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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

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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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