race tracking will cause the pressure measured here to rapidly change from roughly  
Figure 9: MAP estimate (solid line) and 99% confidence intervals (shaded region) for each parameter versus the number of sensors used, in the optimal locations. The true value of each parameter is indicated with a dashed line. 0.2 to roughly 0.35. This effect is more prominent in models with a small exit vent, where there is a large change in boundary conditions as the resin reaches the end. The scenario in Figure 10 is one with a very small but realistic (5 mm wide) exit vent--smaller than what is used in the numerical examples in this paper. However, the effect still exists, to a lesser extent, for larger vents. Figure 11 shows the true posterior (without using the Laplace approximation) for a much simpler formulation of the problem, where the only parameter to predict is \(K=K_{xx}=K_{yy}\) for a homogeneous domain, with \(K_{xy}=0\). Again, a central block is included, as in Section 4.1. The true value for this parameter is one. The inverse problem is solved with four different outlet boundary conditions: a single node, the top 0.1 m of the right boundary, the top 0.25 m of the right boundary, and the entire right boundary. As the outlet vent becomes smaller, a nondifferentiable point develops around \(K=1.02\). \(K=1.02\) is a point that exhibits the behaviour discussed above, where increasing this value a small amount causes one measurement to be taken after the resin has reached the end of the mould, instead of before. This point still exists even when the outlet is as large as 0.25 m. The roughness of the posterior shape aside from this discontinuity should be ignored, it is the product of using a coarse mesh in order to develop the true posterior within an acceptable time frame. The non-differentiability causes two key limitations to our results:  
* It increases the discrepancy between the Laplace approximation and the true posterior, as seen in Figure 11 and in the results for Model 1, where the true values were outside of the Laplace approximation. * Finding the MAP estimate through optimisation techniques becomes much more difficult, as the function becomes nonsmooth. Figure 10: An example of pressure measurements over time from a sensor located at (0.45 m, 0.175 m) in Model 1 (blue). This simulation uses a small 5 mm vent. For Model 1, the nondifferentiable points are far away from the MAP estimate, so there is no noticeable effect on the posterior. However, the optimisation to find the MAP estimate is long and difficult due to these nondifferentiable regions.