Article

The Lagrange method for the regularization of discrete ill-posed problems

(Impact Factor: 1.32). 01/2008; 39(3):347-368. DOI: 10.1007/s10589-007-9059-3

ABSTRACT

In many science and engineering applications, the discretization of linear ill-posed problems gives rise to large ill-conditioned
linear systems with the right-hand side degraded by noise. The solution of such linear systems requires the solution of minimization
problems with one quadratic constraint, depending on an estimate of the variance of the noise. This strategy is known as regularization.
In this work, we propose a modification of the Lagrange method for the solution of the noise constrained regularization problem.
We present the numerical results of test problems, image restoration and medical imaging denoising. Our results indicate that
the proposed Lagrange method is effective and efficient in computing good regularized solutions of ill-conditioned linear
systems and in computing the corresponding Lagrange multipliers. Moreover, our numerical experiments show that the Lagrange
method is computationally convenient. Therefore, the Lagrange method is a promising approach for dealing with ill-posed problems.

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• "Los diferentes métodos para la selección del parámetro de regularización suelen clasificarse en dos categorías: aquellos que requieren conocimiento previo de la varianza del ruido y aquellos métodos que no la necesitan. Así mismo, la mayoría de los métodos de selección de parámetros de regularización han sido diseñados para su uso con el funcional de Tikhonov, y pocos esfuerzos se han desarrollado para otros funcionales como el de variación total (Landi, 2008). Algunos métodos comúnmente estudiados son: "
Conference Paper: Revisión de Métodos de Regularización Directa y sus Aplicaciones en las Ciencias Atmosféricas
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L Convención Nacional del Instituto Mexicano de Ingenieros Químicos; 10/2010
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• Article: An iterative Lagrange method for the regularization of discrete ill-posed inverse problems
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ABSTRACT: In this paper, an iterative method is presented for the computation of regularized solutions of discrete ill-posed problems. In the proposed method, the regularization problem is formulated as an equality constrained minimization problem and an iterative Lagrange method is used for its solution. The Lagrange iteration is terminated according to the discrepancy principle. The relationship between the proposed approach and classical Tikhonov regularization is discussed. Results of numerical experiments are presented to illustrate the effectiveness and usefulness of the proposed method.
Computers & Mathematics with Applications 09/2010; 60(6):1723-1738. DOI:10.1016/j.camwa.2010.07.003 · 1.70 Impact Factor