Non-stationary and non-linear signals are ubiquitous in real life. Their decomposition and analysis is an important research topic of signal processing. Recently a new technique, called Iterative Filtering, has been developed with the goal of decomposing such signals into simple oscillatory components. Several papers have been devoted to the investigation of this technique from a mathematical point of view. All these works start with the assumption that each compactly supported signal is extended periodically outside the boundaries. In this work, we tackle the problem of studying the influence of different boundary conditions on the decompositions produced by the Iterative Filtering method. In particular, the choice of boundary conditions gives rise to different types of structured matrices. Thus, we describe their spectral properties and then convergence properties of Iterative Filtering algorithm (in which such matrices are involved). Numerical results provide an interesting overview on important aspects (such as accuracy and error propagation) of the techniques proposed and show the way of further promising developments.

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It is well known that iterative algorithms for image deblurring that involve the normal equations show usually a slow convergence. A variant of the normal equations which re- places the conjugate transpose A^H of the system matrix A with a new matrix is proposed. This approach, which is linked with regularization preconditioning theory and reblurring processes, can be applied to a wide set of iterative methods; here we examine Landweber, Steepest descent, Richardson-Lucy and Image Space Reconstruction Algorithm. Several computational tests show that this strategy leads to a significant improvement of the convergence speed of the methods. Moreover it can be naturally combined with other widely used acceleration techniques

The Alternating Direction Multipliers Method (ADMM) is a very popular algorithm for computing the solution of convex constrained minimization problems. Such problems are important from the application point of view, since they occur in many fields of science and engineering. ADMM is a powerful numerical tool, but unfortunately its main drawback is that it can exhibit slow convergence. Several approaches for its acceleration have been proposed in the literature and in this paper we present a new general framework devoted to this aim. In particular, we describe an algorithmic framework that makes possible the application of any acceleration step while still having the guarantee of convergence. This result is achieved thanks to a guard condition that ensures the monotonic decrease of the combined residual. The proposed strategy is applied to image deblurring problems. Several acceleration techniques are compared; to the best of our knowledge, some of them are investigated for the first time in connection with ADMM. Numerical results show that the proposed framework leads to a faster convergence with respect to other acceleration strategies recently introduced for ADMM.

We analyse the practical efficiency of multi-iterative techniques for the numerical solution of graph-structured large linear systems. In particular we evaluate the effectiveness of several combinations of coarser-grid operators which preserve the graph structure of the projected matrix at the inner levels and smoothers. We also discuss and evaluate some possible strategies (inverse projection and dense projection) to connect coarser-grid operators and graph-based preconditioners. Our results show that an appropriate choice of adaptive projectors and tree-based preconditioned conjugate gradient methods result in highly effective and robust approaches, that are capable to efficiently solve large-scale, difficult systems, for which the known iterative solvers alone can be rather slow