A novel characteristic value correction iteration method

Zhai, Min; Liu, Guolin; Qing, Tao; Xin, Mingzhen

doi:10.1080/03610918.2017.1390124

Cited by 6 publications

(5 citation statements)

References 5 publications

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“…If c k tends to zero, d k tends to the Gauss-Newton direction. ere are many methods of selecting damping factor c k [40]. After obtaining c k , the search direction d k can be obtained by solving (17).…”

Section: Determination Of the Nonlinear Parametersmentioning

confidence: 99%

Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems

Guo

2019

Complexity

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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.

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Section: Determination Of the Nonlinear Parametersmentioning

confidence: 99%

Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems

Guo

2019

Complexity

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“…Regularization is a commonly used method when ill-conditioned problems are solved. e use of regularization can make the regression coefficients have lower variance values, thereby reducing potential ill-conditioned problems [11,12]. Commonly used regularization methods include the Tikhonov regularization method, the truncated singular value method, the kernel-based regularization method, and the norm-based regularization method [13][14][15].…”

Section: Introductionmentioning

confidence: 99%

An Efficient Algorithm for Ill-Conditioned Separable Nonlinear Least Squares

Wang

Guo

et al. 2021

Journal of Mathematics

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For separable nonlinear least squares models, a variable projection algorithm based on matrix factorization is studied, and the ill-conditioning of the model parameters is considered in the specific solution process of the model. When the linear parameters are estimated, the Tikhonov regularization method is used to solve the ill-conditioned problems. When the nonlinear parameters are estimated, the QR decomposition, Gram–Schmidt orthogonalization decomposition, and SVD are applied in the Jacobian matrix. These methods are then compared with the method in which the variables are not separated. Numerical experiments are performed using RBF neural network data, and the experimental results are analyzed in terms of both qualitative and quantitative indicators. The results show that the proposed algorithms are effective and robust.

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“…In addition, Chen et al [41] proposed a weighted generalized crossvalidation method to determine Tikhonov regularization parameters for the regularization of separable nonlinear least squares ill-posed problems based on the VP algorithm and verified its effectiveness experimentally. Aiming at the randomness of parameter selection in the iteration by correcting characteristic values in the process of linear least squares solution, Zhai et al [42] constructed the L-curve [43,44] of the relationship between the norm of the parameter solution and the residual. ey selected the maximum curvature point as the regularization parameter and verified the correctness of the method through numerical experiments.…”

Section: Introductionmentioning

confidence: 99%

Efficient Parameters Estimation Method for the Separable Nonlinear Least Squares Problem

et al. 2020

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In this work, we combine the special structure of the separable nonlinear least squares problem with a variable projection algorithm based on singular value decomposition to separate linear and nonlinear parameters. en, we propose finding the nonlinear parameters using the Levenberg-Marquart (LM) algorithm and either solve the linear parameters using the least squares method directly or by using an iteration method that corrects the characteristic values based on the L-curve, according to whether or not the nonlinear function coefficient matrix is ill posed. To prove the feasibility of the proposed method, we compared its performance on three examples with that of the LM method without parameter separation. e results show that (1) the parameter separation method reduces the number of iterations and improves computational efficiency by reducing the parameter dimensions and (2) when the coefficient matrix of the linear parameters is well-posed, using the least squares method to solve the fitting problem provides the highest fitting accuracy. When the coefficient matrix is ill posed, the method of correcting characteristic values based on the L-curve provides the most accurate solution to the fitting problem.

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A novel characteristic value correction iteration method

Cited by 6 publications

References 5 publications

Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems

Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems

An Efficient Algorithm for Ill-Conditioned Separable Nonlinear Least Squares

Efficient Parameters Estimation Method for the Separable Nonlinear Least Squares Problem

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