# Optimisation of Segregation Distances between Electric Cable Bundles Embedded in a Structure
## 4 Sampling and Surrogate Modelling Joint Approach
### Surrogate Model Strategy  
The EMC probability with Monte Carlo sampling presented in the previous section may require many calls to the costly \(G\) function. An idea to reduce this cost is to learn a surrogate model \(\widehat{G}\) of the \(G\) function from an input-output \(n\)-sample of this function and then estimate the constraint probability on the surrogate model \(\widehat{G}\) instead of the \(G\) function. If the surrogate model \(\widehat{G}\) is not accurate enough on some regions of the input space, it is possible to improve it by calling the true \(G\) function for some relevant input positions. The most well-known strategy for probability estimation with the surrogate model and Monte Carlo sampling is notably active learning reliability method combining Kriging and Monte Carlo simulation (AK-MCS) [8], and various evolutions of this algorithms have been proposed recently [9; 10; 11]. The probability estimation with a surrogate model relies mainly on four elements:  
* The type of surrogate model. Throughout the article, the surrogate model \(\widehat{G}\) is assumed to be a conditioned Gaussian process (GP) \(\mathcal{G}_{n}\). Hence, the distribution \(\mathcal{G}_{n}\) knowing the \(n\) input-output observations \(\{(\mathbf{d},\mathbf{x})_{doe}=((d,\mathbf{x}_{1}),\ldots,(d_{1},\mathbf{x}_{n})),\mathbf{y}=G(d _{n},\mathbf{x}_{doe})\}\) is Gaussian \(\mathcal{G}_{n}=\mathcal{G}|((\mathbf{d},\mathbf{x}_{doe},y)\sim GP(\mu_{n}(\cdot), \sigma_{n}^{2}(\cdot))\). The initialisation of the design of experiments (DOE) is often performed with Latin hypercube sampling [12]. The mean \(\mu_{n}\) and the variance \(\sigma_{n}^{2}\) are estimated from \(((\mathbf{d},\mathbf{x})_{doe},\mathbf{y})\). The mean \(\mu_{n}\) is an approximation of \(G\), whereas the term \(\sigma_{n}^{2}\) evaluates the surrogate model error;
* The sampling approach to estimate the probability with the surrogate model. In this article, we only considered Monte-Carlo-based sampling approaches. The constraint probability is estimated with: \[\mathbb{P}(G(d,\mathbf{X})<0)\approx\mathbb{P}(\mu_{n}(d,\mathbf{X})<0)\approx \widehat{\mathbb{P}_{d}^{\widehat{MC}}}=\frac{1}{N}\sum_{i=1}^{N}\mathbf{1}_{ \mu_{n}(d,\mathbf{X}_{i})<0}\] (9) where \(\mathbf{X}_{i}\) are i.i.d. samples with distribution \(f_{\mathbf{X}}\). An inferior bound of the probability estimate can be estimated with: \[\mathbb{P}(\mu_{n}(d,\mathbf{X})-1.96\sigma_{n}(d,\mathbf{X})<0)\approx \widehat{\mathbb{P}_{d,\text{min}}^{\widehat{MC}}}=\frac{1}{N}\sum_{i=1}^{N} \mathbf{1}_{\mu_{n}(d,\mathbf{X}_{i})-1.96\sigma_{n}(d,\mathbf{X}_{i})<0}\] (10)and in the same way, a superior bound is given with: \[\mathbb{P}(\mu_{n}(d,\mathbf{X})+1.96\sigma_{n}(d,\mathbf{X})<0)\approx\widehat{ \mathbb{P}_{d,\max}^{\text{MC}}}=\frac{1}{N}\sum_{i=1}^{N}\mathbf{1}_{\mu_{n}(d \left(\mathbf{X}_{i}\right)+1.96\sigma_{n}(d,\mathbf{X}_{i})<0}\] (11)
* The surrogate model enrichment criterion to properly enrich the surrogate model in order to achieve an accurate approximation of the probability.