{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T10:05:44Z","timestamp":1776852344908,"version":"3.51.2"},"reference-count":46,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,12,23]],"date-time":"2021-12-23T00:00:00Z","timestamp":1640217600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51775472"],"award-info":[{"award-number":["51775472"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Science &amp;Technology Innovation Talents in Universities of Henan Province","award":["21HASTIT027"],"award-info":[{"award-number":["21HASTIT027"]}]},{"name":"Henan Excellent Youth Fund Project","award":["202300410346"],"award-info":[{"award-number":["202300410346"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Kriging-based modeling has been widely used in computationally intensive simulations. However, the Kriging modeling of high-dimensional problems not only takes more time, but also leads to the failure of model construction. To this end, a Kriging modeling method based on multidimensional scaling (KMDS) is presented to avoid the \u201cdimensional disaster\u201d. Under the condition of keeping the distance between the sample points before and after the dimensionality reduction unchanged, the KMDS method, which mainly calculates each element in the inner product matrix due to the mapping relationship between the distance matrix and the inner product matrix, completes the conversion of design data from high dimensional to low dimensional. For three benchmark functions with different dimensions and the aviation field problem of aircraft longitudinal flight control, the proposed method is compared with other dimensionality reduction methods. The KMDS method has better modeling efficiency while meeting certain accuracy requirements.<\/jats:p>","DOI":"10.3390\/a15010003","type":"journal-article","created":{"date-parts":[[2021,12,23]],"date-time":"2021-12-23T21:40:21Z","timestamp":1640295621000},"page":"3","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["An Efficient Kriging Modeling Method Based on Multidimensional Scaling for High-Dimensional Problems"],"prefix":"10.3390","volume":"15","author":[{"given":"Yu","family":"Ge","sequence":"first","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Xuchang University, Xuchang 461000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Junjun","family":"Shi","sequence":"additional","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Xuchang University, Xuchang 461000, China"},{"name":"College of Science, Huazhong Agricultural University, Wuhan 430070, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3832-9894","authenticated-orcid":false,"given":"Yaohui","family":"Li","sequence":"additional","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Xuchang University, Xuchang 461000, China"},{"name":"College of Science, Huazhong Agricultural University, Wuhan 430070, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jingfang","family":"Shen","sequence":"additional","affiliation":[{"name":"College of Science, Huazhong Agricultural University, Wuhan 430070, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1007\/s12572-017-0200-y","article-title":"Seismic fragility analysis in the probabilistic performance-based earthquake engineering framework: An overview","volume":"13","author":"Ghosh","year":"2021","journal-title":"Int. J. Adv. Eng. Sci. Appl. Math."},{"key":"ref_2","unstructured":"Samui, P., Tien Bui, D., Chakraborty, S., and Deo, R.C. (2020). Chapter 13\u2014Novel concepts for reliability analysis of dynamic structural systems. Handbook of Probabilistic Models, Butterworth-Heinemann."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1316","DOI":"10.1080\/13632469.2019.1570395","article-title":"Kriging Metamodeling-Based Monte Carlo Simulation for Improved Seismic Fragility Analysis of Structures","volume":"25","author":"Ghosh","year":"2021","journal-title":"J. Earthq. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"853","DOI":"10.2514\/1.8650","article-title":"Use of Kriging Models to Approximate Deterministic Computer Models","volume":"43","author":"Martin","year":"2005","journal-title":"AIAA J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ejor.2016.06.041","article-title":"Regression and Kriging metamodels with their experimental designs in simulation: A review","volume":"256","author":"Kleijnen","year":"2017","journal-title":"Eur. J. Oper. Res."},{"key":"ref_6","first-page":"97","article-title":"Kriging-based infill sampling criterion for constraint handling in multi-objective optimization","volume":"64","year":"2015","journal-title":"J. Glob. Optim."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"31689","DOI":"10.1016\/j.ijhydene.2020.08.174","article-title":"An adaptive Kriging method with double sampling criteria applied to hydrogen preparation case","volume":"45","author":"Li","year":"2020","journal-title":"Int. J. Hydrogen Energy"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Li, Y., Shen, J., Cai, Z., Wu, Y., and Wang, S. (2021). A Kriging-Assisted Multi-Objective Constrained Global Optimization Method for Expensive Black-Box Functions. Mathematics, 9.","DOI":"10.3390\/math9020149"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1287\/ijoc.1110.0465","article-title":"Robust Optimization in Simulation: Taguchi and Krige Combined","volume":"24","author":"Dellino","year":"2012","journal-title":"INFORMS J. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1007\/s00158-020-02495-6","article-title":"A global optimization strategy based on the Kriging surrogate model and parallel computing","volume":"62","author":"Xing","year":"2020","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"778","DOI":"10.1007\/s10489-019-01549-7","article-title":"Cluster-based Kriging approximation algorithms for complexity reduction","volume":"50","author":"Wang","year":"2020","journal-title":"Appl. Intell."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Kleijnen, J.P.C. (2020). Simulation Optimization through Regression or Kriging Metamodels. High-Performance Simulation-Based Optimization, Springer.","DOI":"10.1007\/978-3-030-18764-4_6"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Bonnot, J., Menard, D., and Desnos, K. (2020, January 9\u201313). Fast Kriging-based Error Evaluation for Approximate Computing Systems. Proceedings of the 2020 Design, Automation & Test in Europe Conference & Exhibition (DATE), Grenoble, France.","DOI":"10.23919\/DATE48585.2020.9116320"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"782","DOI":"10.1016\/j.cma.2018.12.026","article-title":"Filter-based adaptive Kriging method for black-box optimization problems with expensive objective and constraints","volume":"347","author":"Shi","year":"2019","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Bae, S.-H., Choi, J.Y., Qiu, J., and Fox, G.C. (2010, January 21\u201325). Dimension reduction and visualization of large high-dimensional data via interpolation. Proceedings of the 19th ACM International Symposium on High Performance Distributed Computing, Association for Computing Machinery, Chicago, IL, USA.","DOI":"10.1145\/1851476.1851501"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1090\/conm\/115\/1117048","article-title":"High-dimensional numerical integration and massively parallel computing","volume":"115","author":"Mascagni","year":"1991","journal-title":"Contemp. Math."},{"key":"ref_17","unstructured":"De Luca, P., Galletti, A., and Marcellino, L. (2020, January 23\u201325). Parallel solvers comparison for an inverse problem in fractional calculus. In Proceeding of the 9th International Conference on Theory and Practice in Modern Computing (TPMC 2020), Online."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1016\/j.compchemeng.2017.09.017","article-title":"Advances in surrogate based modeling, feasibility analysis, and optimization: A review","volume":"108","author":"Bhosekar","year":"2018","journal-title":"Comput. Chem. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"908","DOI":"10.1002\/2014WR015963","article-title":"A Bayesian kriging approach for blending satellite and ground precipitation observations","volume":"51","author":"Verdin","year":"2015","journal-title":"Water Resour. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1016\/j.apm.2021.04.005","article-title":"Coupling of Homotopy Perturbation Method and Kriging surrogate model for an efficient fuzzy linear buckling analysis: Application to additively manufactured lattice structures","volume":"97","author":"Doan","year":"2021","journal-title":"Appl. Math. Model."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Schabenberger, O., and Gotway, C.A. (2017). Statistical Methods for Spatial Data Analysis, CRC Press.","DOI":"10.1201\/9781315275086"},{"key":"ref_22","first-page":"323","article-title":"An incremental Kriging method for sequential optimal experimental design","volume":"97","author":"Li","year":"2014","journal-title":"Comput. Model. Eng. Sci. (CMES)"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s11831-017-9226-3","article-title":"An Overview of Gradient-Enhanced Metamodels with Applications","volume":"26","author":"Laurent","year":"2019","journal-title":"Arch. Comput. Methods Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"112861","DOI":"10.1016\/j.cma.2020.112861","article-title":"Optimization of expensive black-box problems via Gradient-enhanced Kriging","volume":"362","author":"Chen","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Liu, W., and Batill, S. (2002, January 4\u20136). Gradient-Enhanced Response Surface Approximations Using Kriging Models. Proceedings of the 9th AIAA\/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Atlanta, Georgia.","DOI":"10.2514\/6.2002-5456"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"5256","DOI":"10.1016\/j.apm.2015.12.033","article-title":"High dimensional Kriging metamodelling utilising gradient information","volume":"40","author":"Ulaganathan","year":"2016","journal-title":"Appl. Math. Model."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4330","DOI":"10.2514\/1.J055842","article-title":"Weighted Gradient-Enhanced Kriging for High-Dimensional Surrogate Modeling and Design Optimization","volume":"55","author":"Han","year":"2017","journal-title":"AIAA J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.apm.2018.11.048","article-title":"A screening-based gradient-enhanced Kriging modeling method for high-dimensional problems","volume":"69","author":"Chen","year":"2019","journal-title":"Appl. Math. Model."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1007\/s00158-020-02705-1","article-title":"Efficient high-dimensional metamodeling strategy using recursive decomposition coupled with sequential sampling method","volume":"63","author":"Kang","year":"2021","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"6723410","DOI":"10.1155\/2016\/6723410","article-title":"An Improved Approach for Estimating the Hyperparameters of the Kriging Model for High-Dimensional Problems through the Partial Least Squares Method","volume":"2016","author":"Bouhlel","year":"2016","journal-title":"Math. Probl. Eng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1007\/s00158-015-1395-9","article-title":"Improving kriging surrogates of high-dimensional design models by Partial Least Squares dimension reduction","volume":"53","author":"Bouhlel","year":"2016","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/s00366-018-0590-x","article-title":"Gradient-enhanced kriging for high-dimensional problems","volume":"35","author":"Bouhlel","year":"2019","journal-title":"Eng. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/s00158-019-02342-3","article-title":"An efficient kriging modeling method for high-dimensional design problems based on maximal information coefficient","volume":"61","author":"Zhao","year":"2020","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"106356","DOI":"10.1016\/j.knosys.2020.106356","article-title":"A distance correlation-based Kriging modeling method for high-dimensional problems","volume":"206","author":"Fu","year":"2020","journal-title":"Knowl. Based Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1109\/TEVC.2013.2248012","article-title":"A Gaussian Process Surrogate Model Assisted Evolutionary Algorithm for Medium Scale Expensive Optimization Problems","volume":"18","author":"Liu","year":"2014","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1109\/T-C.1969.222678","article-title":"A Nonlinear Mapping for Data Structure Analysis","volume":"18","author":"Sammon","year":"1969","journal-title":"IEEE Trans. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"106687","DOI":"10.1016\/j.ymssp.2020.106687","article-title":"An enhanced Kriging surrogate modeling technique for high-dimensional problems","volume":"140","author":"Zhou","year":"2020","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Liu, B., and Xie, L. (2020). An Improved Structural Reliability Analysis Method Based on Local Approximation and Parallelization. Mathematics, 8.","DOI":"10.3390\/math8020209"},{"key":"ref_39","unstructured":"Nielsen, H.B. (2002). Aspects of the Matlab Toolbox DACE, The Technical University of Denmark."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1092","DOI":"10.1214\/17-EJS1250","article-title":"Geometric foundations for scaling-rotation statistics on symmetric positive definite matrices: Minimal smooth scaling-rotation curves in low dimensions","volume":"11","author":"Groisser","year":"2017","journal-title":"Electron. J. Stat."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Entezami, A., Sarmadi, H., Behkamal, B., and Mariani, S. (2021). Health Monitoring of Large-Scale Civil Structures: An Approach Based on Data Partitioning and Classical Multidimensional Scaling. Sensors, 21.","DOI":"10.3390\/s21051646"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1016\/j.acha.2020.11.006","article-title":"Multidimensional scaling of noisy high dimensional data","volume":"51","author":"Erez","year":"2021","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_43","first-page":"125561","article-title":"Geometric multidimensional scaling: A new approach for data dimensionality reduction","volume":"409","author":"Gintautas","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1413","DOI":"10.1007\/s11222-016-9696-4","article-title":"Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC","volume":"27","author":"Vehtari","year":"2017","journal-title":"Stat. Comput."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1017\/S0001924000001494","article-title":"Longitudinal flight control design with handling quality requirements","volume":"110","author":"Saydy","year":"2006","journal-title":"Aeronaut. J."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"012026","DOI":"10.1088\/1742-6596\/1130\/1\/012026","article-title":"Longitudinal flight control laws for high aspect ratio light utility aircraft","volume":"1130","author":"Bahri","year":"2018","journal-title":"J. Phys. Conf. Ser."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/1\/3\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:52:28Z","timestamp":1760169148000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/1\/3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,23]]},"references-count":46,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["a15010003"],"URL":"https:\/\/doi.org\/10.3390\/a15010003","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,23]]}}}