Two-level preconditioners for regularized ill-posed problems (original) (raw)
The goal of this thesis is the solution of large symmetric positive definite linear systems which arise when Tikhonov regularization is applied to solve an ill-posed problem. The coefficient matrix for these systems is the sum of two terms. The first term comes from the discretization of a compact operator and is a dense matrix, i.e., not sparse. The second term, which is called the regularization matrix, is a sparse matrix that is either the identity or the discretization of a diffusion operator. In addition, the regularization matrix is scaled by a small positive parameter, which is called the regularization parameter. In practice, these systems are moderately ill-conditioned. To solve such systems, we apply the preconditioned conjugate gradient algorithm with two-level preconditioners. These preconditioners were previously developed by Hanke and Vogel for positive definite regularization matrices. The contribution of this thesis is extension to the case where the regularization matrix is positive semidefinite. Also there is a compilation of the two-level preconditioning algorithms, and an examination of computational cost issues. To evaluate performance the preconditioners are applied to a problem from image processing known as image deblurring.
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