In parameter identification problem, errors are common in measurement data, resulting in uncertainty in the identified parameters. Traditional deterministic methods cannot address this uncertainty. A novel approach, which integrates an advanced particle swarm optimization algorithm (APSO) and the stochastic perturbation collocation method (SPC), is proposed to address this issue, called APSO-SPC for short. The APSO algorithm improves the heterogeneous comprehensive learning particle swarm optimization algorithm (HCLPSO) based on the dynamic evolution sequence (DES), improving computational efficiency for each deterministic parameter identification process. Furthermore, the SPC method accurately estimates the means and standard deviations of uncertain parameters. Three numerical examples demonstrate the accuracy and efficiency of the APSO-SPC method in assessing parameter uncertainties caused by random measurement errors.
Citation: Peng Zhong, Xuanlong Wu, Li Zhu, Aohao Yang. A new APSO-SPC method for parameter identification problem with uncertainty caused by random measurement errors[J]. AIMS Mathematics, 2025, 10(2): 3848-3865. doi: 10.3934/math.2025179
In parameter identification problem, errors are common in measurement data, resulting in uncertainty in the identified parameters. Traditional deterministic methods cannot address this uncertainty. A novel approach, which integrates an advanced particle swarm optimization algorithm (APSO) and the stochastic perturbation collocation method (SPC), is proposed to address this issue, called APSO-SPC for short. The APSO algorithm improves the heterogeneous comprehensive learning particle swarm optimization algorithm (HCLPSO) based on the dynamic evolution sequence (DES), improving computational efficiency for each deterministic parameter identification process. Furthermore, the SPC method accurately estimates the means and standard deviations of uncertain parameters. Three numerical examples demonstrate the accuracy and efficiency of the APSO-SPC method in assessing parameter uncertainties caused by random measurement errors.
[1] |
W. C. Xing, Y. Q. Wang, A unified nonlinear dynamic model for bolted flange joint disk-drum structures under different interface states: theory and experiment, Appl. Math. Model. , 137 (2025), 115695. https://doi.org/10.1016/j.apm.2024.115695 doi: 10.1016/j.apm.2024.115695
![]() |
[2] |
D. W. Huang, Y. L. Zhao, K. Q. Ye, F. Wu, H. W. Zhang, W. X. Zhong, The efficient calculation methods for stochastic nonlinear transient heat conduction problems, J. Comput. Sci. , 67 (2023), 101939. https://doi.org/10.1016/j.jocs.2022.101939 doi: 10.1016/j.jocs.2022.101939
![]() |
[3] |
Y. R. Hong, Nonlinear modeling of measurement errors in gateway energy meters, Measurement: Sensors, 35 (2024), 101286. https://doi.org/10.1016/j.measen.2024.101286 doi: 10.1016/j.measen.2024.101286
![]() |
[4] |
F. Wu, L. Zhu, Y. L. Zhao, C. F. Ai, X. Wang, F. Cai, et al., Wave spectrum fitting with multiple parameters based on optimization algorithms and its application, Ocean Eng. , 312 (2024), 119073. https://doi.org/10.1016/j.oceaneng.2024.119073 doi: 10.1016/j.oceaneng.2024.119073
![]() |
[5] |
H. Y. Zeng, Z. F. Lin, G. H. Huang, X. Q. Yang, Y. F. Li, J. B. Su, et al., Parameter identification of DEM-FEM coupling model to simulate traction behavior of tire-soil interaction, J. Terramechanics, 117 (2025), 101012. https://doi.org/10.1016/j.jterra.2024.101012 doi: 10.1016/j.jterra.2024.101012
![]() |
[6] |
F. Han, X. L. Guo, C. Mo, H. Y. Gao, P. J. Hou, Parameter identification of nonlinear rotor-bearing system based on improved kriging surrogate model, J. Vib. Control, 23 (2017), 794–807. https://doi.org/10.1177/1077546315585242 doi: 10.1177/1077546315585242
![]() |
[7] |
F. Wang, C. J. Zhao, Y. H. Zhou, H. W. Zhou, Z. P. Liang, F. Wang, et al., Multiple thermal parameter inversion for concrete dams using an integrated surrogate model, Appl. Sci. , 13 (2023), 5407. https://doi.org/10.3390/app13095407 doi: 10.3390/app13095407
![]() |
[8] |
E. Roux, Y. Tillier, S. Kraria, P. Bouchard, An efficient parallel global optimization strategy based on Kriging properties suitable for material parameters identification, Arch. Mech. Eng. , 67 (2020), 131689. https://doi.org/10.24425/ame.2020.131689 doi: 10.24425/ame.2020.131689
![]() |
[9] |
C. Ding, S. X. Pei, H. Q. Chen, Y. Huang, B. Meng, L. Liu, Effect of clearance on measuring accuracy in two-dimensional piston flowmeter, Flow Meas. Instrum. , 99 (2024), 102673. https://doi.org/10.1016/j.flowmeasinst.2024.102673 doi: 10.1016/j.flowmeasinst.2024.102673
![]() |
[10] |
S. Dorvash, S. N. Pakzad, Effects of measurement noise on modal parameter identification, Smart Mater. Struct. , 21 (2012), 065008. https://doi.org/10.1088/0964-1726/21/6/065008 doi: 10.1088/0964-1726/21/6/065008
![]() |
[11] |
F. Wu, W. X. Zhong, A hybrid approach for the time domain analysis of linear stochastic structures, Comput. Method. Appl. M. , 265 (2013), 71–82. https://doi.org/10.1016/j.cma.2013.06.006 doi: 10.1016/j.cma.2013.06.006
![]() |
[12] |
D. W. Huang, F. Wu, Y. L. Zhao, J. Yan, H. W. Zhang, Application of high-credible statistical results calculation scheme based on least squares Quasi-Monte Carlo method in multimodal stochastic problems, Comput. Method. Appl. M. , 418 (2024), 116576. https://doi.org/10.1016/j.cma.2023.116576 doi: 10.1016/j.cma.2023.116576
![]() |
[13] |
H. T. Podeh, A. Parsaie, B. Shahinejad, A. Arshia, Z. Shamsi, Development and uncertainty analysis of infiltration models using PSO and Monte Carlo method, Irrig. Drain. , 72 (2023), 38–47. https://doi.org/10.1002/ird.2769 doi: 10.1002/ird.2769
![]() |
[14] |
F. Wu, D. W. Huang, X. M. Xu, K. Zhao, N. Zhou, An adaptive divided-difference perturbation method for solving stochastic problems, Struct. Saf. , 103 (2023), 102346. https://doi.org/10.1016/j.strusafe.2023.102346 doi: 10.1016/j.strusafe.2023.102346
![]() |
[15] |
M. Kamiński, Uncertainty analysis in solid mechanics with uniform and triangular distributions using stochastic perturbation-based Finite Element Method, Finite Elem. Anal. Des. , 200 (2022), 103648. https://doi.org/10.1016/j.finel.2021.103648 doi: 10.1016/j.finel.2021.103648
![]() |
[16] |
L. Zhu, K. Q. Ye, D. W. Huang, F. Wu, W. X. Zhong, An adaptively filtered precise integration method considering perturbation for stochastic dynamics problems, Acta Mech. Solida Sin. , 36 (2023), 317–326. https://doi.org/10.1007/s10338-023-00381-4 doi: 10.1007/s10338-023-00381-4
![]() |
[17] |
A. Pavone, A. Merlo, S. Kwak, J. Svensson, Machine learning and Bayesian inference in nuclear fusion research: an overview, Plasma Phys. Control. Fusion, 65 (2023), 053001. https://doi.org/10.1088/1361-6587/acc60f doi: 10.1088/1361-6587/acc60f
![]() |
[18] | M. Welandawe, M. R. Andersen, A. Vehtari, J. H. Huggins, A framework for improving the reliability of black-box variational inference, J. Mach. Learn. Res. , 25 (2024), 1–71. |
[19] |
J. C. Rodrigues, J. Facão, M. J. Carvalho, Parameter identification and uncertainty evaluation in Quasi-Dynamic test of solar thermal collectors with Monte Carlo method, Renew. Energ. , 236 (2024), 121403. https://doi.org/10.1016/j.renene.2024.121403 doi: 10.1016/j.renene.2024.121403
![]() |
[20] |
P. Z. Pan, F. S. Su, H. J. Chen, S. L. Yan, X. T. Feng, F. Yan, Uncertainty analysis of rock failure behaviour using an integration of the probabilistic collocation method and elasto-plastic cellular automaton, Acta Mech. Solida Sin. , 28 (2015), 536–555. https://doi.org/10.1016/S0894-9166(15)30048-3 doi: 10.1016/S0894-9166(15)30048-3
![]() |
[21] |
M. Y. Feng, T. J. Sun, Adaptive perturbation method for optimal control problem governed by stochastic elliptic PDEs, Comp. Appl. Math. , 43 (2024), 100. https://doi.org/10.1007/s40314-024-02607-8 doi: 10.1007/s40314-024-02607-8
![]() |
[22] |
F. Wu, K. Zhao, L. L. Zhao, C. Y. Chen, W. X. Zhong, Uncertainty analysis of the control rod drop based on the adaptive collocation stochastic perturbation method, Ann. Nucl. Energy, 190 (2023), 109873. https://doi.org/10.1016/j.anucene.2023.109873 doi: 10.1016/j.anucene.2023.109873
![]() |
[23] |
F. Wu, Q. Gao, X. M. Xu, W. X. Zhong, A modified computational scheme for the stochastic perturbation finite element method, Lat. Am. J. Solids Struct. , 12 (2015), 2480–2505. https://doi.org/10.1590/1679-78251772 doi: 10.1590/1679-78251772
![]() |
[24] |
Z. J. Shao, X. M. Li, P. Xiang, A new computational scheme for structural static stochastic analysis based on Karhunen-Loève expansion and modified perturbation stochastic finite element method, Comput. Mech. , 71 (2023), 917–933. https://doi.org/10.1007/s00466-022-02259-7 doi: 10.1007/s00466-022-02259-7
![]() |
[25] |
S. Chakraborty, S. Adhikari, R. Ganguli, The role of surrogate models in the development of digital twins of dynamic systems, Appl. Math. Model. , 90 (2021), 662–681. https://doi.org/10.1016/j.apm.2020.09.037 doi: 10.1016/j.apm.2020.09.037
![]() |
[26] |
Z. J. Shao, Q. Xia, P. Xiang, H. Zhao, L. Z. Jiang, Stochastic free vibration analysis of FG-CNTRC plates based on a new stochastic computational scheme, Appl. Math. Model. , 127 (2024), 119–142. https://doi.org/10.1016/j.apm.2023.11.016 doi: 10.1016/j.apm.2023.11.016
![]() |
[27] |
S. Z. Feng, Q. J. Sun, S. Xiao, X. Han, Y. B. Li, Z. X. Li, A novel multi-physics coupling model for the stochastic analysis of phased arrays considering material spatial uncertainty, Appl. Math. Model. , 128 (2024), 707–722. https://doi.org/10.1016/j.apm.2024.01.042 doi: 10.1016/j.apm.2024.01.042
![]() |
[28] |
Y. X. Yang, K. Zhao, Y. L. Zhao, F. Wu, C. Y. Chen, J. Yan, et al., UA-CRD, a computational framework for uncertainty analysis of control rod drop with time-variant epistemic uncertain parameters, Ann. Nucl. Energy, 195 (2024), 110171. https://doi.org/10.1016/j.anucene.2023.110171 doi: 10.1016/j.anucene.2023.110171
![]() |
[29] |
F. Wu, Y. L. Zhao, Y. X. Yang, X. P. Zhang, N. Zhou, A new discrepancy for sample generation in stochastic response analyses of aerospace problems with uncertain parameter, Chinese J. Aeronaut. , 37 (2024), 192–211. https://doi.org/10.1016/j.cja.2024.09.044 doi: 10.1016/j.cja.2024.09.044
![]() |
[30] |
N. Lynn, P. N. Suganthan, Heterogeneous comprehensive learning particle swarm optimization with enhanced exploration and exploitation, Swarm Evol. Comput. , 24 (2015), 11–24. https://doi.org/10.1016/j.swevo.2015.05.002 doi: 10.1016/j.swevo.2015.05.002
![]() |
[31] |
E. Zhang, Z. H. Nie, Q. Yang, Y. Q. Wang, D. Liu, S. Jeon, et al., Heterogeneous cognitive learning particle swarm optimization for large-scale optimization problems, Inform. Sciences, 633 (2023), 321–342. https://doi.org/10.1016/j.ins.2023.03.086 doi: 10.1016/j.ins.2023.03.086
![]() |
[32] | T. Jeyaraman, D. Joelpraveenkumar, M. Kaliraj, M. K. Chandar, M. W. Iruthayarajan, Tuning of MIMO PID controller using HCLPSO algorithm, In: Proceedings of international conference on power electronics and renewable energy systems, Singapore: Springer, 2022,377–385. https://doi.org/10.1007/978-981-16-4943-1_35 |
[33] |
O. Hachana, B. Aoufi, G. M. Tina, M. A. Sid, Photovoltaic mono and bifacial module/string electrical model parameters identification and validation based on a new differential evolution bee colony optimizer, Energ. Convers. Manage. , 248 (2021), 114667. https://doi.org/10.1016/j.enconman.2021.114667 doi: 10.1016/j.enconman.2021.114667
![]() |
[34] |
J. J. Zhu, M. Huang, Z. R. Lu, Bird mating optimizer for structural damage detection using a hybrid objective function, Swarm Evol. Comput. , 35 (2017), 41–52. https://doi.org/10.1016/j.swevo.2017.02.006 doi: 10.1016/j.swevo.2017.02.006
![]() |
[35] |
D. Yousri, D. Allam, M. B. Eteiba, P. N. Suganthan, Static and dynamic photovoltaic models' parameters identification using chaotic heterogeneous comprehensive learning particle swarm optimizer variants, Energ. Convers. Manage. , 182 (2019), 546–563. https://doi.org/10.1016/j.enconman.2018.12.022 doi: 10.1016/j.enconman.2018.12.022
![]() |
[36] |
F. Wu, Y. L. Zhao, K. Zhao, W. X. Zhong, A multi-body dynamical evolution model for generating the point set with best uniformity, Swarm Evol. Comput. , 73 (2022), 101121. https://doi.org/10.1016/j.swevo.2022.101121 doi: 10.1016/j.swevo.2022.101121
![]() |
[37] |
F. Wu, W. X. Zhong, A modified stochastic perturbation method for stochastic hyperbolic heat conduction problems, Comput. Method. Appl. M. , 305 (2016), 739–758. https://doi.org/10.1016/j.cma.2016.03.032 doi: 10.1016/j.cma.2016.03.032
![]() |
[38] |
K. Zhao, X. M. Xu, C. Y. Chen, F. Wu, D. W. Huang, Y. Y. Xi, et al., Nonlinear state equation and adaptive symplectic algorithm for the control rod drop, Ann. Nucl. Energy, 179 (2022), 109402. https://doi.org/10.1016/j.anucene.2022.109402 doi: 10.1016/j.anucene.2022.109402
![]() |
[39] |
J. Kudela, R. Matousek, Recent advances and applications of surrogate models for finite element method computations: a review, Soft Comput. , 26 (2022), 13709–13733. https://doi.org/10.1007/s00500-022-07362-8 doi: 10.1007/s00500-022-07362-8
![]() |
[40] | F. Wu, Y. L. Zhao, J. H. Pang, J. Yan, W. X. Zhong, Low-discrepancy sampling in the expanded dimensional space: an acceleration technique for particle swarm optimization, 2023, arXiv: 2303.03055. |
[41] | Y. L. Zhao, F. Wu, J. H. Pang, W. X. Zhong, Updating velocities in heterogeneous comprehensive learning particle swarm optimization with low-discrepancy sequences, 2022, arXiv: 2209.09438. |
[42] |
Q. X. Zhuang, N. Wan, Y. H. Guo, Z. Y. Chang, Z. Wang, Uncertainty propagation and assessment of five-axis on-machine measurement error, J. Manuf. Sci. Eng. , 144 (2022), 101004. https://doi.org/10.1115/1.4054286 doi: 10.1115/1.4054286
![]() |
[43] |
C. C. Wen, P. Zhang, J. Wang, S. W. Hu, Influence of fibers on the mechanical properties and durability of ultra-high-performance concrete: a review, J. Build. Eng. , 52 (2022), 104370. https://doi.org/10.1016/j.jobe.2022.104370 doi: 10.1016/j.jobe.2022.104370
![]() |
[44] |
N. Ankur, N. Singh, Performance of cement mortars and concretes containing coal bottom ash: a comprehensive review, Renew. Sust. Energ. Rev. , 149 (2021), 111361. https://doi.org/10.1016/j.rser.2021.111361 doi: 10.1016/j.rser.2021.111361
![]() |
[45] |
H. Huang, Y. J. Yuan, W. Zhang, L. Zhu, Property assessment of high-performance concrete containing three types of fibers, Int. J. Concr. Struct. Mater. , 15 (2021), 39. https://doi.org/10.1186/s40069-021-00476-7 doi: 10.1186/s40069-021-00476-7
![]() |
[46] |
G. Y. Zhao, J. Ma, K. Peng, Q. Yang, L. Zhou, Mix ratio optimization of alpine mine backfill based on the response surface method, Chinese Journal of Engineering, 35 (2013), 559–565. https://doi.org/10.13374/j.issn1001-053x.2013.05.003 doi: 10.13374/j.issn1001-053x.2013.05.003
![]() |
[47] |
F. Wu, K. Zhao, X. L. Wu, H. J. Peng, L. L. Zhao, W. X. Zhong, A time-averaged method to analyze slender rods moving in tubes, Int. J. Mech. Sci. , 279 (2024), 109510. https://doi.org/10.1016/j.ijmecsci.2024.109510 doi: 10.1016/j.ijmecsci.2024.109510
![]() |
[48] |
F. M. A. AL-Gaadi, F. Alemdar, Dynamic parameter identification of cold-formed storage rack systems using shaking table test, Results in Engineering, 22 (2024), 102160. https://doi.org/10.1016/j.rineng.2024.102160 doi: 10.1016/j.rineng.2024.102160
![]() |
[49] |
F. Wu, W. X. Zhong, Constrained Hamilton variational principle for shallow water problems and Zu-class symplectic algorithm, Appl. Math. Mech. , 37 (2016), 1–14. https://doi.org/10.1007/s10483-016-2051-9 doi: 10.1007/s10483-016-2051-9
![]() |
[50] |
Q. Gao, F. Wu, H. W. Zhang, W. X. Zhong, W. P. Howson, F. W. Williams, A fast precise integration method for structural dynamics problems, Struct. Eng. Mech. , 43 (2012), 1–13. https://doi.org/10.12989/sem.2012.43.1.001 doi: 10.12989/sem.2012.43.1.001
![]() |