# Surrogate-Based Robust Design for a Non-Smooth Radiation Source Detection Problem
## 5 Numerical Examples  
As detailed in Section 3, the problem under investigation consists of identifying the location and intensity of a radiation source in a simulated downtown Washington, DC block with minimum error with respect to the true source location and intensity. The solution to this problem can be obtained by applying optimal sensor location strategies [15]. Instead of applying a local optimal design method, whose solution depends on some _a priori_ estimate of the true source, we propose a robust design strategy to remove this dependency. Specifically, we propose a 'compromise' design where the obtained network is good enough (in a least-error sense) to identify any possible source from the admissible domain \(\mathcal{X}\). As detailed in (4), we take the admissible parameter space to be \(\mathcal{X}=[0,250]\) m \(\times[0,180]\) m \(\times[5\cdot 10^{8},5\cdot 10^{10}]\) Bq. Next we specify the set of all possible detector locations to be a discrete set of 30 spatial positions. By sampling from a uniform distribution, we generate the possible locations of detectors in the domain denoted by diamond marks in Figure 2. In this way, we avoid the problem of overlapping sensors encountered for a continuous formulation. The specific dispersal pattern was selected to spread the detectors evenly throughout the area. We assume that detectors have facial areas \(A_{i}\), with 3-inch diameters and 3-inch lengths, for incident gamma energy of 662 KeV. This is standard packaging for sodium iodide (NaI) scintillators that possess intrinsic efficiency of \(\varepsilon_{i}=62\%\) for 662 keV gammas. The dwell time \(\Delta t_{i}\) for all detectors was chosen to be 1 s.  
Finally, we set the size of the network to 10 detectors and formulate the robust design problem in the average sense:  
Find the network \(\mathbf{\xi}_{10}^{*}=\{\mathbf{D}_{i},\;i=1,\ldots,10\}\) consisting of 10 detectors out of the 30 possible detectors locations depicted in Figure 2 that solves  
\[\mathbf{\xi}_{10}^{*}=argmax_{\mathbf{\xi}_{10}}\Gamma_{ED},\;\;\;\Gamma_{ED}(\mathbf{\xi }_{10})=\int_{\mathcal{X}}\det\big{(}\mathbf{M}(\mathbf{\xi}_{10},\mathbf{\theta})\big{)} p(\mathbf{\theta})d\mathbf{\theta}. \tag{27}\]  
Figure 2: Network formed by 10 sensors with 30 possible locations. By using a sufficiently large number of sources, the integral in (27) can be accurately approximated. To evaluate the efficiency of the robust design network, we employ the metric  
\[\sqrt{\frac{\sum_{\ell=1}^{\mathcal{M}}\|\hat{\mathbf{\theta}}_{\mathbf{\xi} _{10}}^{\ell}-\mathbf{\theta}_{0}^{\ell}\|_{2}^{2}}{\mathcal{M}}}, \tag{28}\]  
where \(\mathbf{\theta}_{0}^{\ell},\ \ell=1,\cdots\mathcal{M}\), are \(\mathcal{M}\) distinct true radiation sources. For each true source \(\mathbf{\theta}_{0}^{\ell}\), we can compute the associated estimate \(\hat{\mathbf{\theta}}_{\mathbf{\xi}_{10}}\) as the solution of the problem (15) using the network \(\mathbf{\xi}_{10}\). The score (28) corresponding to the robust design \(\mathbf{\xi}_{10}^{*}\) will be tested against scores obtained by randomly selected networks \(\mathbf{\xi}_{10}\) of 10 detectors. The smallest score should be obtained by the robust design \(\mathbf{\xi}_{10}^{*}\) thus validating the approach. The discrete nature of the space of all possible detectors locations transforms (27) into a combinatorial optimization problem. The number of possible networks is 30,045,015 as given by (30 choose 10) which is equivalent to combination of 30 possible sensor locations taken 10. By imposing that each possible network \(\mathbf{\xi}_{10}\) contains only one detector out of the three possible choices from each of the ten rectangular areas shown in Figure 2, we decrease the number of possible networks to 59,049. This makes the combinatorial problem computationally feasible. In Figure 3, we plot the model response \(F_{8}\) (7) corresponding to the sensor location \(\mathbf{D}_{8}\) in Figure 2 and a source intensity of \(3.5\times 10^{9}\) Bq.