Methods for large-scale discrete ill-posed problems

Author(s)

Leonardo Silveira Borges

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2013-07-02

Examining board members

Maria Cristina de Castro Cunha; Fermin Sinforiano Viloche Bazan; Fabio Antonio Dorini; Luiz Mariano Paes de Carvalho Filho; Márcia Aparecida Gomes Ruggiero; Eduardo Cardoso de Abreu

• Advisor: Maria Cristina de Castro Cunha; Fermin Sinforiano Viloche Bazan

Abstract

Discrete ill-posed problems need to be regularized in order to be stably solved. Amongst several regularization methods, perhaps the most used is the method of Tikhonov whose effectiveness depends on a proper choice of the regularization parameter. There are considerable amount of parameter choice rules in the literature; these include the Discrepancy Principle by Morozov and heuristic methods like the L-curve criterion by Hansen, Generalized Cross Validation by Golub, Heath and Wahba, and a fixed point method due to Bazán. Large-scale discrete ill-posed problems can be solved by iterative methods like CGLS and LSQR provided that the iterations are stopped before the noise starts deteriorating the quality of the iterates. This is a difficult task which has not yet been addressed satisfactorily in the literature. In an attempt to alleviate the difficulty associated with selecting the regularization parameter, iterative methods can be combined with Tikhonov regularization giving rise to the so-called hybrid methods such as GKB-FP and W-GCV (both using the identity matrix as regularization matrix). The contributions of this thesis include further results concerning the theoretical properties of GKB-FP algorithm as well as the extension of GKB-FP to Tikhonov regularization using a general regularization matrix. Apart from this, as a second contribution, we propose an automatic stopping rule for iterative methods for large-scale problems, including the case where the methods are preconditioned via smoothing norms. Tikhonov regularization has been widely applied to solve linear ill-posed problems, but almost always confined to a single regularization parameter. Nevertheless, some problems have solutions with distinctive characteristics that must be included in the regularized solution. This leads to multi-parameter Tikhonov regularization problems. The third contribution of the thesis is the development of a fixed point method to select the regularization parameters in this multi-parameter case as well as a GKB-FP type algorithm which is well suited for large-scale problems. The proposed algorithms are numerically illustrated by solving several problems such as reconstruction and super-resolution image problems, scattering problems and others from Fredholm integral equations (AU)