# Surrogate-Based Robust Design for a Non-Smooth Radiation Source Detection Problem
## 4 Robust Design in the Average Sense
### Solution of the Inverse Problem  
The solution of a nonlinear inverse problem is typically more difficult than in the linear case since one regularly does not have analytic solutions. Under the assumption that the model (10) behaves linearly with respect to variables \(\boldsymbol{\theta}\), one can derive an analytic solution for the weighted least-square estimate as described in [15; 25]. For nonlinear model responses, it is customary to linearize the system response  
\[\bar{F}_{i}(\boldsymbol{\theta})=\bar{F}_{i}(\boldsymbol{\theta}^{0})+\frac{ \partial\bar{F}_{i}}{\partial\boldsymbol{\theta}}(\boldsymbol{\theta}^{0})^ {T}(\boldsymbol{\theta}-\boldsymbol{\theta}^{0})+\mathcal{O}(\|\boldsymbol{ \theta}-\boldsymbol{\theta}^{0}\|_{2}^{2}),\ i=1,\ldots,N \tag{16}\]  
about a prior estimate \(\boldsymbol{\theta}^{0}\), where \(\|\cdot\|_{2}\) is the Euclidean norm.  
By neglecting the higher order terms, the statistical model (13) can be rewritten as  
\[\bar{\Upsilon}_{i}=\frac{\partial\bar{F}_{i}}{\partial\boldsymbol{\theta}}( \boldsymbol{\theta}^{0})^{T}\boldsymbol{\theta}_{0}+\bar{\varepsilon}_{i},\ i=1,\ldots,N. \tag{17}\]  
Here a realization \(\bar{v}_{i}\) of \(\bar{\Upsilon}_{i}\) can be expressed as \(\bar{v}_{i}=v_{i}-\bar{F}_{i}(\boldsymbol{\theta}^{0})+\frac{\partial\bar{F} _{i}}{\partial\boldsymbol{\theta}}(\boldsymbol{\theta}^{0})^{T}\boldsymbol{ \theta}^{0}\), where \(v_{i}\) is sampled from the distribution defined in (13). Consequently, the problem (15) can be reformulated as  
\[\boldsymbol{\hat{\theta}}=\arg\min_{\boldsymbol{\theta}}\mathcal{J}( \boldsymbol{\theta}),\ \mathcal{J}(\boldsymbol{\theta})=\frac{1}{2}\sum_{i=1}^{N}\frac{1}{\sigma_{i}^{ 2}}\bigg{[}\bar{v}_{i}-\frac{\partial\bar{F}_{i}}{\partial\boldsymbol{\theta} }(\boldsymbol{\theta}^{0})^{T}\boldsymbol{\theta}\bigg{]}^{2}. \tag{18}\]  
By differentiating \(J\) with respect to \(\boldsymbol{\theta}\), we obtain  
\[\
abla J(\boldsymbol{\hat{\theta}})=-\sum_{i=1}^{N}\frac{1}{\sigma_{i}^{2}} \frac{\partial\bar{F}_{i}}{\partial\boldsymbol{\theta}}(\boldsymbol{\theta}^{0 })\bigg{[}\bar{v}_{i}-\frac{\partial\bar{F}_{i}}{\partial\boldsymbol{\theta} }(\boldsymbol{\theta}^{0})^{T}\boldsymbol{\theta}\bigg{]}. \tag{19}\]  
By imposing \(\
abla J(\boldsymbol{\hat{\theta}})=0\), the estimate  
\[\hat{\boldsymbol{\theta}}=\boldsymbol{M}^{-1}\sum_{i=1}^{N}\frac{1}{\sigma_{i} ^{2}}\frac{\partial\bar{F}_{i}}{\partial\boldsymbol{\theta}}(\boldsymbol{ \theta}^{0})\bar{v}_{i}, \tag{20}\]  
is unique only if the Fisher information matrix  
\[\boldsymbol{M}=\sum_{i=1}^{N}\frac{1}{\sigma_{i}^{2}}\frac{\partial\bar{F}_{i} }{\partial\boldsymbol{\theta}}(\boldsymbol{\theta}^{0})\left[\frac{\partial \bar{F}_{i}}{\partial\boldsymbol{\theta}}(\boldsymbol{\theta}^{0})\right]^{T} \tag{21}\]  
is nonsingular.  
A formula for the estimator \(\boldsymbol{\theta}\) of the problem (15) is obtained based on the linearization of the surrogate model response (10) and by assuming that the high-order terms in the expansion (16) are negligible. In the next subsection, we highlight the role of Fisher information matrix in defining the D-optimality criterion widely used in the optimum experimental design problems as a quantitative measure of the 'goodness' of different networks of sensors.