scholarly journals Efficient Parameters Estimation Method for the Separable Nonlinear Least Squares Problem

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16 ◽  
Author(s):  
Ke Wang ◽  
Guolin Liu ◽  
Qiuxiang Tao ◽  
Min Zhai
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Coefficient Matrix ◽  
Nonlinear Parameters ◽  
Nonlinear Least Squares Problem ◽  
Ill Posed ◽  
Characteristic Values ◽  
Parameter Separation ◽  
Fitting Problem

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. Then, 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. The 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.

Estimation of a Regularisation Parameter for a Robin Inverse Problem

2017 ◽  
Vol 7 (2) ◽  
pp. 325-342
Author(s):  
Xi-Ming Fang ◽  
Fu-Rong Lin ◽  
Chao Wang
Keyword(s):  
Inverse Problem ◽  
White Noise ◽  
Least Squares ◽  
Laplace Equation ◽  
Noise Level ◽  
Nonlinear Least Squares ◽  
Measured Data ◽  
Residual Vector ◽  
Nonlinear Least Squares Problem ◽  
Ill Posed

AbstractWe consider the nonlinear and ill-posed inverse problem where the Robin coefficient in the Laplace equation is to be estimated using the measured data from the accessible part of the boundary. Two regularisation methods are considered — viz. L2 and H1 regularisation. The regularised problem is transformed to a nonlinear least squares problem; and a suitable regularisation parameter is chosen via the normalised cumulative periodogram (NCP) curve of the residual vector under the assumption of white noise, where information on the noise level is not required. Numerical results show that the proposed method is efficient and competitive.

scholarly journals A Method for Solving LiDAR Waveform Decomposition Parameters Based on a Variable Projection Algorithm

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Ke Wang ◽  
Guolin Liu ◽  
Qiuxiang Tao ◽  
Luyao Wang ◽  
Yang Chen
Keyword(s):  
Parameter Estimation ◽  
Least Squares ◽  
Gaussian Function ◽  
Nonlinear Least Squares ◽  
Projection Algorithm ◽  
Nonlinear Parameters ◽  
Least Squares Problem ◽  
Variable Projection ◽  
Nonlinear Least Squares Problem ◽  
Lidar Waveform

Light detection and ranging (LiDAR) is commonly used to create high-resolution maps; however, the efficiency and convergence of parameter estimation are difficult. To address this issue, we evaluated the structural characteristics of received LiDAR signals by decomposing them into Gaussian functions and applied the variable projection algorithm of the separable nonlinear least-squares problem to the process of waveform fitting. First, using a variable projection algorithm, we separated the linear (amplitude) and nonlinear (center position and width) parameters in the Gaussian function model; the linear parameters are expressed with nonlinear parameters by the function. Thereafter, the optimal estimation of the characteristic parameters of the Gaussian function components was transformed into a least-squares problem only comprising nonlinear parameters. Finally, the Levenberg–Marquardt algorithm was used to solve these nonlinear parameters, whereas the linear parameters were calculated simultaneously in each iteration, and the estimation results satisfying the nonlinear least-square criterion were obtained. Five groups of waveform decomposition simulation data and ICESat/GLAS satellite LiDAR waveform data were used for the parameter estimation experiments. During the experiments, for the same accuracy, the separable nonlinear least-squares optimization method required fewer iterations and lesser calculation time than the traditional method of not separating parameters; the maximum number of iterations was reached before the traditional method converged to the optimal estimate. The method of separating variables only required 14 iterations to obtain the optimal estimate, reducing the computational time from 1128 s to 130 s. Therefore, the application of the separable nonlinear least-squares problem can improve the calculation efficiency and convergence speed of the parameter solution process. It can also provide a new method for parameter estimation in the Gaussian model for LiDAR waveform decomposition.

A Regularization Homotopy Iterative Method for Ill-Posed Nonlinear Least Squares Problem and its Application

2011 ◽  
Vol 90-93 ◽  
pp. 3268-3273 ◽  
Author(s):  
Li Min Tang
Keyword(s):  
Iterative Method ◽  
Least Squares ◽  
Regularization Parameter ◽  
Nonlinear Least Squares ◽  
Least Squares Problems ◽  
Least Squares Problem ◽  
Nonlinear Least Squares Problem ◽  
Calculation Results ◽  
Ill Posed ◽  
Defor Mation

A regularization homotopy iterative method established for ill-posed nonlinear least squares problem. Two new regularization parameter selecting strategies are proposed, which are called direct search method and interval division method. The calculation results of nonlinear least squares problems show that the regularization homotopy iterative method and parameter selecting strategies proposed in this paper are correctly and applicable. And also calculation results of nonlinear adjustment of free networks with rank deficiency of Jianglong bridge pier displacement defor-mation monitoring control network show that the method not only decrease the iterative matrix con-dition number, but also make the condition number small fluctuation in full iterative process.

scholarly journals MIM and nonlinear least‐squares inversions of AEM data in Barataria basin, Louisiana

Geophysics ◽  
2003 ◽  
Vol 68 (4) ◽  
pp. 1126-1131 ◽  
Author(s):  
Melissa Whitten Bryan ◽  
Kenneth W. Holladay ◽  
Clyde J. Bergeron ◽  
Juliette W. Ioup ◽  
George E. Ioup
Keyword(s):  
Least Squares ◽  
Water Depth ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Image Method ◽  
Water Conductivity ◽  
Near Surface ◽  
Barataria Basin ◽  
Airborne Electromagnetic Survey ◽  
Nonlinear Least Squares Method

An airborne electromagnetic survey was performed over the marsh and estuarine waters of the Barataria basin of Louisiana. Two inversion methods were applied to the measured data to calculate layer thicknesses and conductivities: the modified image method (MIM) and a nonlinear least‐squares method of inversion using two two‐layer forward models and one three‐layer forward model, with results generally in good agreement. Uniform horizontal water layers in the near‐shore Gulf of Mexico with the fresher (less saline, less conductive) water above the saltier (more saline, more conductive) water can be seen clearly. More complex near‐surface layering showing decreasing salinity/conductivity with depth can be seen in the marshes and inland areas. The first‐layer water depth is calculated to be 1–2 m, with the second‐layer water depth around 4 m. The first‐layer marsh and beach depths are computed to be 0–3 m, and the second‐layer marsh and beach depths vary from 2 to 9 m. The first‐layer water conductivity is calculated to be 2–3 S/m, with the second‐layer water conductivity around 3 to 4 S/m and the third‐layer water conductivity 4–5 S/m. The first‐layer marsh conductivity is computed to be mainly 1–2 S/m, and the second‐ and third‐layer marsh conductivities vary from 0.5 to 1.5 S/m, with the conductivities decreasing as depth increases except on the beach, where layer three has a much higher conductivity, ranging up to 3 S/m.

The Strength Evaluation and Reliability Analysis of Periodic Symmetric Struts Support

2011 ◽  
Vol 462-463 ◽  
pp. 1164-1169
Author(s):  
Jing Xiang Yang ◽  
Ya Xin Zhang ◽  
Mamtimin Gheni ◽  
Ping Ping Chang ◽  
Kai Yin Chen ◽  
...  
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Stress Distribution ◽  
Least Squares ◽  
Reliability Analysis ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Distribution Model ◽  
Different Types ◽  
Nonlinear Least Squares Method

In this paper, strength evaluations and reliability analysis are conducted for different types of PSSS(Periodically Symmetric Struts Supports) based on the FEA(Finite Element Analysis). The numerical models are established at first, and the PMA(Prestressed Modal Analysis) is conducted. The nodal stress value of all of the gauss points in elements are extracted out and the stress distributions are evaluated for each type of PSSS. Then using nonlinear least squares method, curve fitting is carried out, and the stress probability distribution function is obtained. The results show that although using different number of struts, the stress distribution function obeys the exponential distribution. By using nonlinear least squares method again for the distribution parameters a and b of different exponential functions, the relationship between number of struts and distribution function is obtained, and the mathematical models of the stress probability distribution functions for different supports are established. Finally, the new stress distribution model is introduced by considering the DSSI(Damaged Stress-Strength Interference), and the reliability evaluation for different types of periodically symmetric struts supports is carried out.

A NONLINEAR LEAST‐SQUARES METHOD FOR SEISMIC REFRACTION MAPPING—PART I: ALGORITHM AND PROCEDURE

Geophysics ◽  
1972 ◽  
Vol 37 (2) ◽  
pp. 260-272 ◽  
Author(s):  
Leonidas C. Ocola
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Seismic Refraction ◽  
Vertical Plane ◽  
Nonlinear Least Squares ◽  
Inversion Method ◽  
Almost Everywhere ◽  
Isotropic Media ◽  
Time Term ◽  
Nonlinear Least Squares Method

An iterative inversion method (Reframap) based on the kinematic properties of critically refracted waves is developed. The method is based on ray tracing and assumes homogeneous and isotropic media and ray paths confined to a vertical plane through each source‐detector pair. Unlike the earlier Profile or Time‐Term Methods, no restrictions are imposed on interface topography except that it be continuous almost everywhere (in the mathematical sense). As in the preexisting methods, more observations than unknowns are assumed. The algorithm and procedure, on which the Reframap Method is based, generate apparent dips for each source detector pair at the noncritical interfaces from the slope of a least‐squares line approximation to the interface functional in the neighborhood of each refraction point. In turn, the dip and path along the critical refractor is, at every iteration, pairwise approximated by a line through the critical refracting points. The incidence angles are computed recursively by Snell’s law. The solution of the overdetermined, nonlinear multiple refractor time‐distance system of simultaneous equations is sought by Marquardt’s algorithm for least‐squares estimation of critical refractor velocity and vertical thickness under each element.

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