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Simultaneous estimation of nonlinear parameters in parabolic partial differential equation using quantum-behaved particle swarm optimization with Gaussian mutation

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Abstract

In this paper, an improved quantum-behaved particle swarm optimization with Gaussian mutation is proposed to simultaneously estimate nonlinear parameters in a one-dimensional parabolic partial differential equation (PDE). No a priori information about the functional form is available, therefore the problems may be treated as function estimation which is difficult to estimate using traditional gradient-based methods. Measurements on the boundary are used in the least square modelling. Tikhonov regularization technique is used to stabilize the ill-posed problem. The numerical benchmark and experiment results demonstrate the validity and efficiency of the proposed method to solve inverse problems of estimating nonlinear parameters in parabolic PDEs.

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Abbreviations

K(T):

Thermal conductivity

C(T):

Heat capacity per unit volume

ρ :

Density

T(x, t):

Is the temperature distribution at a spatial location x and time t

T j i , C j i and K j i :

Are temperature, heat capacity and thermal conductivity at the jth time step along the ith grid point

Δx :

Is the mesh size

Δt :

Is the time incremental size

λ :

Are the regularization parameter

References

  1. Alifanov OM (1994) Inverse heat transfer problems. Springer, Berlin

    Book  MATH  Google Scholar 

  2. Beck JV, Blackwell B, St-Clair CR Jr (1985) Inverse heat conduction: Ill-posed problems. Wiley, New York

    MATH  Google Scholar 

  3. Kim S, Chuang BJ, Kim MC, Kim KY (2003) Inverse estimation of temperature-dependent thermal conductivity and heat capacity per unit volume with the direct integration approach. Numer Heat Transf Part A Appl 44:521–535

    Article  Google Scholar 

  4. Huang CH, Ozisik MN (1991) Direct integration approach for simultaneously estimating temperature dependent thermal conductivity and heat capacity. Numer Heat Transf Part A Appl 20:95–110

    Article  Google Scholar 

  5. Huang CH, Yan JY (1995) An inverse problem in simultaneously measuring temperature-dependent thermal conductivity and heat capacity. Int J Heat Mass Transf 38:3433–3441

    Article  Google Scholar 

  6. Flach GP, Ozisik MN (1989) Inverse heat conduction problem of simultaneously estimating spatially varying thermal conductivity and heat capacity per unit volume. Numer Heat Transf Part A Appl 16:249–266

    Article  Google Scholar 

  7. Chen HT, Lin JY (1998) Simultaneous estimation of temperature-dependent thermal conductivity and heat capacity. Int J Heat Mass Transf 41:2237–2244

    Article  MATH  Google Scholar 

  8. Imani A, Ranjbar AA, Esmkhani M (2006) Simultaneous estimation of temperature-dependent thermal conductivity and heat capacity based on modified genetic algorithm. Inverse Probl Sci Eng 14:767–783

    Article  MATH  MathSciNet  Google Scholar 

  9. Ranjbar AA, Famouri M, Imani A (2010) A transient inverse problem in simultaneous estimation of TDTP based on MEGA. Int J Numer Methods Heat Fluid Flow 20:201–217

    Article  Google Scholar 

  10. Sun J, Feng B, Xu WB (2004) Particle swarm optimization with particles having quantum behaviour. IEEE Congress Evolutionary Computation, Portland

    Google Scholar 

  11. Sun J, Xu WB, Liu J (2005) Parameter selection of quantum-behaved particle swarm optimization. Lecture notes in computer science. Springer, vol 3612, pp 543–552

  12. Sun J, Xu WB, Feng B (2004) A global search strategy of quantum-behaved particle swarm optimization. IEEE Conf. on Cybernetics and Intelligent Systems, Singapore

    Google Scholar 

  13. Kennedy J, Eberhart RC (1995) Particle swarm optimization. IEEE Int. Conf. Neural Networks, Perth

    Google Scholar 

  14. Eberhart RC, Shi Y (1998) Comparison between genetic algorithm and particle swarm optimization. Evolutionary programming VII. Lecture notes in computer science, vol 1447. Springer Berlin, Heidelberg, pp 611–616

  15. Van den Bergh F (2001) An analysis of particle swarm optimizers. Ph.D. diss., University of Pretoria, South Africa

  16. Clerc M, Kennedy J (2002) The particle swarm: explosion, stability, and convergence in a multi-dimensional complex space. IEEE Trans. Evolutionary Comput 6:58–73

    Article  Google Scholar 

  17. Dumitrescu D, Lazzerini B, Jain LC, Dumitrescu A (2000) Evolutionary computation. MIT Press Journals, Boca Raton

  18. Lu ZS, Hou ZR, Du J (2006) Particle swarm optimization with adaptive mutation. Front Electr Electron Eng China 1:99–104

    Article  Google Scholar 

  19. Stacey A, Jancic M, Grundy I (2003) Particle swarm optimization with mutation. IEEE Cong. Evolutionary Comput, vol 2, pp 1425–1430

  20. Gao YL, Ren ZH (2007) Adaptive particle swarm optimization algorithm with genetic mutation operation. Third Int Conf Nat Comput 2:211–215

    Google Scholar 

  21. Andrews PS (2006) An investigation into mutation operators for particle swarm optimization, IEEE Cong. Evolutionary Computation. Vancouver, pp 1044–1051

  22. Liu J, Sun J, Xu WB (2006) Quantum-behaved particle swarm optimization with adaptive mutation operator. LNCS 4221:959–967

    Google Scholar 

  23. Fang W, Sun J, Xu WB (2009) Analysis of mutation operators on quantum-behaved particle swarm optimization algorithm. New Math Nat Comput 05:487–496

    Article  Google Scholar 

  24. Morozov VA (1966) On the solution of functional equations by the method of regularization. Soviet Math Dokl 7:414–417

    MATH  MathSciNet  Google Scholar 

  25. Hansen PC (2001) The L—curve and its use in the numerical treatment of inverse problems, computational inverse problem in electrocardiology, Advances in computational bioengineering. WIT Press, Holland

    Google Scholar 

  26. Taormina R et al (2012) Artificial neural network simulation of hourly groundwater levels in a coastal aquifer system of the Venice lagoon. Eng Appl Artif Intell 25(8):1670–1676

    Article  Google Scholar 

  27. Muttil N et al (2006) Neural network and genetic programming for modelling coastal algal blooms. Int J Environ Pollut 28(3–4):223–238

    Article  Google Scholar 

  28. Wu CL et al (2009) Predicting monthly stream-flow using data-driven models coupled with data-preprocessing techniques. Water Resour Res 45:W08432

    Article  Google Scholar 

  29. Cheng CT et al. (2005) Long-term prediction of discharges in Manwan Reservoir using artificial neural network models. Lecture notes in computer science, vol 3498, p 1040–1045

  30. Zhang J et al (2009) Multilayer ensemble pruning via novel multi-sub-swarm particle swarm optimization. J Univ Comput Sci 15(4):840–858

    Google Scholar 

  31. Chau KW (2007) Application of a PSO-based neural network in analysis of outcomes of construction claims. Autom Constr 16(5):642–646

    Article  MathSciNet  Google Scholar 

  32. Wang X, He Y, Dong L, Zhao H (2011) Particle swarm optimization for determining fuzzy measures from data. Inf Sci 181(19):4230–4252

    Article  MATH  Google Scholar 

  33. Rana S, Jasola S, Kumar R (2013) A boundary restricted adaptive particle swarm optimization for data clustering. Int J Mach Learn Cybernet 4(4):391–400

    Article  Google Scholar 

  34. Ma W, Wang M, Zhu X (2014) Improved particle swarm optimization based approach for bilevel programming problem-an application on supply chain model. Int J Mach Learn Cybernet 5(2):281–292

    Article  Google Scholar 

  35. Tian Na, Lai Choi-Hong (2014) Parallel quantum-behaved particle swarm optimization. Int J Mach Learn Cybernet 5(2):309–318

    Article  MathSciNet  Google Scholar 

  36. Jin X, Liang Y, Tian D, Zhuang F (2013) Particle swarm optimization using dimension selection methods. Appl Math Comput 219:5185–5197

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This work is supported by the innovative research of Jiangnan University (Project Number: 1245210382130120, 1242050205142810), by National High-Technology Research Development Plan Project (Project Number: 2013AA040405).

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Authors and Affiliations

  1. Department of Educational Technology, Jiangnan University, Wuxi, 214122, China

    Na Tian

  2. Institute of Electrical Automation, Jiangnan University, Wuxi, 214122, China

    Na Tian & Zhicheng Ji

  3. School of Computing and Mathematical Sciences, University of Greenwich, London, SE10 9LS, UK

    Choi-Hong Lai

Authors
  1. Na Tian
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  2. Zhicheng Ji
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  3. Choi-Hong Lai
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Correspondence to Na Tian.

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Tian, N., Ji, Z. & Lai, CH. Simultaneous estimation of nonlinear parameters in parabolic partial differential equation using quantum-behaved particle swarm optimization with Gaussian mutation. Int. J. Mach. Learn. & Cyber. 6, 307–318 (2015). https://doi.org/10.1007/s13042-014-0261-1

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  • DOI: https://doi.org/10.1007/s13042-014-0261-1

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