Marco Donatelli

Marco Donatelli, Stefano Serra-Capizzano, Debora Sesana

Starting from the spectral analysis of g-circulant matrices, we consider a new multigrid method for circulant and Toeplitz matrices with given generating function. We assume that the size n of the coefficient matrix is divisible by g \geq 2 such that at the lower level the system is reduced to one of size n/g by employing g-circulant based projectors. We perform a rigorous two-grid convergence analysis in the circulant case and we extend experimentally the results to the Toeplitz setting, by employing structure preserving projectors. The optimality of the proposed two-grid method and of the multigrid method is proved, when the number theta \in N of recursive calls is such that 1 < theta < g. The previous analysis is used also to overcome some pathological cases, in which the generating function has zeros located at "mirror points" and the standard two-grid method with g = 2 is not optimal. The numerical experiments show the correctness and applicability of the proposed ideas both for circulant and Toeplitz matrices.

Zheng-Jian Bai, Marco Donatelli, Stefano Serra-Capizzano

In recent works several authors have proposed the use of precise boundary conditions (BCs) for blurring models and they proved that the resulting choice (Neumann or reflective, anti-reflective) leads to fast algorithms both for deblurring and for detecting the regularization parameters in presence of noise. When considering a symmetric point spread function, the crucial fact is that such BCs are related to fast trigonometric transforms. In this paper we combine the use of precise BCs with the Total Variation (TV) approach in order to preserve the jumps of the given signal (edges of the given image) as much as possible. We consider a classic fixed point method with a preconditioned Krylov method (usually the conjugate gradient method) for the inner iteration. Based on fast trigonometric transforms, we propose some preconditioning strategies which are suitable for reflective and anti-reflective BCs. A theoretical analysis motivates the choice of our preconditioners and an extensive numerical experimentation is reported and critically discussed. The latter shows that the TV regularization with anti-reflective BCs implies not only a reduced analytical error, but also a lower computational cost of the whole restoration procedure over the other BCs.

Marco Donatelli

The Local Fourier analysis (LFA) is a classic tool to prove convergence theorems for multigrid methods (MGMs). In particular, we are interested in optimality that is a convergence speed independent of the size of the involved matrices. For elliptic partial differential equations (PDEs), a well known optimality result requires that the sum of the orders of the grid transfer operators is not lower than the order of the PDE to solve. Analogously, when dealing with MGMs for Toeplitz matrices in the literature an optimality condition on the position and on the order of the zeros of the symbols of the grid transfer operators has been found. In this work we show that in the case of elliptic PDEs with constant coefficients, the two different approaches lead to an equivalent condition. We argue that the analysis for Toeplitz matrices is an algebraic generalization of the LFA, which allows to deal not only with differential problems but also for instance with integral problems. The equivalence of the two approaches gives the possibility of using grid transfer operators with different orders also for MGMs for Toeplitz matrices. We give also a class of grid transfer operators related to the B-spline's refinement equation and we study their geometric properties. This analysis suggests further links between wavelets and multigrid methods. A numerical experimentation confirms the correctness of the proposed analysis.

Andrea Angino, Matteo Aurina, Alena Kopaničáková et al.

We introduce two multifidelity trust-region methods based on the Magical Trust Region (MTR) framework. MTR augments the classical trust-region step with a secondary, informative direction. In our approaches, the secondary ``magical'' directions are determined by solving coarse trust-region subproblems based on low-fidelity objective models. The first proposed method, Sketched Trust-Region (STR), constructs this secondary direction using a sketched matrix to reduce the dimensionality of the trust-region subproblem. The second method, SVD Trust-Region (SVDTR), defines the magical direction via a truncated singular value decomposition of the dataset, capturing the leading directions of variability. Several numerical examples illustrate the potential gain in efficiency.

Carlo Santambrogio, Monica Pragliola, Alessandro Lanza et al.

We consider an unsupervised bilevel optimization strategy for learning regularization parameters in the context of imaging inverse problems in the presence of additive white Gaussian noise. Compared to supervised and semi-supervised metrics relying either on the prior knowledge of reference data and/or on some (partial) knowledge on the noise statistics, the proposed approach optimizes the whiteness of the residual between the observed data and the observation model with no need of ground-truth data.We validate the approach on standard Total Variation-regularized image deconvolution problems which show that the proposed quality metric provides estimates close to the mean-square error oracle and to discrepancy-based principles.

Giuseppe Scarlato, Antonio Cicone, Marco Donatelli

In the analysis of real-world data, extracting meaningful features from signals is a crucial task. This is particularly challenging when signals contain non-stationary frequency components. The Iterative Filtering (IF) method has proven to be an effective tool for decomposing such signals. However, such a technique cannot handle directly data that have been sampled non-uniformly. On the other hand, graph signal processing has gained increasing attention due to its versatility and wide range of applications, and it can handle data sampled both uniformly and non-uniformly. In this work, we propose two algorithms that extend the IF method to signals defined on graphs. In addition, we provide a unified convergence analysis for the different IF variants. Finally, numerical experiments on a variety of graphs, including real-world data, confirm the effectiveness of the proposed methods. In particular, we test our algorithms on seismic data and the total electron content of the ionosphere. Those data are by their nature non-uniformly sampled, and, therefore, they cannot be directly analyzed by the standard IF method.

Costanza Conti, Marco Donatelli, Lucia Romani et al.

Convergence and normal continuity analysis of a bivariate non-stationary (level-dependent) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, in this paper we derive new sufficient conditions for establishing convergence and normal continuity of any rotationally symmetric, non-stationary, subdivision scheme near an extraordinary vertex/face.

Maria Charina, Marco Donatelli, Lucia Romani et al.

In this paper, we present a family of multivariate grid transfer operators appropriate for anisotropic multigrid methods. Our grid transfer operators are derived from a new family of anisotropic interpolatory subdivision schemes. We study the minimality, polynomial reproduction and convergence properties of these interpolatory schemes and link their properties to the convergence and optimality of the corresponding multigrid methods. We compare the performance of our interpolarory grid transfer operators with the ones derived from a family of corresponding approximating subdivision schemes.

Hamid Moghaderi, Mehdi Dehghan, Marco Donatelli et al.

Fractional diffusion equations (FDEs) are a mathematical tool used for describing some special diffusion phenomena arising in many different applications like porous media and computational finance. In this paper, we focus on a two-dimensional space-FDE problem discretized by means of a second order finite difference scheme obtained as combination of the Crank-Nicolson scheme and the so-called weighted and shifted Grünwald formula. By fully exploiting the Toeplitz-like structure of the resulting linear system, we provide a detailed spectral analysis of the coefficient matrix at each time step, both in the case of constant and variable diffusion coefficients. Such a spectral analysis has a very crucial role, since it can be used for designing fast and robust iterative solvers. In particular, we employ the obtained spectral information to define a Galerkin multigrid method based on the classical linear interpolation as grid transfer operator and damped-Jacobi as smoother, and to prove the linear convergence rate of the corresponding two-grid method. The theoretical analysis suggests that the proposed grid transfer operator is strong enough for working also with the V-cycle method and the geometric multigrid. On this basis, we introduce two computationally favourable variants of the proposed multigrid method and we use them as preconditioners for Krylov methods. Several numerical results confirm that the resulting preconditioning strategies still keep a linear convergence rate.

Maria Charina, Marco Donatelli, Lucia Romani et al.

The convergence rate of a multigrid method depends on the properties of the smoother and the so-called grid transfer operator. In this paper we define and analyze new grid transfer operators with a generic cutting size which are applicable for high order problems. We enlarge the class of available geometric grid transfer operators by relating the symbol analysis of the coarse grid correction with the approximation properties of univariate subdivision schemes. We show that the polynomial generation property and stability of a subdivision scheme are crucial for convergence and optimality of the corresponding multigrid method. We construct a new class of grid transfer operators from primal binary and ternary pseudo-spline symbols. Our numerical results illustrate the behavior of the new grid transfer operators.

Yuantao Cai, Marco Donatelli, Davide Bianchi et al.

Thresholding iterative methods are recently successfully applied to image deblurring problems. In this paper, we investigate the modified linearized Bregman algorithm (MLBA) used in image deblurring problems, with a proper treatment of the boundary artifacts. We consider two standard approaches: the imposition of boundary conditions and the use of the rectangular blurring matrix. The fast convergence of the MLBA depends on a regularizing preconditioner that could be computationally expensive and hence it is usually chosen as a block circulant circulant block (BCCB) matrix, diagonalized by discrete Fourier transform. We show that the standard approach based on the BCCB preconditioner may provide low quality restored images and we propose different preconditioning strategies, that improve the quality of the restoration and save some computational cost at the same time. Motivated by a recent nonstationary preconditioned iteration, we propose a new algorithm that combines such method with the MLBA.We prove that it is a regularizing and convergent method. A variant with a stationary preconditioner is also considered. Finally, a large number of numerical experiments shows that our methods provide accurate and fast restorations, when compared with the state of the art.

Davide Bianchi, Alessandro Buccini, Marco Donatelli et al.

Fractional Tikhonov regularization methods have been recently proposed to reduce the oversmoothing property of the Tikhonov regularization in standard form, in order to preserve the details of the approximated solution. Their regularization and convergence properties have been previously investigated showing that they are of optimal order. This paper provides saturation and converse results on their convergence rates. Using the same iterative refinement strategy of iterated Tikhonov regularization, new iterated fractional Tikhonov regularization methods are introduced. We show that these iterated methods are of optimal order and overcome the previous saturation results. Furthermore, nonstationary iterated fractional Tikhonov regularization methods are investigated, establishing their convergence rate under general conditions on the iteration parameters. Numerical results confirm the effectiveness of the proposed regularization iterations.

Matteo Semplice, Marco Donatelli, Stefano Serra-Capizzano

The novel contribution of this paper relies in the proposal of a fully implicit numerical method designed for nonlinear degenerate parabolic equations, in its convergence/stability analysis, and in the study of the related computational cost. In fact, due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required and every step implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate iterative or multi-iterative solvers, with special attention to preconditioned Krylov methods and to multigrid procedures: in particular we investigate the mutual benefit of combining in various ways suitable preconditioners with V-cycle algorithms. Numerical experiments in one and two spatial dimensions for the validation of our multi-facet analysis complement this contribution.

Marco Donatelli

We study strategies for increasing the precision in the blurring models by maintaining a complexity in the related numerical linear algebra procedures (matrix-vector product, linear system solution, computation of eigenvalues etc.) of the same order of the celebrated Fast Fourier Transform. The key idea is the choice of a suitable functional basis for representing signals and images. Starting from an analysis of the spectral decomposition of blurring matrices associated to the antireflective boundary conditions introduced in [S. Serra Capizzano, SIAM J. Sci. Comput. 25-3 pp. 1307--1325], we extend the model for preserving polynomials of higher degree and fast computations also in the nonsymmetric case. We apply the proposed model to Tikhonov regularization with smoothing norms and the generalized cross validation for choosing the regularization parameter. A selection of numerical experiments shows the effectiveness of the proposed techniques.