## 2 Materials and Methods
### Expected Improvement Multi-Points Infill Criterion  
Kriging not only predicts the objective function values at unknown points but also provides the uncertainty of the predictions. This makes kriging well-suited to be paired with an acquisition function to enhance optimization efficiency. The kriging predictor treats the predicted objective function value at an unknown point as a realization of a stochastic process, which follows a normal distribution with a mean of \(\hat{y}(x)\) and a variance of \(s^{2}(x)\). Based on this, the criterion for identifying unknown points with potential for improvement can be expressed as [10]  
\[EI(x)=(y_{min}-\hat{y}(x))\Phi\left(\frac{y_{min}-\hat{y}(x)}{s(x)}\right)+s(x )\phi\left(\frac{y_{min}-\hat{y}(x)}{s(x)}\right) \tag{21}\]  
Here, \(\Phi(x)\) and \(\phi(x)\) represent the cumulative distribution function and the probability density function of the standard normal distribution, respectively. The EI function consists of two terms: the first term represents local exploitation near the current optimal value, while the second term represents global exploration in sparsely sampled regions. Therefore, the EI function maintains a perfect balance between local exploitation and global exploration. However, a significant limitation of the EI function is that it can only identify one candidate point within a single optimization cycle. This prevents it from fully utilizing external parallel computing resources to reduce the wall-clock time cost during the optimization process. To overcome this limitation, the PEI criterion proposes using an additional influence function to simulate the impact of the updated points on the behavior of the original EI function, thereby eliminating the costly evaluation of the update points and the Kriging model update within a single optimization cycle. As illustrated in Figure 4, a comparative analysis of the approximation and discrepancies between the EI and PEI functions during point infill on a one-dimensional Griewank's function reveals several key observations. Figure 4a demonstrates that the EI function derived from the updated kriging model and the PEI function obtained using the IF function exhibit generally consistent variation trends. The IF function effectively captures the characteristic behavior of the EI function, where the influence of updated points on the EI function diminishes with increasing distance from the update location in the design space. More notably, in this particular case, the PEI function demonstrates superior convergence performance compared to its EI counterpart. This enhanced performance stems precisely from the subtle differences introduced by the approximation process, which enable the PEI function to overcome the well-documented limitation of conventional EI functions--their tendency to become trapped in local optima. The observed phenomenon suggests that the intentional approximation employed in the PEI formulation not only maintains the essential properties of the EI function but also introduces beneficial characteristics that improve global search capability. The expression for the PEI criterion is given by [24]  
\[PEI(x,q-1)=EI(x)\times IF\Big{(}x,x^{n+1}\Big{)}\times\cdots\times IF\Big{(}x,x ^{n+q-1}\Big{)} \tag{22}\]  
where \(q\) represents the number of candidate points to be identified within a single optimization cycle. The IF function is used to simulate the impact of the updated points on the initial EI function and can be computed using the following expression:  
\[IF\Big{(}x,\ x^{n+1}\Big{)}=1-Corr\Big{[}Y(x),Y\Big{(}x^{n+1}\Big{)}\Big{]} \tag{23}\]  
Here, \(x^{n+1}\) denotes the position of the candidate point, and \(Corr[. ]\) is the spatial correlation function used in the construction of kriging. The value of the PEI function at the update points is zero, and as the distance between the unknown point and the update point increases, the value of the PEI function gradually increases to one.