Numerical Analysis of the Non-uniform Sampling Problem

We give an overview of recent developments in the problem of reconstructing a band-limited signal from non-uniform sampling from a numerical analysis view point. It is shown that the appropriate design of the finite-dimensional model plays a key role in the numerical solution of the non-uniform sampling problem. In the one approach (often proposed in the literature) the finite-dimensional model leads to an ill-posed problem even in very simple situations. The other approach that we consider leads to a well-posed problem that preserves important structural properties of the original infinite-dimensional problem and gives rise to efficient numerical algorithms. Furthermore a fast multilevel algorithm is presented that can reconstruct signals of unknown bandwidth from noisy non-uniformly spaced samples. We also discuss the design of efficient regularization methods for ill-conditioned reconstruction problems. Numerical examples from spectroscopy and exploration geophysics demonstrate the performance of the proposed methods

Multilevel Approach For Signal Restoration Problems With Toeplitz Matrices

We present a multilevel method for discrete ill-posed problems arising from the discretization of Fredholm integral equations of the first kind. In this method, we use the Haar wavelet transform to define restriction and prolongation operators within a multigrid-type iteration. The choice of the Haar wavelet operator has the advantage of preserving matrix structure, such as Toeplitz, between grids, which can be exploited to obtain faster solvers on each level where an edge-preserving Tikhonov regularization is applied. Finally, we present results that indicate the promise of this approach for restoration of signals and images with edges

A multi-level preconditioned Krylov method for the efficient solution of algebraic tomographic reconstruction problems

Classical iterative methods for tomographic reconstruction include the class of Algebraic Reconstruction Techniques (ART). Convergence of these stationary linear iterative methods is however notably slow. In this paper we propose the use of Krylov solvers for tomographic linear inversion problems. These advanced iterative methods feature fast convergence at the expense of a higher computational cost per iteration, causing them to be generally uncompetitive without the inclusion of a suitable preconditioner. Combining elements from standard multigrid (MG) solvers and the theory of wavelets, a novel wavelet-based multi-level (WMG) preconditioner is introduced, which is shown to significantly speed-up Krylov convergence. The performance of the WMG-preconditioned Krylov method is analyzed through a spectral analysis, and the approach is compared to existing methods like the classical Simultaneous Iterative Reconstruction Technique (SIRT) and unpreconditioned Krylov methods on a 2D tomographic benchmark problem. Numerical experiments are promising, showing the method to be competitive with the classical Algebraic Reconstruction Techniques in terms of convergence speed and overall performance (CPU time) as well as precision of the reconstruction.Comment: Journal of Computational and Applied Mathematics (2014), 26 pages, 13 figures, 3 table

Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems