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arxiv: 2604.14975 · v1 · submitted 2026-04-16 · 📊 stat.CO · cs.NA· math.NA· stat.AP· stat.ML

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Theta-regularized Kriging: Modelling and Algorithms

Xuelin Xie , Xiliang Lu
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Pith reviewed 2026-05-10 09:28 UTC · model grok-4.3

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The pith

Theta-regularized Kriging penalizes the theta hyperparameter in Gaussian stochastic processes using Lasso, Ridge, or Elastic-net, yielding higher accuracy and stability than prior penalized Kriging variants on numerical tests and engineering cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Kriging predicts unknown values from scattered data by assuming the underlying function behaves like a smooth random process. A key control knob called theta sets how fast the influence between data points fades with distance. The new method adds a penalty term to theta during model fitting, much like shrinking coefficients in regression to avoid wild guesses. This comes from rewriting the usual maximum-likelihood objective with extra regularization. The authors give step-by-step algorithms for solving the penalized problem and for choosing the penalty strength via a geometric search version of cross-validation. When run on nine standard math test functions and two real engineering problems, the regularized versions gave lower prediction errors and more consistent results than earlier penalized Kriging approaches.

Core claim

Compared with other penalized Kriging models, the proposed model performs better in terms of accuracy and stability.

Load-bearing premise

That penalizing theta specifically, derived from the maximum-likelihood perspective, produces genuine gains that generalize beyond the nine numerical functions and two engineering examples shown, without the improvements arising from post-hoc tuning or limited test diversity.

Figures

Figures reproduced from arXiv: 2604.14975 by Xiliang Lu, Xuelin Xie.

Figure 1
Figure 1. Figure 1: Uncertainty estimation for Kriging prediction. The Kriging model is defined as a weighted linear combination of known points [30]: 𝐲̂ (𝐱) = ∑𝑛 𝑖=1 𝜔𝑖 (𝐱)𝐲𝑖 = 𝜔(𝐱) 𝑇 𝐲, (1) where 𝜔𝑖 (𝐱) is the weight coefficient of 𝐲𝑖 , 𝐲 = ( 𝐲1 , 𝐲2 ,…, 𝐲𝑛 )𝑇 is the response value matrix, and 𝜔(𝐱) = [𝜔1 (𝐱), 𝜔2 (𝐱), ..., 𝜔𝑛 (𝐱)]𝑇 is the weighting vector. To calculate the weighting coefficients, the Kriging model assumes th… view at source ↗
Figure 2
Figure 2. Figure 2: Interpolation results of Kriging models with different 𝜽 To demonstrate the importance of Theta-regularization, we first use the Forrester function to verify the impact of parameter 𝜽 on the prediction results of the Kriging model. The expression of Forrester function [38] is shown in equation (10). 𝑓(𝐱) = (6𝐱 − 1)2 sin(12𝐱 − 4), 𝐱 ∈ [0, 1]. (10) The Latin Hypercube Sampling (LHS) method [39] was employed … view at source ↗
Figure 3
Figure 3. Figure 3: Universal Kriging and Theta-regularized Kriging model Unfortunately, the complexity of the Kriging model increases significantly as the problem dimension increases, leading to greater computational cost but lower efficiency. For example, when considering the 8-dimensional Trid function: 𝑓 (𝐱) = ∑ 8 𝑖=1 (𝐱𝑖 − 1)2 − ∑ 8 𝑖=2 𝐱𝑖𝐱𝑖−1. (13) where 𝐱𝑖 ∈ [-1, 1] , 𝑖 = 1, 2,...,8. We calculate its Hessian matrix, an… view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the proposed TRK model. Firstly, samplings are obtained from the sample space. Subsequently, regression basis functions, correlation functions, and penalty functions are defined. Following this, a nested optimization problem needs to be solved, ultimately building the TRK model. 3.2. Parameter sensitivity analysis of the TRK model Due to the addition of regularization penalties in the TRK m… view at source ↗
Figure 5
Figure 5. Figure 5: Sensitivity of TRK models to different parameters. 3.3. GSCV algorithm for solving optimal regularization parameters According to the results of parameter sensitivity analysis, in practical applications, we can typically manually select the regularization coefficient of the TRK model in the form of a geometric progression. However, considering that manually selected regularization coefficients are subjecti… view at source ↗
Figure 6
Figure 6. Figure 6: Schematic diagram of the two-dimensional test functions [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Results of 10 repetitions (Borehole simulation) UK TR-LK TR-RK TR-EK PB-LK PB-RK PB-EK MPBK 0.7 0.8 0.9 Means of R 2 UK TR-LK TR-RK TR-EK PB-LK PB-RK PB-EK MPBK 3000 4000 5000 Means of RMSE UK TR-LK TR-RK TR-EK PB-LK PB-RK PB-EK MPBK 2000 3000 4000 Means of MAE Methods Methods Methods [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Boxplot of Borehole simulation From [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Results of 10 repetitions (Steel columns simulation) Xuelin Xie et al.: Preprint submitted to Elsevier Page 20 of 26 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Boxplot of Steel columns simulation [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Iterative curves of TRK models on different test functions In addition, we investigated the convergence behavior of the TRK algorithm [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
read the original abstract

To obtain more accurate model parameters and improve prediction accuracy, we proposed a regularized Kriging model that penalizes the hyperparameter theta in the Gaussian stochastic process, termed the Theta-regularized Kriging. We derived the optimization problem for this model from a maximum likelihood perspective. Additionally, we presented specific implementation details for the iterative process, including the regularized optimization algorithm and the geometric search cross-validation tuning algorithm. Three distinct penalty methods, Lasso, Ridge, and Elastic-net regularization, were meticulously considered. Meanwhile, the proposed Theta-regularized Kriging models were tested on nine common numerical functions and two practical engineering examples. The results demonstrate that, compared with other penalized Kriging models, the proposed model performs better in terms of accuracy and stability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes Theta-regularized Kriging, which penalizes the hyperparameter theta in the Gaussian process correlation function using Lasso, Ridge, or Elastic-net penalties. This is derived from a penalized maximum likelihood objective, and the authors detail iterative optimization algorithms and a geometric-search cross-validation procedure for tuning the regularization parameter lambda. The method is tested on nine numerical functions and two engineering examples, where it is claimed to outperform other penalized Kriging models in terms of prediction accuracy and stability.

Significance. Should the empirical advantages prove robust under controlled comparisons, this regularization strategy on theta could improve the reliability of Kriging surrogates in engineering design and optimization tasks by mitigating overfitting in hyperparameter estimation. The explicit algorithmic contributions, including the CV tuning method, add practical value and support reproducibility.

major comments (2)
  1. [§5 (Numerical Experiments)] §5 (Numerical Experiments): The central claim that the proposed model outperforms other penalized Kriging models in accuracy and stability is not fully supported, as there is no explicit confirmation that baseline methods were tuned using the identical geometric-search cross-validation procedure described for Theta-regularized Kriging. Without this, observed differences may stem from unequal hyperparameter search effort rather than the specific penalization of theta.
  2. [§5.1, Tables 1-2] §5.1, Tables 1-2: The reported RMSE and stability metrics lack error bars, standard deviations from repeated runs, or formal statistical significance tests, which undermines the ability to verify that the claimed improvements are reliable rather than artifacts of the specific test functions or initialization.
minor comments (2)
  1. [Introduction] Introduction: Expand the discussion of why penalizing theta (as opposed to the nugget or other parameters) is particularly effective, with additional citations to existing sensitivity analyses in Gaussian process literature.
  2. [Algorithm 1] Algorithm 1: Clarify the convergence criteria and initialization strategy for the iterative regularized optimization to improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments on our manuscript. We address each major comment below, indicating the revisions we will make to improve the clarity and rigor of the numerical experiments section.

read point-by-point responses

  1. Referee: §5 (Numerical Experiments): The central claim that the proposed model outperforms other penalized Kriging models in accuracy and stability is not fully supported, as there is no explicit confirmation that baseline methods were tuned using the identical geometric-search cross-validation procedure described for Theta-regularized Kriging. Without this, observed differences may stem from unequal hyperparameter search effort rather than the specific penalization of theta.

    Authors: We acknowledge that the manuscript does not explicitly confirm that the baseline penalized Kriging models were tuned with the identical geometric-search cross-validation procedure. In the original experiments, baselines followed the hyperparameter settings and tuning approaches reported in their respective source papers (typically grid search over theta with fixed or default regularization). To address this, we will revise §5 to include a dedicated subsection detailing the uniform application of the geometric-search CV procedure to all compared methods, including the baselines. Where necessary, we will re-run the baseline comparisons under this consistent tuning protocol to ensure the performance differences can be attributed to the theta penalization. revision: yes

  2. Referee: §5.1, Tables 1-2: The reported RMSE and stability metrics lack error bars, standard deviations from repeated runs, or formal statistical significance tests, which undermines the ability to verify that the claimed improvements are reliable rather than artifacts of the specific test functions or initialization.

    Authors: We agree that the current presentation of results in Tables 1-2 would be strengthened by quantitative measures of variability and statistical testing. We will revise the numerical experiments section to report means and standard deviations computed over 10 independent runs with randomized initializations for each method and test function. Additionally, we will include results from paired statistical tests (e.g., Wilcoxon signed-rank or t-tests with appropriate corrections) to assess the significance of observed differences in RMSE and stability metrics. revision: yes

Circularity Check

0 steps flagged

No circularity: regularization is an explicit additive penalty on the standard Kriging MLE; performance claims rest on external benchmarks.

full rationale

The derivation begins from the ordinary maximum-likelihood objective for Kriging and inserts an explicit penalty term on the correlation-length vector theta (Lasso, Ridge, or Elastic-net). This construction is not self-referential; the penalized objective is a standard regularized extension rather than a re-expression of its own fitted values. The geometric-search CV procedure is described as a separate tuning step for the penalty strength, not as a post-hoc fit that is then relabeled a prediction. All reported accuracy and stability gains are measured on nine independent numerical test functions plus two engineering examples, none of which are used to define the model itself. No self-citation chain, uniqueness theorem, or ansatz smuggling appears in the load-bearing steps. The central empirical claim therefore remains falsifiable against external data and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on the standard Gaussian-process assumption for Kriging plus the addition of a tunable regularization strength that is chosen by cross-validation; no new physical entities are introduced.

free parameters (1)
  • regularization strength lambda
    Penalty coefficient for each of Lasso, Ridge, and Elastic-net terms; chosen via the geometric-search cross-validation procedure.
axioms (1)
  • domain assumption The observed responses follow a Gaussian stochastic process whose correlation structure is controlled by the hyperparameter theta.
    Standard modeling assumption invoked when the regularized likelihood is written.

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Works this paper leans on

54 extracted references · 46 canonical work pages

  1. [1]

    Hosseini, M

    N. Hosseini, M. Tadjfar, A. Abba, Configuration optimization of two tandem airfoils at low Reynolds numbers, Appl. Math. Model. (102) (2022) 828–846.doi:10.1016/j.apm.2021.10.029

    work page doi:10.1016/j.apm.2021.10.029 2022

  2. [2]

    D. Zhan, H. Xing, A fast Kriging-assisted evolutionary algorithm based on incremental learning, IEEE Trans. Evol. Comput. 25 (5) (2021) 941–955.doi:10.1109/TEVC.2021.3067015. Xuelin Xie et al.:Preprint submitted to Elsevier Page 24 of 26 Theta-regularized Kriging: Modelling and Algorithms

    work page doi:10.1109/tevc.2021.3067015 2021

  3. [3]

    M. A. Alves, P. J. Oliveira, F. T. Pinho, Numerical methods for viscoelastic fluid flows, Annu. Rev. Fluid Mech. 53 (1) (2020) 1–33. doi:10.1146/annurev-fluid-010719-060107

    work page doi:10.1146/annurev-fluid-010719-060107 2020

  4. [4]

    M. A. Sanchez, B. Cockburn, N.-C. Nguyen, J. Peraire, Symplectic Hamiltonian finite element methods for linear elastodynamics, Comput. Meth. Appl. Mech. Eng. 381 (2021) 113843.doi:10.1016/j.cma.2021.113843

    work page doi:10.1016/j.cma.2021.113843 2021

  5. [5]

    Chugh, Y

    T. Chugh, Y. Jin, K. Miettinen, J. Hakanen, et al., A surrogate-assisted reference vector guided evolutionary algorithm for computationally expensive many-objective optimization, IEEE Trans. Evol. Comput. 22 (1) (2018) 129–142.doi:10.1109/TEVC.2016.2622301

    work page doi:10.1109/tevc.2016.2622301 2018

  6. [6]

    X.Peng,J.Kou,W.Zhang,Multi-fidelitynonlinearunsteadyaerodynamicmodelinganduncertaintyestimationbasedonHierarchicalKriging, Appl. Math. Model. 122 (2023) 1–21.doi:10.1016/j.apm.2023.05.031

    work page doi:10.1016/j.apm.2023.05.031 2023

  7. [7]

    R. T. Johnson, The design and analysis of computer experiments, Springer Science & Business Media, New York, 2008

    2008

  8. [8]

    P. V. Arun, A comparative analysis of different DEM interpolation methods, Egypt. J. Remote Sens. Space Sci. 16 (2013) 133–139. doi:10.1016/j.ejrs.2013.09.001

    work page doi:10.1016/j.ejrs.2013.09.001 2013

  9. [9]

    Pavlicek, V

    K. Pavlicek, V. Kotlan, I. Dolezel, Applicability and comparison of surrogate techniques for modeling of selected heating problems, Comput. Math. Appl. 78 (9) (2019) 2897–2910.doi:10.1016/j.camwa.2019.02.013

    work page doi:10.1016/j.camwa.2019.02.013 2019

  10. [10]

    Johari, H

    A. Johari, H. Fooladi, System reliability analysis of site slope using the conditional spatial variability of soil properties, Iran. J. Sci. Technol.- Trans. Civ. Eng. 47 (6) (2023) 3743–3761.doi:10.1007/s40996-023-01200-z

    work page doi:10.1007/s40996-023-01200-z 2023

  11. [11]

    Johari, H

    A. Johari, H. Fooladi, Simulation of the conditional models of borehole’s characteristics for slope reliability assessment, Transp. Geotech. 35 (2022). doi:10.1016/j.trgeo.2022.100778

    work page doi:10.1016/j.trgeo.2022.100778 2022

  12. [12]

    Pouladi, A

    N. Pouladi, A. B. Moller, S. Tabatabai, M. H. Greve, Mapping soil organic matter contents at field level with Cubist, Random Forest and Kriging, Geoderma 342 (2019) 85–92.doi:10.1016/j.geoderma.2019.02.019

    work page doi:10.1016/j.geoderma.2019.02.019 2019

  13. [13]

    X. Wu, Q. Lin, W. Lin, Y. Ye, et al., A Kriging model-based evolutionary algorithm with support vector machine for dynamic multimodal optimization, Eng. Appl. Artif. Intell. 122 (2023) 106039.doi:10.1016/j.engappai.2023.106039

    work page doi:10.1016/j.engappai.2023.106039 2023

  14. [14]

    E. Park, B. W. Brorsen, A. Harri, Using Bayesian Kriging for spatial smoothing in crop insurance rating, Am. J. Agr. Econ. 101 (1) (2019) 330–351.doi:10.1093/ajae/aay045

    work page doi:10.1093/ajae/aay045 2019

  15. [15]

    doi:10.1016/j.rse.2019.111456

    K.Wu,G.A.Rodriguez,M.Zajc,E.Jacquemin,etal.,Anewdrone-borneGPRforsoilmoisturemapping,RemoteSens.Environ.235(2019) 111456. doi:10.1016/j.rse.2019.111456

    work page doi:10.1016/j.rse.2019.111456 2019

  16. [16]

    L. Chen, H. Qiu, L. Gao, C. Jiang, et al., A screening-based gradient-enhanced Kriging modeling method for high-dimensional problems, Appl. Math. Model. 69 (2019) 15–31.doi:10.1016/j.apm.2018.11.048

    work page doi:10.1016/j.apm.2018.11.048 2019

  17. [17]

    X. Cai, L. Gao, X. Li, Efficient generalized surrogate-assisted evolutionary algorithm for high-dimensional expensive problems, IEEE Trans. Evol. Comput. 24 (2) (2020) 365–379.doi:10.1109/TEVC.2019.2919762

    work page doi:10.1109/tevc.2019.2919762 2020

  18. [18]

    Dietterich, Overfitting and undercomputing in machine learning, ACM Comput

    T. Dietterich, Overfitting and undercomputing in machine learning, ACM Comput. Surv. 27 (3) (1995) 326–327.doi:10.1145/212094. 212114

    work page doi:10.1145/212094 1995

  19. [19]

    Z. Luo, S. Cai, G. Chen, J. Gao, et al., Improving data analytics with fast and adaptive regularization, IEEE Trans. Knowl. Data Eng. 33 (2) (2021) 551–568.doi:10.1109/TKDE.2019.2916683

    work page doi:10.1109/tkde.2019.2916683 2021

  20. [20]

    Calatroni, G

    L. Calatroni, G. Garrigos, L. Rosasco, S. Villa, Accelerated iterative regularization via dual diagonal descent, SIAM J. Optim. 31 (1) (2021) 754–784.doi:10.1137/19M1308888

    work page doi:10.1137/19m1308888 2021

  21. [21]

    Z. Sun, M. Dai, Y. Wang, S.-B. Lin, Nystrom regularization for time series forecasting, J. Mach. Learn. Res. 23 (312) (2022) 1–42

    2022

  22. [22]

    W. Gong, Z. Huang, L. Yang, Accurate regularized Tucker decomposition for image restoration, Appl. Math. Model. 123 (2023) 75–86. doi:10.1016/j.apm.2023.06.031

    work page doi:10.1016/j.apm.2023.06.031 2023

  23. [23]

    Y. Fan, W. Yang, A backpropagation learning algorithm with graph regularization for feedforward neural networks, Inf. Sci. 607 (2022) 263–277.doi:10.1016/j.ins.2022.05.121

    work page doi:10.1016/j.ins.2022.05.121 2022

  24. [24]

    Hung, Penalized blind Kriging in computer experiments, Stat

    Y. Hung, Penalized blind Kriging in computer experiments, Stat. Sin. 21 (3) (2011) 1171–1190.doi:10.5705/ss.2009.226

    work page doi:10.5705/ss.2009.226 2011

  25. [25]

    Zhang, W

    Y. Zhang, W. Yao, S. Ye, X. Chen, A regularization method for constructing trend function in Kriging model, Struct. Multidiscip. Optim. 59 (4) (2019) 1221–1239.doi:10.1007/s00158-018-2127-8

    work page doi:10.1007/s00158-018-2127-8 2019

  26. [26]

    Park, Lasso Kriging for efficiently selecting a global trend model, Struct

    I. Park, Lasso Kriging for efficiently selecting a global trend model, Struct. Multidiscip. Optim. 64 (3) (2021) 1527–1543.doi:10.1007/ s00158-021-02939-7

    2021

  27. [27]

    Y.Zhao,Z.Feng,M.Li,X.Li,ModifiedPenalizedblindKrigingforefficientlyselectingaglobaltrendmodel,J.Stat.Comput.Simul.93(17) (2023) 3052–3066.doi:10.1080/00949655.2023.2216335

    work page doi:10.1080/00949655.2023.2216335 2023

  28. [28]

    D. G. Krige, A statistical approach to some basic mine valuation problems on the Witwatersrand, J. S. Afr. Inst. Min. Metall. 52 (6) (1952) 119–139

    1952

  29. [29]

    Matheron, The intrinsic random functions and their applications, Adv

    G. Matheron, The intrinsic random functions and their applications, Adv. Appl. Probab. 5 (3) (1973) 439–468.doi:10.2307/1425829

    work page doi:10.2307/1425829 1973

  30. [30]

    Alex Shtoff, Elie Abboud, Rotem Stram, and Oren Somekh

    J. Sacks, W. J. Welch, T. J. Mitchell, H. P. Wynn, Design and analysis of computer experiments, Stat. Sci. 4 (4) (1989) 409–423.doi: 10.1214/ss/1177012413

    work page doi:10.1214/ss/1177012413 1989

  31. [31]

    Lophaven, H

    S. Lophaven, H. Nielsen, S. J., DACE - A MATLAB Kriging Toolbox (version 2.0) [software], Technical University of Denmark, August 1,

  32. [32]

    http://www2.imm.dtu.dk/pubdb/p.php?1460

  33. [33]

    Y. Pang, X. Lai, S. Zhang, Y. Wang, et al., A Kriging-assisted global reliability-based design optimization algorithm with a reliability- constrained expected improvement, Appl. Math. Model. 121 (2023) 611–630.doi:10.1016/j.apm.2023.05.018

    work page doi:10.1016/j.apm.2023.05.018 2023

  34. [34]

    Y.Song,X.Wang,G.Wright,D.Thatcher,etal.,Trafficvolumepredictionwithsegment-basedregressionKriginganditsimplementationin assessing the impact of heavy vehicles, IEEE Trans. Intell. Transp. Syst. 20 (1) (2019) 232–243.doi:10.1109/TITS.2018.2805817

    work page doi:10.1109/tits.2018.2805817 2019

  35. [35]

    X.Zhang,L.Wang,J.D.Sorensen,REIF:Anovelactive-learningfunctiontowardadaptiveKrigingsurrogatemodelsforstructuralreliability analysis, Reliab. Eng. Syst. Saf. 185 (2019) 440–454.doi:10.1016/j.ress.2019.01.014

    work page doi:10.1016/j.ress.2019.01.014 2019

  36. [36]

    Huang, T

    D. Huang, T. Allen, W. Notz, N. Zeng, Global optimization of stochastic black-box systems via sequential Kriging meta-models, J. Glob. Optim. 34 (3) (2006) 441–466.doi:10.1007/s10898-005-2454-3. Xuelin Xie et al.:Preprint submitted to Elsevier Page 25 of 26 Theta-regularized Kriging: Modelling and Algorithms

    work page doi:10.1007/s10898-005-2454-3 2006

  37. [37]

    62 (12) (2016) 4308–4322.doi:10.1002/aic.15352

    O.T.Kajero,R.B.Thorpe,T.Chen,B.Wang,etal.,Krigingmeta-modelassistedcalibrationofcomputationalfluiddynamicsmodels,AICHE J. 62 (12) (2016) 4308–4322.doi:10.1002/aic.15352

    work page doi:10.1002/aic.15352 2016

  38. [38]

    Y. Zhou, Z. Lu, An enhanced Kriging surrogate modeling technique for high-dimensional problems, Mech. Syst. Signal Proc. 140 (2020) 106687. doi:10.1016/j.ymssp.2020.106687

    work page doi:10.1016/j.ymssp.2020.106687 2020

  39. [39]

    Forrester, A

    A. Forrester, A. Sobester, A. Keane, Engineering design via surrogate modelling: a practical guide, John Wiley & Sons, Chichester, 2008

    2008

  40. [40]

    Roshanian, M

    J. Roshanian, M. Ebrahimi, Latin hypercube sampling applied to reliability-based multidisciplinary design optimization of a launch vehicle, Aerosp. Sci. Technol. 28 (1) (2013) 297–304.doi:10.1016/j.ast.2012.11.010

    work page doi:10.1016/j.ast.2012.11.010 2013

  41. [41]

    Gupta, A

    S. Gupta, A. Paudel, M. Thapa, S. B. Mulani, et al., Optimal sampling-based neural networks for uncertainty quantification and stochastic optimization, Aerosp. Sci. Technol. 133 (2023).doi:10.1016/j.ast.2023.108109

    work page doi:10.1016/j.ast.2023.108109 2023

  42. [42]

    Y. Li, J. Shi, J. Shen, H. Cen, et al., An adaptive Kriging method with double sampling criteria applied to hydrogen preparation case, Int. J. Hydrog. Energy 45 (56) (2020) 31689–31705.doi:10.1016/j.ijhydene.2020.08.174

    work page doi:10.1016/j.ijhydene.2020.08.174 2020

  43. [43]

    S.S.Sathya,M.V.Radhika,Convergenceofnomadicgeneticalgorithmonbenchmarkmathematicalfunctions,Appl.SoftComput.13(2013) 2759–2766.doi:10.1016/j.asoc.2012.11.011

    work page doi:10.1016/j.asoc.2012.11.011 2013

  44. [44]

    Morokoff, R

    W. Morokoff, R. Caflisch, Quasi-Monte Carlo Integration, J. Comput. Phys. 122 (2) (1995) 218–230.doi:10.1006/jcph.1995.1209

    work page doi:10.1006/jcph.1995.1209 1995

  45. [45]

    C. Yang, W. Gao, N. Liu, C. Song, Low-discrepancy sequence initialized particle swarm optimization algorithm with high-order nonlinear time-varying inertia weight, Appl. Soft. Comput. 29 (2015) 386–394.doi:10.1016/j.asoc.2015.01.004

    work page doi:10.1016/j.asoc.2015.01.004 2015

  46. [46]

    3 (4) (1990) 467–483.doi:10.1016/0893-6080(90)90029-K

    M.Styblinski,T.Tang,Experimentsinnonconvexoptimization:Stochasticapproximationwithfunctionsmoothingandsimulatedannealing, Neural Netw. 3 (4) (1990) 467–483.doi:10.1016/0893-6080(90)90029-K

    work page doi:10.1016/0893-6080(90)90029-k 1990

  47. [47]

    doi:10.1016/j.jcp.2015.01.034

    P.Kersaudy,B.Sudret,N.Varsier,O.Picon,etal.,AnewsurrogatemodelingtechniquecombiningKrigingandpolynomialchaosexpansions- Applicationtouncertaintyanalysisincomputationaldosimetry,J.Comput.Phys.286(2015)103–117. doi:10.1016/j.jcp.2015.01.034

    work page doi:10.1016/j.jcp.2015.01.034 2015

  48. [48]

    S.Xiong,P.Z.G.Qian,C.F.J.Wu,Sequentialdesignandanalysisofhigh-accuracyandlow-accuracycomputercodes,Technometrics55(1) (2013) 37–46.doi:10.1080/00401706.2012.723572

    work page doi:10.1080/00401706.2012.723572 2013

  49. [49]

    Okoro, F

    A. Okoro, F. Khan, S. Ahmed, Dependency effect on the reliability-based design optimization of complex offshore structure, Reliab. Eng. Syst. Saf. 231 (2023).doi:10.1016/j.ress.2022.109026

    work page doi:10.1016/j.ress.2022.109026 2023

  50. [50]

    Eldred, C

    M. Eldred, C. Webster, P. G. Constantine, Evaluation of non-intrusive approaches for Wiener-Askey generalized polynomial chaos, in: 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2008, pp. 1–22

    2008

  51. [51]

    A. H. Amjadi, A. Johari, Stochasticnonlinear ground response analysis considering existing boreholes locations by the geostatistical method, Bull. Earthq. Eng. 20 (5) (2022) 2285–2327.doi:10.1007/s10518-022-01322-1

    work page doi:10.1007/s10518-022-01322-1 2022

  52. [52]

    Johari, H

    A. Johari, H. Fooladi, Comparative study of stochastic slope stability analysis based on conditional and unconditional random field, Comput. Geotech. 125 (2020) 103707.doi:10.1016/j.compgeo.2020.103707

    work page doi:10.1016/j.compgeo.2020.103707 2020

  53. [53]

    Y. Cho, Y. Cao, Y. Zagayevskiy, T. Wong, et al., Kriging-based monitoring of reservoir gas saturation distribution using time-lapse multicomponent borehole gravity measurements: Case study, Hastings Field, J. Pet. Sci. Eng. 190 (2020).doi:10.1016/j.petrol.2020. 107054

    work page doi:10.1016/j.petrol.2020 2020

  54. [54]

    Fischer, C

    M. Fischer, C. Proppe, Enhanced universal Kriging for transformed input parameter spaces, Probab. Eng. Mech. 74 (2023).doi:10.1016/ j.probengmech.2023.103486. Xuelin Xie et al.:Preprint submitted to Elsevier Page 26 of 26

    work page arXiv 2023