# Surrogate-Based Robust Design for a Non-Smooth Radiation Source Detection Problem
## 5 Numerical Examples
The dependencies associated with the Fisher information matrix are the derivatives of the surrogate models with respect to the source characteristics and variances \((\sigma_{i})^{2}=(\sigma_{i}^{o})^{2}+(\sigma_{i}^{m})^{2},\ i=1,\ldots,30.\) The gradients \(\frac{\partial\bar{F}_{i}}{\partial\boldsymbol{\theta}_{I}}\) are computed from (10) knowing that \(\psi\) is the Gaussian radial basis function and \(\|\cdot\|_{2}\) is the Euclidean norm. These sources \(\boldsymbol{\theta}_{\ell},\ell=1,\ldots,9880\), differ from those used for constructing the surrogate models and are uniformly distributed over the entire domain. This spatial distribution was employed since we selected \(p(\theta)\) to be the uniform distribution over \(\mathcal{X}\) in score \(\Gamma_{ED}\) (27). The values \(\Gamma_{ED}(\boldsymbol{\xi}_{10}^{\ell})\) are computed for all possible networks and the results are shown in Figure 6a. Allowing each network to include only one detector out of the three possible choices over each of the ten rectangular areas--see Figure 2--likely explains the periodic behavior. The optimization problem does not have a unique solution as seen from the expected values. Three different networks produce the largest score. Figure 6b shows the detectors locations of one of these three networks corresponding to the index 35,714. To test the obtained robust design network, we use the Formula (28) with the 11 randomly selected networks of sensors plotted in Figure 7. Next we set \(\mathcal{M}=50\), and uniform randomly select 50 true sources \(\boldsymbol{\theta}_{0}^{\ell}\) from \(\mathcal{X}\). Observations were then generated using the statistical model (9) for each source \(\boldsymbol{\theta}_{0}^{\ell},\ \ell=1,\ldots,50\) and network of sensors including the optimal one. We then solve the inverse problem (15) using the Particle Swarm algorithm [10] deta  
Figure 6: (**a**) Scores \(\Gamma_{ED}\) for all possible networks. The solution of the robust design problem is not unique; (**b**) The optimal network obtained as the solution of the robust design in the average sense strategy. The Particle Swarm approach is a global, meta-heuristic optimization algorithm motivated by social-psychological principles [27]. It was originally introduced in [10] and it was designed to imitate a social behavior such as the movements of birds in a flock or fishes in a shoal. Later the algorithm was simplified and its performance for solving optimization problems were reported in [28]. For our example, we set the inertia parameter to be \(1.1\) and the neighborhood of each particle is set to \(4\). The self and social adjustment coefficients \(y_{1}\) and \(y_{2}\) are set to \(1.49\). We select the swarm size to \(70\), and for each given source \(\boldsymbol{\theta}_{0}^{\ell},\ \ell=1,\ldots,50\), and network out of the \(11\) randomly selected networks plus the optimal one, we compute the inverse problem solution. The errors of the inverse problem solutions (i.e., the estimated sources characteristics) are shown in Figure 8 for all possible networks and sources. We note that the solution obtained using the optimal network does not have the smallest error for all the sources. For example, for source number \(28\), the source characteristics errors obtained using the optimal network are larger than all the estimates errors associated with the random networks except Network \(4\). This is not unexpected, since the optimal design was obtained following an average sense formulation. Next, the errors of the inverse problem solutions are averaged over the entire set of sources and the results of Formula (28) are illustrated in Figure 9 for all \(12\) considered networks. The index associated with the optimal network is \(12\) and corresponds to the smallest RMSE. This result suggests that we were able to identify the robust design in the average sense for the nuclear transport inverse problem.