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Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems

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  • Zhengqing Fu
  • Lanlan Guo

Abstract

This paper considers the classical separable nonlinear least squares problem. Such problems can be expressed as a linear combination of nonlinear functions, and both linear and nonlinear parameters are to be estimated. Among the existing results, ill-conditioned problems are less often considered. Hence, this paper focuses on an algorithm for ill-conditioned problems. In the proposed linear parameter estimation process, the sensitivity of the model to disturbance is reduced using Tikhonov regularisation. The Levenberg–Marquardt algorithm is used to estimate the nonlinear parameters. The Jacobian matrix required by LM is calculated by the Golub and Pereyra, Kaufman, and Ruano methods. Combining the nonlinear and linear parameter estimation methods, three estimation models are obtained and the feasibility and stability of the model estimation are demonstrated. The model is validated by simulation data and real data. The experimental results also illustrate the feasibility and stability of the model.

Suggested Citation

  • Zhengqing Fu & Lanlan Guo, 2019. "Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems," Complexity, Hindawi, vol. 2019, pages 1-9, November.
    • Handle: RePEc:hin:complx:4861708
    DOI: 10.1155/2019/4861708
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    References listed on IDEAS

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    1. Jianguang Zhu & Binbin Hao, 2014. "A New Noninterior Continuation Method for Solving a System of Equalities and Inequalities," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-6, April.
    2. Guoying Ma & Binbing Huang, 2014. "Optimization of Process Parameters of Stamping Forming of the Automotive Lower Floor Board," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-9, August.
    3. Zeng-Shun Zhao & Xiang Feng & Yan-yan Lin & Fang Wei & Shi-Ku Wang & Tong-Lu Xiao & Mao-Yong Cao & Zeng-Guang Hou & Min Tan, 2013. "Improved Rao-Blackwellized Particle Filter by Particle Swarm Optimization," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-7, August.
    4. Li, Hongjie & Zhu, Yinglian & jing, Liu & ying, Wang, 2018. "Consensus of second-order delayed nonlinear multi-agent systems via node-based distributed adaptive completely intermittent protocols," Applied Mathematics and Computation, Elsevier, vol. 326(C), pages 1-15.
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