An Adaptive Parallel EI Infilling Strategy Extended by Non-Parametric PMC Sampling Scheme for Efficient Global Optimization

This paper presents a novel adaptive parallel Expected Improvement (EI) infilling strategy for Efficient Global Optimization (EGO) by introducing a two-staged Non-parametric Population Monte Carlo Sampling (NPMS) scheme. The samples are uniformly generated from EI function in the first stage and converge to sub-domains of high EI values thresholded by a non-parametric sampling selection method in Population Monte Carlo (PMC) iterative succession. In the second stage, learning from potential information, Density-Based Spatial Clustering (DBSCAN) method is used to cluster samples and converge to candidate points. Compared to the original EI strategy, NPMS improves the minimum result by 14.6% and reduces the number of candidate points by 15.8% on our benchmark cases of EGO. Furthermore, 13 test functions involving different input space sizes, difficulties, and dimensions are conducted on six strategies including NPMS, and the results showed that NPMS achieves the highest ranking in terms of result finding and cost savings but slightly decreases optimization efficiency. Benefiting from broad sampling and dynamic clustering, especially in large input space size cases, NPMS not only guarantees high result accuracy but also reduces optimization costs by up to 34.9% compared to other parallel methods. Finally, our proposed NPMS-extended EI strategy has successfully reduced the number of candidate points, which is expected to provide a cost-practical approach to more complex problems.


I. INTRODUCTION
Efficient Global Optimization (EGO) is a model-based sequential optimization technique that effectively solves the The associate editor coordinating the review of this manuscript and approving it for publication was Kuo-Ching Ying .
classical machine intelligence problem in decision theory [1], [2], [3], [4]. EGO is based on the kriging model and Expected Improvement (EI) infilling strategy. Kriging model provides cheap evaluation (prediction µ(x) and uncertainty measure σ (x)). And EI infilling strategy provides promising candidate points (maximum EI function value), VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ which effectively addresses the balance between optimization results and optimization costs in decision making [5], [6], [7]. Since proposed by Jones et al. [8] in 1998, EGO has been used in complex engineering. References [9], [10], [11], [12], [13], and [14] and has become the most popular approach. As a candidate point selection credential, infilling strategy determines the kriging performance and EGO efficiency. Effective infilling strategies based on different criteria are proposed, such as the probability of improvement (PI) [15], goal seeking [16], upper confidence bound (UCB) [17]. Among them, the most famous and widely used is EI infilling strategy due to a good balance between global exploration and local exploitation in EGO [18], [19].
Since EI function is a greedy improvement of heuristics, EI-based EGO often falls into unbalanced problems such as over-exploration and over-exploitation, resulting in slow optimization convergence [20], [21], [22], which requires infilling strategies to make further enhancements to balance and sampling efficiency. Schonlau et al. [23] and Sóbester et al. [24] introduced additional parameters in EI function, and proposed the Generalized Expected Improvement (GEI) and the Weighted Expected Improvement (WEI), respectively. Recently, Lv et al. [25] made up for the limitation of EI strategy by mixing multiple individual infilling strategies and proposed a Go-Inspired Hybrid (GO-HI) infilling strategy. Compared with independent infilling strategies, GO-HI saves computational costs and generates global points. However, single-point infilling strategy is not economically viable for the multiple CPU cores of modern computers, especially if the simulations or physical experiments are significant time consuming [26], [27], [28], [29].
Parallel infilling strategy focuses on inefficiency of single-point selection and generates multi-point in each optimization iteration. Although sampling accuracy is inevitably reduced, it improves computation utilization and convergence speed [30]. Based on different multi-point selection schemes, researchers developed many parallel strategies. Sóbester et al. [31] proposed a parallel strategy to select n local maxima of EI as new candidate points assuming given n processors. Compared with single-point strategy, multipoint of EI function local maxima can build an effective global kriging model faster. Ginsbourger et al. [32] derived q−EI based on the work by Schonlau et al. [26] and pointed out that q−EI strategy needs to be optimized one time to produce q candidate points in each iteration. Still, the dimension of optimization problem increases to q × d (d is the dimension of the design variables). Researchers relaxed the exact EI multi-peak calculation and replaced it with fake values or the points from high EI values subdomains. Ginsbourger et al. [33] proposed a heuristic strategy, named Kriging Believer (KB) strategy, which uses the kriging prediction replaces the actual calculation and cycle q times in a turn to obtain q samples. The KB strategy depends on the performance of the kriging model and produces candidate points clustered around false values. To solve this problem, Ginsbourger et al. [33] used a constant to replace the actual calculation of the selected samples named the Constant Liar (CL) strategy. However, low observation points lead to high uncertainty in constant selection. Inspired by the KB and CL strategies, Zhan et al. [34] used a fake function to replace the actual EI function and cycle q times, named Pseudo EI (PEI) strategy. PEI infilling strategy has a fast convergence speed in constructing a good kriging model. As EI function tends to converge, the fixation of q makes it easier to generate meaningless points, which increases the cost of optimization.
EI function changes with EGO iteration, and sequential select q multi-point often leads to additional optimization burdens by falling into aggregation. Xie et al. [35] removed duplicate points by calculating candidate point correlations. Gobert et al. [36] proposed an acquisition process technique based on design space partitioning to select valuable candidate points from kriging model. Since EI function has a closed-form expression, the computational cost is negligible compared to the optimization evaluation [27]. Researchers devise adaptive parallel infilling strategies to reduce the optimization burden by sacrificing EI function calculation. Xiao et al. [3] focused on selecting global multi-point and used a refined sampling/importance resampling to search the points with large EI values. Moreover, the of points needs to be set in advance. Zhan et al. [37] constructed EI multi-peak using Latin hypercube sampling and dynamic generation of points by each non-contiguous sub-domain. However, EI function multi-peak selection judgments not be discussed much. Selecting low-information peaks may not be helpful during the optimization process.
For an excellent infilling strategy, points generated should be global and dynamic while avoiding clustering with each other. The difficulty lies in capturing and selecting EI multi-peak for each EGO iteration. Population Monte Carlo (PMC) method has the potential to solve these problems. PMC is an unbiased sampling method consisting of iteratively generated importance samples from global space. Beaumont et al. [38] extended the sampling method to Bayesian inference for approximating target distribution P(θ | X) when likelihood function P(X | θ) hardly computed or even unknown. As defined in PMC, given a series of decreasing sampling thresholds ϵ 1 > ϵ 2 > · · · > ϵ ℓ (ℓ being the final iteration) in advance, the intermediate samples are sequentially updated in order to approximate true distribution. The sampling threshold controls range of samples produced and is a key to PMC generating samples closer to target multi-peak distribution.
PMC generates approximate samples to construct unknown distribution. For converging samples and selecting global candidate points, density-based spatial clustering (DBSCAN) approach is simple and effective. DBSCAN method enables dynamic partition and reorganize samples with density properties. The intuition for clustering is to divide the samples as sub-domains that satisfy the minimum density. Moreover, these sub-domains are constrained by a minimum distance threshold to ensure the diversity of information [39]. Samples in same distribution or adjacent to each other are considered to be in same cluster. In contrast, samples far apart from each other are considered in different distributions, which effectively solves the problem of final candidate points falling into aggregation due to proximity. This paper proposes an adaptive parallel sampling scheme based on EI infilling strategy, called the parallel Non-parametric Population Monte Carlo sampling scheme (NPMS). The NPMS scheme follows a native idea: taking EI function as a probability density function with a multi-peak form. And in order to reduce parallel strategy burden caused by points aggregation, NPMS scheme obtaines multiple candidate points in staged manner. In the first stage, PMC method samples from global input space and converges samples to individual high EI value sub-domains. A nonparametric sampling threshold method is used to determine whether the areas are selected and save more sampling costs. In the second stage, the neighborhood distance threshold and the sample size threshold are adjusted according to sample density and EI convergence information, and then DBSCAN clustering is performed. The candidate points come from PMC samples with the best EI value in each cluster. Above, NPMS scheme uses the currently known information and makes adaptive decisions by sampling EI function more in return for a lower optimization casts. At the same time, candidate points from interest sub-domains and far from each other, which improving optimization benefits.
The rest of this paper is organized as follows. Section II details the NPMS scheme framework. Section III validates the characteristics and feasibility of the NPMS scheme through a benchmark case. The effect of the parameters on the model is then analyzed. Section IV provides a detailed performance comparison and analysis of the NPMS strategy with other infilling strategies (i.e., EI, GO-HI, CL, KB, and PEI) through benchmark function tests. Section V summarizes the conclusion and future work.

II. NON-PARAMETRIC POPULATION MONTE CARLO SAMPLING SCHEME (NPMS)
This section details the Non-parametric Population Monte Carlo sampling (NPMS) scheme. After kriging model is constructed, NPMS scheme acts on EGO candidate selection process. Defining EI function as a probability density function, NPMS scheme uses two stages: non-parametric population Monte Carlo sampling and density clustering. The flowchart of NPMS applied to EGO as shown in Figure 1.
In the paper, we assume that EGO is applied in continuous space, and the goal is to find a point with the minimum value of unknown objective function f : where X denotes input space of x. For convenience, the obtained EI values are converted into negative numbers shown in Eq.(3).
whereŷ is kriging prediction value andσ is corresponding standard deviation. The and φ are the cumulative density function and probability density function of a normal distribution, respectively. The multi-peak in EI function represents the minimum value regions.

A. EI MULTI-PEAK SAMPLING
Define multi-peak sub-domains in EI function as the P(x). Samples X = {x 1 , x 2 , . . . , x N } are obtained by non-parametric method sampling from high EI value sub-domains and used to construct an approximate distribution of EI function P(x | X). For each EGO iteration step, sampling method and sampling threshold selection are performed by following two steps: Step 1: PMC Sampling Framework Construction: PMC method initially begins with sufficient statistics. At the initial step ℓ = 1, Latin Hypercube Sampling (LHS) method is used to generate a set of N global sample pool X 1 = {x 1 , x 2 , . . . , x N }, with corresponding EI value EI (X 1 ). VOLUME 11, 2023 For each sample pool, reject the ineligible samples based on the approximate constraint rule Eq.(4), and form prior distribution P(x) from the accepted samples.
where ϵ ℓ is defined as PMC sampling threshold in range min(EI) ≤ ϵ ℓ ≤ 0. For approximate constraint rule, samples with the EI value greater than ϵ ℓ will be rejected, while samples in high EI values (less than ϵ ℓ ) are accepted. Use the prior distribution P(x) to complete the sample pool X 1 to number N , and give initial sample pool weight W 1 .
In arbitrary step (ℓ > 1) in PMC sampling iteration, iterative sample pool X ℓ is derived from X ℓ−1 via a particle filter methodology under approximate constraints ϵ ℓ . Specifically, a particle x i is randomly samples from In Gaussian Markov transition kernel, the mean is x i and the variance σ 2 is twice the weighted empirical variance of sample pool X ℓ−1 as shown in Eq. (6).
ℓ ∈ X ℓ assigned in the previous iteration is shown in Eq. (7). As sampling threshold ϵ ℓ shrinks, the distribution constructed by final sampling pool X ℓ accurately approximates current EI function.
Step 2: Non-Parametric Sampling Threshold Selection: During iterative sampling, all coefficients are adaptive except sampling threshold ϵ ℓ . Sampling threshold directly affects cost and accuracy in PMC: (i) a strict threshold allows samples to be distributed in fewer areas with higher EI values, accompanied by high rejection sample sizes and sampling times, (ii) a loose threshold preserves a large number of samples and shortens sampling time in PMC, but inevitably contains meaningless samples. Suppose that at any iteration step ℓ with sampling threshold ϵ ℓ , another sample pool with N p (N < N p ) samples is sampled from the ℓ−1 (using the LHS method in ℓ = 1). Based on approximate constraint rule in Eq.(4), the number of accepted samples is N acc . The acceptable ratio P accept is defined as: For accepted N actual samples after using approximate constraint rule, the required accumulated sample size N need is calculated by acceptable ratio and N as shown in Eq. (9). PMC method should balance sampling cost and information gain limit in multi-peak areas in each sampling iteration, which means the sampling threshold ϵ ℓ and the number total of samples N need should both be minimum.
We propose to minimize the Euclidean distance from ϵ ℓ − N need relationship curve to the origin as sampling threshold location, as shown in Figure 2 elbow area. When sampling threshold does not stable, each sampling iteration ℓ is performed according to following steps: (1) Predefined another sample pool with N p sample size, The sample pool is sampled by PMC method without approximate constraint rule in Eq.(4), where each sample based on Eq.(6) and Eq.(7) from the previous generation sample pool X ℓ−1 . The sampling threshold values EI (X N p ) correspond to each sample calculated by the EI function. Sort samples according to the EI (X N p ) from small to large. Let the sorting indexes I = 1, . . . , N p be N accept values.
(2) N need range is [N , N × N p ] obtained by Eq. (9). The curve between EI (X N p )-N need is inversely proportional as discussed above. Due to the large data difference between the EI value and N need , the logarithmic processing of N need is defined as N Lneed . Cost and accuracy have same status in the balance. Normalised EI (X N p ) and the N Lneed by Eq.(10) (defined as Nor(EI (X N p )) and Nor(N Lneed )).
(3) Final sampling threshold is in elbow area of the Nor(EI (X N p )) -Nor(N Lneed ) relationship. Calculate Euclidean distance to the origin for each sample in the relationship curve. And the sampling threshold position corresponds to the minimum distance, with threshold value ϵ ℓ corresponds to the EI value. (4) Samples with EI values greater than epsilon ℓ in N p sample pool are rejected. And replenish the number of accepted samples to N as true iterative sample pool.

B. DENSITY CLUSTERING
Non-parametric PMC sampling method finally generates samples with EI multi-peak characteristic distribution. The DBSCAN method redistributes the final sample pool X: (1) Define the minimum neighborhood distance threshold Eps and the minimum sample size threshold minPts, (2) Select each sample in sample pool X, according to Eps and minPts to determine whether it is a core point, if it is a core point, select all vertices whose density is reachable to form a cluster, (3) Find all clusters in X where the density of all core samples are reachable, and the samples that are not assigned to any cluster are samples as noise points (4) Select samples in each cluster with the minimum EI value as final candidate points. The workflow is shown in Figure 3.
Not each peak in EI function corresponds to a cluster, and emphasis should be placed on the diversity of information for candidate points. The minimum neighborhood distance threshold Eps and the minimum sample size threshold minPts constrain the clustering behavior. Lower Eps and minPts correspond to more candidate points selection. On the contrary, there will be a single-choice phenomenon. NPMS scheme proposes adaptive Eps and minPts. The Eps is shown as follows:

Algorithm 1 The Framework of NPMS Scheme
Input: Constructed kriging model, N p , N , γ , β Output: Candidate points X next 1: Sampling initial pool X N p from LHS method, and sorting by EI value 2: Establishing the relationship between the N Lneed and EI (X N p ) using Eq. (9) and Eq.(10) 3: Calculating the sampling threshold ϵ 1 with the minimum distance 4: Retaining the samples using Eq.(4), and getting the prior distribution P(x) 5: Completing X N p to N as first sample pool X 1 6: Setting weight W 1 = 1 N , . . . , 1 N 7: while sampling threshold does not stable do 8: Sampling X N p from X ℓ−1 by PMC method using Eq. (6) 9: Getting the sampling threshold ϵ ℓ using step 2-3 10: Completing the iterative sample pool X ℓ to number N 11: Setting σ 2 ℓ ← 2Cov( ℓ ) 12: Setting W ℓ through Eq.(7) 13: end while 14: Calculating the cluster threshold Eps and minPts by Eq. (12) and Eq.(13) 15: Clustering the final sample pool X 16: Selecting the minimum EI value in each cluster as final candidate points X next where d(·) represents the distance from each sample to the origin, σ is the standard deviation of X. Eps defines a distance threshold based on overall sample dispersion, regulated by γ parameter. minPts varies as EI converges and is defined as follows:  where ϵ is the final sampling threshold value in NPMS. min(eis) represents the minimum sampling threshold of EGO iteration history. In the end, NPMS scheme pseudocode can be expressed as Algorithm.1.

III. BENCHMARK
In this section, a benchmark case is used to analyze the properties of NPMS scheme. First, we validate characteristics of scheme itself and its performance in EGO applications.
And then compare it with the original EI strategy to illustrate the feasibility and advantages. Finally, model parameters' effect is analyzed, and reasonable values are given. The benchmark is named Three-hump camel-back (Camel3) function, and formula is shown in Eq. (14). Camel3 has a regular U-shaped structure in space shown in Figure 4, with one global minimum, two local minimums, and one global maximum. The difficulty of capturing changes in function values around global and local minima makes optimization take a long to converge.
Five indicators are involved in quantifying the impact of strategy. The standardized average LOO CV (CV ) and coefficient of determination (R 2 ) [40] are used to evaluate the prediction performance of kriging model: where n and T are the numbers of observable points and test points. y i andŷ i denote the true responses and predicted responses.ȳ is the mean of true responses. CV < 0.1 is used as the EGO stopping condition in the benchmark.
To evaluate the performance of NPMS applied in EGO, I s , N s , and Res are proposed. I s is the total number of EGO sequential iterations and also measures the speed of EGO optimization convergence. N s is the total number of candidate points required by infilling strategy when EGO meets stopping condition. N s determines the computational cost optimization. Res is defined as residual of global minimum and optimal result, with more accurate optimization results implying a smaller Res. For the above metrics, higher R 2 and lower CV , I s , N s , Res are usually desired. During benchmark, all initial points [41] and test points [25] are generated by the LHS method following the maximin criterion. And all benchmarks are independently repeated 30 times.

A. CONVERGENCE AND ADAPTIVE
First, we verify the convergence of the NPMS scheme itself. In the non-parametric PMC sampling stage, the NPMS scheme aims to converge the sample pool to high EI value sub-domains. The initial NPMS scheme generates a uniform sample pool containing multi-peak information. As sampling iterative, samples inherit from the previous sample pool and converge to target multi-peak constrained by sampling threshold, as shown in Figure 5 (b). Until sampling threshold becomes stable, PMC sampling process is complete. Samples are finally distributed among the EI function multi-peak from entire space, while the areas with EI values smaller than sampling threshold are considered meaningless. NPMS aims to converge the samples to candidate points during clustering stage. Based on density features, the sample pool is re-divided into new clusters, ensuring each cluster adequately expresses information gain of EI function. The candidate points are selected with the best EI values in each new cluster, as shown in Figure (c) red points. Non-parametric sampling can sample from high EI value areas, while clustering can select global points. Thus, the NPMS scheme itself is convergent.
When applied to EGO, NPMS scheme has good sampling threshold convergence and dynamics candidate points selection. For each EGO iteration, the fixed EI function ensures that sampling threshold relationship in NPMS scheme does not change. Furthermore, the actual sampling threshold position will not change either. Due to sampling uncertainty, the threshold will fluctuate during sampling iterative and tend to be stable. Figure 6 shows the history of sampling threshold in NPMS scheme when at 20 th , 40 th and 60 th EGO iterations, respectively. Each NPMS scheme can obtain a stable final sampling threshold. And as EGO iteration proceeds, sampling threshold tends to zero as shown in Figure 6 (d). Secondly, adaptive clustering based on sample density ensures NPMS scheme dynamics candidate points selection. With the requirement that each cluster maintains    its distance, avoiding the problem of selected candidates falling into aggregation. Figure 7 shows the NPMS scheme clustering and points selection in the EGO iteration. Samples come from global sub-domains with high EI values, and the candidate points are derived from these promising samples, which effective collaboration in sampling and clustering. As shown in Figure 7 (d), NPMS scheme maintains the candidate point selection dynamics when EI function changes and tends to single-point as EI function convergence.
To illustrate the performance of NPEI scheme, we compare it with original EI strategy. Figure 8 and Figure 9 show the convergence histories of EI strategy and NPMS scheme for the Camel3 case. Benchmark is conducted in 30 independent trials, and the average results are shown in Table 1, where the best results are marked in bold. Result shows that samples selected by the two strategies can construct useful models. In the iterative history of EGO, NPMS scheme is faster in building kriging models in the initial steps and has a significant tendency to converge in finding the optimal values. The performance improvement of EGO comes from the broader range of EI function calls and efficient multi-point selection in NPMS scheme. In the first stage, the PMC method generates more samples and performs EI calculations during the iterative process, resulting in the sample size that can be selected being much larger than that of the original EI strategy. The average time consumed internally in each NPMS scheme increased by 77.1% compared to the EI strategy. A large number of samples is more likely to contain optimal values, which results in a residual reduction of 14.6% compared to EI strategy. Meanwhile, the clustering in NPMS second stage can efficiently filter out representative candidate points, reducing 15.8% candidate point number compared to EI strategy. The cost of computing candidate points in EGO is often much higher than calling EI functions, and it is worthwhile for NPMS solutions to improve optimization results and reduce the number of evaluations by calling more EI functions.
For demonstrating global and efficient nature of NPMS, the candidate points distribution of EI and NPMS is analyzed, as shown in Figure 10 (a) and (b). The objective function VOLUME 11, 2023  has one global minimum and two local minima on the same line A-B, as shown in red dashed line. Both EI and NPMS strategies can converge to a global minimum during optimization. However, EI strategy is prone to falling into local optima and tends to fall into a state of point aggregation. In contrast, NPMS scheme is more decentralized in the objective function space. To better represent the distribution, candidate points within selected range (solid lines) are projected onto dashed line. Their distances to the global minimum are counted, yielding the statistics shown in Figure 10 (c). EI strategy produces a fluctuation distribution of candidate points and often higher relative frequencies of points in the two local minima regions. In contrast, NPMS distribution is more uniform, with no significant low peaks, while focusing more on the global minimum area.

B. MODEL PARAMETER ANALYSIS
The NPMS scheme involves four parameters in optimization process : N , N p , γ , and β. N and N p are sampling parameters of NPMS first stage, representing the sampling number in actual sample pool and pre-sample pool, respectively. N p involves generating a non-parametric sampling threshold. N significantly influences the NPMS scheme, which determines sampling cost and candidate selection accuracy. Large sample sizes can be stably distributed over multi-peak regions, resulting in a high sampling cost. Small sample size can maintain a low sampling cost but cannot accurately capture the multiple peaks of the EI function, resulting in an unstable candidate selection capability. γ and β are the clustering parameters, which influence the ability of the NPMS scheme to balance informativeness and diversity at candidate points. High γ and β can cause NPMS scheme to fail to select enough multiple points and degrade to a singlepoint strategy. Conversely, low γ and β will cause NPMS scheme to produce aggregated candidate points, resulting in redundancy. As mentioned above, since scheme does not require the highest sampling accuracy, setting N p = 200d as a recommended reference value.   standard deviation of Res decrease as N increases until N is 0.3. When N is greater than 0.5, NPMS scheme could find a good optimization result, but the average of Res tends to increase. In contrast, N is more optimally stable between 0.3 and 0.5. The mean value of total optimization iterations I s tends to increase as N increases, while the standard deviation of I s does not change significantly during the change. In detail, the iteration I s increases slightly between N = 0.3 and N = 0.4. Overall, N = 0.3 is a good choice for the trade-off between optimization results and iteration speed. Table 2 shows the results of γ and β in different combinations, and values after ''±'' represent the standard deviations. Average rank is calculated by Res, I s , and N s . Parameters are taken at 0.2 intervals, and N is fixed at the suggested value. The γ and β have synergistic effects. When both γ and β are less than 0.3, NPMS scheme had fewer EGO iterations but produced more candidate points. When both γ and β are greater than 0.5, EGO results are stable, but I s and N s tend to rise. NPMS scheme loses advantage of adaptive multi-point, resulting in poorer rankings. When a parameter is too small, and the other is dominant, too large value choices can lead to poor I s and N S and unstable optimization results. It is difficult to make a set of parameters that will give NPMS the best I s , N s , and Res in the optimization process. Based on VOLUME 11, 2023 the average ranking, γ = 0.5 and β = 0.5 are reasonable choices and then applied in the subsequent sections.

IV. PERFORMANCE COMPARISON WITH OTHER STRATEGIES
This section compares NPMS with two single-point infilling strategies (i.e., EI and GO-HI) and three parallel strategies (i.e., CL, KB, and PEI) under four test groups. NPMS scheme uses the parameters discussed above. The tests are conducted independently 30 times, with kriging reaching CV < 0.1 first time as EGO convergence condition.
The benchmark cases are provided in Table 3. Nine cases of the same dimension are divided into three groups according to the input space size (V), which is defined by its Euclidean volume: where ub i and lb i denote the input space's upper and lower bound. The input space size is categorized into three classes, small (V ≤ 100), medium (100 < V ≤ 400), and large (V > 400). And the cases in each group are selected in increasing order of difficulty (modality, separability), as discussed in Jamil and Yang [42]. Group 4 focuses on testing the adaptability of each infilling strategy to dimensional changes, in which DIXON-PRICE, Hartmann, Trid Function, and Levy Function dimensions correspond to 4, 6, 8, and 10, respectively. The α, A and P in Har6 case and ω i in Levy case is defined as follow:  Table 4 and Table 5 show the performance of kriging model for strategies, where the name of parallel infilling strategies are tilted, and the best results marked in bold. The optimization results (Res, N s and I s ) for each strategy are shown in Table 6 and Table 7. The Friedman test results are shown in Table 8. For group 1, due to small input space size for cases, all infilling strategies can build a useful kriging model and find a good minimum at a low computational cost. The parallel infilling strategy requires more candidate point numbers while shorter iterations. In the Leo case, GO-HI infilling strategy shows the lowest candidate point number but needs more iterations than NPMS scheme. The PEI strategy shows the best kriging model performance but has a computational cost close to twice as high as NPMS scheme. In the Alp case, NPMS scheme saves at least 20% of the average optimization computational cost and needs 39.1% of the average number of iterations compared to singlepoint strategies. NPMS scheme has the lowest computational cost compared to other parallel strategies and has unstable optimization and kriging performance. The NPMS scheme demonstrates a cost-saving advantage in Buk case, which needs only 77% of N s compared to other strategies but has the worst kriging model. NPMS scheme has a slight advantage over other infilling strategies for the small input size group and has a poor EGO iteration time. Still, it ranks first in candidate point savings and optimization results.
For group 2 (Cub, Hol, Cro), as optimization difficulty increases, strategies require more candidate points and EGO iterations. The fitting ability of constructed kriging model also decreases. In the Cub case, all strategies keep a R 2 score of kriging above 0.9, and NPMS scheme does not get the best results. However, NPMS scheme can find better results in the same iterations compared to other parallel strategies while saving at least 52.5% of the average candidate point number. In the Hol case and Cro case, all strategies do not build a valid kriging model, and complex functions often slow single-point strategies to reach EGO-stopping conditions. While parallel infilling strategies reach EGO-stopping conditions with fewer iterations. However, apart from the NPMS scheme, other parallel strategies required more candidate points than singlepoint strategies. Compared to single-point strategies, the N s required for NPMS scheme is reduced by at least 16.8% with similar optimization results found. Compared to CL strategy, NPMS scheme saves at least 13% of the computational cost and improves the results by 41.1%. In the medium space case, NPMS scheme provides excellent result accuracy and cost savings in comparison while requiring more optimization iterations.
For group 3 (Boh, Bar, EAVD), due to the increased search range, the optimal value is not always found accurately in each strategy experiment, but NPMS is shown to have the best average optimization results and minor result fluctuations. Figure 12 and Figure 13 show residual results and convergence history for each infilling strategy. The large input size resulted in infilling strategies failing to construct  kriging models with good predictive power. Each strategy required more candidate points and iterations to sample from the uncertainty sub-domains. The parallel strategy has the advantage of reducing optimization iterations. In VOLUME 11, 2023 the case of EAVD, NPMS scheme has a clear tendency towards optimal values in initial EGO iteration. Compared to other parallel strategies, NPMS scheme reduces the cost of optimization by at least 62.3% and achieves 20.6% improvement in result accuracy. In the Boh case, NPMS scheme does not have good iterative convergence. Compared to EI strategy, NPMS scheme requires only 75.4% of I s and 151.5% of N s on average while improving Res by 65.2%. Compared to other parallel infilling strategies, NPMS scheme has the lowest computational cost and reduces at least 36.4%. In the Bar case, compared to other infilling strategies, the NPMS scheme reduces at least 22.9% of N s and 14.6% of I s , while the residual mean is shortened by 54.6%. CL strategy has better kriging model performance in comparison. As shown in Figure 14, at the first EGO iteration, NPMS generates nine candidate points, which are widely distributed. In comparison, the CL strategy is fixed to generate four candidate points. As the EGO iteration   increases, the number of candidate points generated by NPMS decreases and is uniformly distributed, while CL appears as aggregating points. And the cumulative total number of candidate points in the CL strategy is significantly more than NPMS scheme. By comparison, because of its adaptive and global nature, NPMS makes EGO optimization more effective. NPMS scheme can quickly establish a global search at the beginning of optimization, generating more candidate points in uncertain regions and increasing the likelihood of finding the optimal value. During EI function convergence, NPMS scheme dynamically changes the number of points selected, leading to fewer candidate points than CL strategy after 80 th iteration of EGO. In addition, NPMS scheme maintains a balance between the amount of information and diversity of candidate points that each point kept away from each other, which is less likely to cause points aggregation. Above, NPMS scheme has the best result ranking in large input space cases.
For group 4, the situation is similar to group 3, where NPMS outperforms the other strategies in general, and the convergence history of strategy residuals shown in Figure 15. In Dix and Hart's case, CL and KB reach EGO stopping conditions faster, while NPMS scheme has better optimization results and costs. In the Dix case, the convergence of  NPMS scheme residuals has a pronounced downward trend. Compared to single-point infilling strategies, NPMS scheme needs only 56.7% of N s and 51.7% of I s . Compared to CL strategy, NPMS scheme has slower optimization iterations and saves 17.4% of computational cost. In the Hart case, NPMS scheme has the worst convergence compared to other parallel infilling strategies but achieves smaller residual values and saves at least 7.71% of computational cost. In the Trid case, all infilling strategies fail to find good optimal values, and NPMS scheme has well average residual value. Compared to EI strategy, NPMS scheme can find good results in the initial optimization iterations, and needs 112% of N s and 55.6% of I s . In the Levy case, NPMS scheme is more likely to find better values, needs 8.92% less computational cost on average, and 17.3% fewer iterations than other parallel infilling strategies. On average, NPMS scheme has the best ranking, saving on average 45.8% of optimization cost and 25.7% of iterations while improving optimization results by 17.7%.
Overall, Friedman test shows that NPMS scheme is the best at finding minima and cost savings. In small and medium input space size cases, NPMS has a slight advantage in optimization efficiency and performs poorly in constructing kriging models. On the other hand, when dimensionality or input size rise, EI function focuses more on exploring uncertainty regions. NPMS scheme samples from global interest areas with much EI function calculation and reduced optimization evaluation make the scheme save costs and find good results.

V. CONCLUSION
We have proposed an extended adaptive parallel infilling strategy of Expected Improvement (EI) with NPMS scheme consisting of two processing stages, that is, PMC sampling and DBSCAN clustering. The scheme can adaptively generate samples in sub-domains with high EI values and finally obtain the desired multi-point in a convergent manner. When benchmarked against the three-hump hump function with high drop characteristics, our scheme improves the result accuracy by 14.6% and reduces the cost of optimization evaluation by 15.8%, relative to conventional EI strategy. In further tests with other five strategies, NPMS with optimized parameters can achieve an increase in result accuracy and a reduction in the number of candidate points across 13 functions with different input spaces, difficulties, and dimensions. When facing poor kriging prediction, especially in large input spaces and high-dimensional functions, NPMS can surprisingly give an average of 72% higher optimization result accuracy than the other five strategies. Taking advantage of broad sampling and adaptive clustering, in fact, NPMS can ensure that candidate points come from promising sub-domains and dynamically control the number of candidate points, making the extended EI strategy more efficient and less costly. Therefore, our model provides a novel idea for the exploration-exploitation balance in EI strategy, which establishes the basis for solving more complex EGO problems in the future.
ZHEN LIU received the master's degree in materials science and engineering in Grenoble National Institute of Technology, France, in 2014, the doctor degree in material physics and chemistry from Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, in 2019. From 2008 to 2012, he studied nuclear engineering and technology at North China Electric Power University. He is currently an Associate Researcher. At present, he is mainly engaged in the application of molecular dynamics methods and artificial intelligence algorithms in the field of multi-scale coupling simulation of material properties. He has published more than ten papers in Scripta Materialia, Journal of Alloys and Compounds, and Ceramics International.
YIFAN LI is currently pursuing degree with College of Materials Science and Chemical Engineering, Harbin Engineering University. He is mainly engaged in theoretical research in the field of nuclear energy materials, using first principles, molecular dynamics, phase fields, and rate theory to study the microstructure, interfaces, and service properties of nuclear energy materials from different aspects and levels.
ZHEYU HU was born in Zhejiang, China. He received the B.S. degree from Quzhou University, in 2020. He is currently pursuing the M.S. degree in computer technology from Ningbo University.
His research interests include artificial intelligence and data science.