## 3 The Sequential Sampling Method
### Infill Criterion III: Minimum Energy Design  
The minimum energy design (MED) was recently developed as a new space-filling design to explore unknown regions of particular interest [12]. The key idea of the MED originates from the laws of electrostatics. In the MED, samples in the input space are considered as charged particles inside a box. Let these charged particles be positively charged. By setting the same charge for all particles, they will repel each other and occupy the positions inside the box to minimize the total potential energy (Figure 5). Let \(q(\mathbf{x}_{i})\) be the charge of the \(i\)th particle. The potential energy between the \(i\)th particle and \(j\)th particle (\(E_{ij}\)) is proportional to the distance and charge as  
\[E_{ij}=\frac{q(\mathbf{x}_{i})q(\mathbf{x}_{j})}{d^{2}\big{(}\mathbf{x}_{i},\mathbf{x}_{j}\big{)}} \tag{18}\]  
where \(d^{2}\big{(}\mathbf{x}_{i},\mathbf{x}_{j}\big{)}\) is the Euclidean distance between the \(i\)th particle and the \(j\)th particle. The charge function \(q(\mathbf{x})\) will be described later since this function is differently defined depending on the desired value of the output (that is, the maximum or minimum). For \(n\)-charged particles, the total potential energy is defined by  
\[E=\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\frac{q(\mathbf{x}_{i})q(\mathbf{x}_{j})}{d^{2} \big{(}\mathbf{x}_{i},\mathbf{x}_{j}\big{)}}. \tag{19}\]  
In the MED, the box represents the input space, the positions taken by the charged particles are the training samples, and the charge of each particle reflects the value of the target output. Unlike the conventional space-filling designs, the MED tries to place the training samples as far apart from each other in the region of interest by reflecting their values (that is, the numerator of Equation (19)). Figure 4: (**a**) The expected improvement on the minimum (\(EI_{min}\)) and (**b**) the maximum (\(EI_{max}\)). Figure 5: The repulsion principle for minimum potential energy in electrostatics. [PERSON] et al. (1998) proposed a sequential strategy for implementing an MED. Suppose that we have \(n\) observed inputs \(\textbf{X}^{(n)}=\left[\textbf{x}_{1},\cdots,\textbf{x}_{n}\right]^{T}\) in a \(k\)-dimensional input space and the corresponding outputs \(\textbf{Y}^{(n)}=\left[y(\textbf{x}_{1}),\cdots,y(\textbf{x}_{n})\right]^{T}\). The kriging model of the \(n\)th stage (\(KRG^{(n)}(\textbf{x})\)) can be constructed using the training samples (\(\textbf{X}^{(n)}\) and \(\textbf{Y}^{(n)}\)). Since the value of the output can be negative, an auxiliary function (\(\hat{g}(\textbf{x})\)) should be used instead of directly using a value of the output (\(\hat{y}(\textbf{x})\)). \[\hat{g}(\textbf{x})=\hat{y}(\textbf{x})-\hat{y}^{(n)}_{min} \tag{20}\]  
where \(\hat{y}(\textbf{x})\) is the mean value from \(KRG^{(n)}(\textbf{x})\) for the sample \(\textbf{x}\), and \(\hat{y}^{(n)}_{min}\) is the estimated minimum value from \(KRG^{(n)}(\textbf{x})\). Then, the \((n+1)\)th infill sample (\(\textbf{x}^{n+1}_{MED}\)) is obtained by  
\[\textbf{x}^{n+1}_{MED}=\text{argmin}\sum_{\textbf{x}}^{n-1}\sum_{\textbf{j}=i +1}^{n}\frac{q(\textbf{x}_{i})q(\textbf{x}_{j})}{d^{2}(\textbf{x}_{i},\textbf {x}_{j})}+\sum_{i=1}^{n}\left(\frac{q(x)q(\textbf{x}_{j})}{d^{2}(\textbf{x}, \textbf{x}_{j})}\right). \tag{21}\]  
Since the first term of Equation (21) is constant regardless of the sample \(\textbf{x}\), this term can be ignored. Thus, Equation (21) is reformulated as  
\[\textbf{x}^{n+1}_{MED}=\text{argmin}_{\textbf{x}}\sum_{i=1}^{n}\left(\frac{q( \textbf{x})q(\textbf{x}_{j})}{d^{2}(\textbf{x},\textbf{x}_{j})}\right). \tag{22}\]  
The charge function (\(q(\textbf{x})\)) can be divided into two types according to the desired value of the output.