Fabio Di Benedetto

Analysis of preconditioning techniques for ill-conditioned Toeplitz matrices

Several preconditioning techniques for solving Toeplitz systems are known in literature, but their convergence features are completely understood only in the well-conditioned case. We study the application of

A new preconditioner for the parallel solution of positive definite Toeplitz systems

\tau

Preconditioning of Block Toeplitz Matrices by Sine Transforms

, circulant, and Hartley preconditioners to ill-conditioned Toeplitz matrices by proving that only the first class realizes a rate of convergence not depending on the dimension of the system.

Improvement of Space-Invariant Image Deblurring by Preconditioned Landweber Iterations

We introduce a new preconditioner for solving a symmetric Toeplitz system of equations by the conjugate gradient method. This choice leads to an algorithm which is particularly suitable for parallel computations and, compared to the circulant preconditioner of [C33, has a better asymptotic convergence rate and a lower arithmetic cost per iteration.

Superoptimal Preconditioned Conjugate Gradient Iteration for Image Deblurring

The iterative solution of a block Toeplitz linear system by the conjugate gradient method is analyzed, the preconditioning step being solved by means of a discrete sine transform. Convergence properties are established and compared to the behavior of the block circulant preconditioner recently proposed in the literature. As in the scalar case, the new approach takes advantage if the system is ill conditioned.

Optimal multilevel matrix algebra operators

The Landweber method is a simple and flexible iterative regularization algorithm, whose projected variant provides nonnegative image reconstructions. Since the method is usually very slow, we apply circulant preconditioners, exploiting the shift invariance of many deblurring problems, in order to accelerate the convergence. This way reasonable reconstructions can be obtained within a few iterations; the method becomes competitive and more robust than other approaches that, although faster, sometimes lead to lower accuracy. Some theoretical analysis of convergence is given, together with numerical validations.

Preconditioned iterative regularization in Banach spaces

We study the superoptimal Frobenius operators in the two-level circulant algebra. We consider two specific viewpoints: (1) the regularizing properties in imaging and (2) the computational effort in connection with the preconditioned conjugate gradient method. Some numerical experiments illustrating the effectiveness of the proposed technique are given and discussed.

Solving the generalized eigenvalue problem for rational Toeplitz matrices

We study the optimal Frobenius operator in a general matrix vector space and in particular in the multilevel trigonometric matrix vector spaces, by emphasizing both the algebraic and geometric properties. These general results are used to extend the Korovkin matrix theory for the approximation of block Toeplitz matrices via trigonometric vector spaces. The abstract theory is then applied to the analysis of the approximation properties of several sine and cosine based vector spaces. Few numerical experiments are performed to give evidence of the theoretical results.

Shift-invariant approximations of structured shift-variant blurring matrices

Regularization methods for inverse problems formulated in Hilbert spaces usually give rise to over-smoothness, which does not allow to obtain a good contrast and localization of the edges in the context of image restoration.On the other hand, regularization methods recently introduced in Banach spaces allow in general to obtain better localization and restoration of the discontinuities or localized impulsive signals in imaging applications.We present here an expository survey of the topic focused on the iterative Landweber method. In addition, preconditioning techniques previously proposed for Hilbert spaces are extended to the Banach setting and numerically tested.

Toeplitz matrices: Structures, algorithms and applications

The generalized eigenvalue problem