Regularization Methods for Uniformly Rank-Deficient Nonlinear Least-Squares Problems

被引:1
作者
J. Eriksson
P. A. Wedin
M. E. Gulliksson
I. Söderkvist
机构
[1] Umeå University,Lecturer, Department of Computing Science
[2] Umeå University,Professor, Department of Computing Science
[3] Mid-Sweden University,Professor, Department of Engineering, Physics, and Mathematics
[4] Luleå University of Technology,Associate Professor, Department of Mathematics
来源
Journal of Optimization Theory and Applications | 2005年 / 127卷
关键词
Nonlinear least squares; Gauss-Newton Method; rank-deficient matrices; minimum norm problems; truncation problems; stabilization methods; ill-posed problems; Tikhonov regularization;
D O I
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中图分类号
学科分类号
摘要
In solving the nonlinear least-squares problem of minimizing ||f(x)||22, difficulties arise with standard approaches, such as the Levenberg-Marquardt approach, when the Jacobian of f is rank-deficient or very ill-conditioned at the solution. To handle this difficulty, we study a special class of least-squares problems that are uniformly rank-deficient, i.e., the Jacobian of f has the same deficient rank in the neighborhood of a solution. For such problems, the solution is not locally unique. We present two solution tecniques: (i) finding a minimum-norm solution to the basic problem, (ii) using a Tikhonov regularization. Optimality conditions and algorithms are given for both of these strategies. Asymptotical convergence properties of the algorithms are derived and confirmed by numerical experiments. Extensions of the presented ideas make it possible to solve more general nonlinear least-squares problems in which the Jacobian of f at the solution is rank-deficient or ill-conditioned.
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页码:1 / 26
页数:25
相关论文
共 12 条
[1]  
Pruessner A.(2003)Blind Deconvolution Using a Regularized Structured Total Least-Norm Algorithm SIAM Journal on Matrix Analysis and Applications 24 1018-1037
[2]  
O’Leary D.P.(1978)Algorithms for the Solution of the Nonlinear Least–Squares Problem SIAM Journal on Numerical Analysis 15 976-992
[3]  
Gill P.E.(1984)Tensor Methods for Nonlinear Equations SIAM Journal on Numerical Analysis 21 815-843
[4]  
Murray W.(1992)Neural Network and Nonlinear Optimization, Part 1: The Representation of Continuous Functions Optimization Methods and Software 1 141-151
[5]  
Frank P.(2000)The Use and Properties of Tikhonov Filter Matrices SIAM Journal on Matrix Analysis and Applications 22 276-281
[6]  
Schnabel R.B.(1977)A Comparison of Some Algorithms for the Nonlinear Least-Squares Problem, BIT 17 72-90
[7]  
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[8]  
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