Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems
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. H...
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| Format: | Article |
| Language: | English |
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Wiley
2019-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2019/4861708 |
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| _version_ | 1832554559992496128 |
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| author | Zhengqing Fu Lanlan Guo |
| author_facet | Zhengqing Fu Lanlan Guo |
| author_sort | Zhengqing Fu |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-4053552d5c2b479e935ba89503eac5b1 |
| institution | Kabale University |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-4053552d5c2b479e935ba89503eac5b12025-02-03T05:51:06ZengWileyComplexity1076-27871099-05262019-01-01201910.1155/2019/48617084861708Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares ProblemsZhengqing Fu0Lanlan Guo1College of Mathematics and System Sciences, SDUST, Qingdao, Shandong 266590, ChinaCollege of Mechanical and Electronic Engineering, SDUST, Qingdao, Shandong 266590, ChinaThis 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.http://dx.doi.org/10.1155/2019/4861708 |
| spellingShingle | Zhengqing Fu Lanlan Guo Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems Complexity |
| title | Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems |
| title_full | Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems |
| title_fullStr | Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems |
| title_full_unstemmed | Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems |
| title_short | Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems |
| title_sort | tikhonov regularized variable projection algorithms for separable nonlinear least squares problems |
| url | http://dx.doi.org/10.1155/2019/4861708 |
| work_keys_str_mv | AT zhengqingfu tikhonovregularizedvariableprojectionalgorithmsforseparablenonlinearleastsquaresproblems AT lanlanguo tikhonovregularizedvariableprojectionalgorithmsforseparablenonlinearleastsquaresproblems |