Levenberg-Marquardt Algorithm for Orthogonal Fitting of Transition Curves

SONG Zhanfeng; WANG Jian; LI Jun

doi:10.3969/j.issn.0258-2724.20190130

Volume 55 Issue 1

Jan. 2020

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Journal of Southwest Jiaotong University > 2020 > 55(1): 144-149.

SONG Zhanfeng, WANG Jian, LI Jun. Levenberg-Marquardt Algorithm for Orthogonal Fitting of Transition Curves[J]. Journal of Southwest Jiaotong University, 2020, 55(1): 144-149. doi: 10.3969/j.issn.0258-2724.20190130

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SONG Zhanfeng, WANG Jian, LI Jun. Levenberg-Marquardt Algorithm for Orthogonal Fitting of Transition Curves[J]. Journal of Southwest Jiaotong University, 2020, 55(1): 144-149. doi: 10.3969/j.issn.0258-2724.20190130

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Levenberg-Marquardt Algorithm for Orthogonal Fitting of Transition Curves

doi: 10.3969/j.issn.0258-2724.20190130

Received Date: 05 Mar 2019
Rev Recd Date: 11 Jun 2019

Available Online: 18 Sep 2019

Publish Date: 01 Feb 2020

Abstract

Abstract

To identify the parameters of transition curves in as-built alignments by measured points, orthogonal least-squares fitting is studied on the basis of the parameter equation of transition curves. First, eigenvalue analysis has clarified that the Gauss-Newton (GN) algorithm usually fails to converge because of the existence of the ill-condition. Next, a bi-objective optimization model is proposed and the Levenberg-Marquardt (LM) algorithm combining the GN algorithm with the steepest descent method is constructed to fit a transition curve to points orthogonally. The LM parameter is updated dynamically during iterations according to the evaluation of the distance between the current and the optimum locations. Finally, Monte Carlo simulations are employed to test the performances of the GN and LM algorithms with measured points and the same 5 000 initial values. Experimental results show that the GN algorithm diverges while the LM algorithm converges to the same optimum under different initial values. The number of iterations, with an average of 16.8 times and the minimum of 5 times and the maximum of 50 times, is related to the distance between the initial and the optimum locations. The LM algorithm shows a better robustness than the GN algorithm.
- transition curve fitting,
- orthogonal least squares,
- Levenberg-Marquardt algorithm,
- Gauss-Newton algorithm,
- steepest descent method

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References

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