An iterative algorithm for large size least-squares constrained regularization problems

Abstract In this paper we propose an iterative algorithm to solve large size linear inverse ill posed problems. The regularization problem is formulated as a constrained optimization problem. The dual Lagrangian problem is iteratively solved to compute an approximate solution. Before starting the iterations, the algorithm computes the necessary smoothing parameters and the error tolerances from the data. The numerical experiments performed on test problems show that the algorithm gives good results both in terms of precision and computational efficiency.

[1]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[2]  Qin Zhang,et al.  Iterative exponential filtering for large discrete ill-posed problems , 1999, Numerische Mathematik.

[3]  G. Golub,et al.  Quadratically constrained least squares and quadratic problems , 1991 .

[4]  S. Nash,et al.  Linear and Nonlinear Programming , 1987 .

[5]  Gene H. Golub,et al.  Matrix computations , 1983 .

[6]  P. Hansen.  The discrete picard condition for discrete ill-posed problems , 1990 .

[7]  P. Toint,et al.  Trust-region and other regularisations of linear least-squares problems , 2009 .

[8]  Danny C. Sorensen,et al.  A Trust-Region Approach to the Regularization of Large-Scale Discrete Forms of Ill-Posed Problems , 2001, SIAM J. Sci. Comput..

[9]  Per Christian Hansen,et al.  REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems , 1994, Numerical Algorithms.

[10]  J. Nagy,et al.  A weighted-GCV method for Lanczos-hybrid regularization. , 2007 .

[11]  Per Christian Hansen,et al.  Rank-Deficient and Discrete Ill-Posed Problems , 1996 .

[12]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[13]  Donald W. Cooley,et al.  Solving quadratically constrained least squares using black box solvers , 1992 .

[14]  Danny C. Sorensen,et al.  Algorithm 873: LSTRS: MATLAB software for large-scale trust-region subproblems and regularization , 2008, TOMS.

[15]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[16]  M. Hanke.  Conjugate gradient type methods for ill-posed problems , 1995 .

[17]  C. Vogel.  Computational Methods for Inverse Problems , 1987 .