# VoI-Based Optimization of Structural Assessment for Spatially Degrading RC Structures
## 3 Practical Implementation of the Calculation of the VoI
### Optimization of Sensor Locations Using Information Entropy  
The VoI can be used either to optimize a monitoring strategy itself or to choose between different alternative strategies. In the optimization of a monitoring strategy, sensor locations could also be considered as an optimization parameter, resulting in the calculation of the VoI for different sensor configurations. Nevertheless, this requires many calculations of the VoI and hence a very large computational effort. This can be resolved by determining optimized sensor locations for parameter estimation beforehand by minimizing a metric called the information entropy. This information entropy expresses the decrease in uncertainty from the prior distribution of the model parameters to the posterior distribution after updating based on the considered data. The latter depends on the available data \(\mathbf{y}\) and the sensor configuration \(L\). It is necessary to choose the optimal sensor locations such that the best possible information on the model parameters \(\mathbf{\theta}\) is extracted from the data [24]. By optimizing the sensor locations beforehand, only the number of sensors should be taken along in the optimization procedure of the monitoring strategy, and the required number of evaluations of the Vol can be reduced.  
For a large number of data, an asymptotic approximation of the information entropy is given by Equation (3) [19].  
\[h(L;D)\,\sim H(L;\mathbf{\theta}_{0})=\frac{1}{2}N_{\theta}\ln(2\pi)-\frac{1}{2}\ln [\det(Q(L;\mathbf{\theta}_{0})) \tag{3}\]  
Here, \(\mathbf{\theta}_{0}\) represents the optimal value of the parameter set \(\mathbf{\theta}\) that minimizes the measure of fit between the measured and modeled data. \(N_{\theta}\) is the number of model parameters, and the matrix \(Q(L;\mathbf{\theta}_{0})\) is known as the Fisher information matrix and contains information on the posterior uncertainty of the parameters \(\mathbf{\theta}\) based on the data from all sensor positions in \(L\). For the prior estimate, nominal values can be used for \(\mathbf{\theta}_{0}\) which are representative for the system (e.g., the prior mean values).  
The Fisher information matrix \(Q(L;\mathbf{\theta}_{0})\) is given by Equation (4).  
\[Q(L;\mathbf{\theta}_{0})=\sum_{k=1}^{N}(L\underline{\
abla}_{\theta}\underline{x} _{k})^{T}\Big{(}L\underline{\Sigma}L^{T}\Big{)}^{-1}(L\underline{\
abla}_{ \theta}\underline{x}_{k}) \tag{4}\]  
where \(\underline{\
abla}_{\theta}\underline{x}_{k}\) represents the sensitivities of the measured data type to the model parameters, \(\underline{\Sigma}\) is the prediction error covariance matrix, and \(N\) is the number of sampled data.  
The optimal sensor locations can be found by solving a discrete-valued optimization problem. The sensors should be placed such that the resulting measurement data are most informative about the model parameters. Hence, the optimal sensor location \(L_{opt}\) is the one that minimizes the information entropy. This minimization is constrained over the set of \(N_{p}\) measurable degrees of freedom. Two heuristic sequential sensor placement (SSP) algorithms exist in the literature: the forward and the backward SSP [19]. In the forward algorithm, the sensor that results in the highest reduction in information entropy is added in each iteration. Hence, at each iteration, the sensor configuration is selected with the minimum information entropy. The backward algorithm works in the reverse order, starting from sensors at all possible locations and each time removing the sensor that results in the smallest increase in information entropy.  
In this work, the sensitivities required in the evaluation of the Fisher information matrix are calculated based on the method of [PERSON] [25], as for example in [21]. Furthermore, the method is here applied in relation to a finite element model. The sensitivities of the measurement data to the corrosion parameters are derived and the optimal sensor positions are determined based on the forward SSP algorithm, which is a greedy search algorithm.