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.
Highlights
Complexity on potentially ill-conditioned problems with parameters
Is paper proposes a hybrid VP-based algorithm that combines the LM and Tikhonov regularisation (TR) methods and evaluates the effect of different Jacobian matrix algorithms on its accuracy and efficiency. e TR method is used to regularise the linear parameter estimation, and the LM algorithm is used to estimate the nonlinear parameters in a separable nonlinear least squares problem. e Jacobian matrix is calculated using three methods: Golub and Pereyra (GP), Kaufman (KAU), and Ruano (RJF). e algorithm proposed in this paper combines the advantages of the LM algorithm with those of a regularisation method and can improve the estimation efficiency. e model was validated using two examples: exponential fitting and the determination of waveform parameters for airborne radar sounding data
For the separable nonlinear least squares problem, this paper establishes a solution model consisting of an LM algorithm combined with TR
Summary
E Jacobian matrix can be calculated in the following three ways:. (1) e calculation method given by Golub and Pereyra is as follows: JGP DP⊥Φy − P⊥ΦDΦΦ− y − P⊥ΦDΦΦ− Ty, (13). (3) Ruano proposed an even simpler calculation method for the Jacobian matrix based on Kaufman’s method: JRJF − DΦΦ− y. Ruano et al [37] proved that (15) is valid and can be obtained from Kaufman’s Jacobian matrix (14). It can be proved that the same gradient vector can be obtained by the above three Jacobian matrices [38]. E following equation can be used to iterate nonlinear parameters: θkN+1 θkN + βkdk,. Scalar step size that ensures that Φ(θN)θL‖22 is decreasing and dk the objective is the search
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