In this paper we are interested in the solution by multigrid strategies of multilevel linear systems whose coefficient matrices belong to the circulant, Hartley, or τ algebras or to the Toeplitz class and are generated by (the Fourier expansion of) a nonnegative multivariate polynomial f. It is well known that these matrices are banded and have eigenvalues equally distributed as f , so they are ill-conditioned whenever f takes the zero value; they can even be singular and need a low-rank correction. We prove the V-cycle multigrid iteration to have a convergence rate independent of the dimension even in presence of ill-conditioning. If the (multilevel) coefficient matrix has partial dimension nr at level r, r = 1,. .. , d, then the size of the algebraic system is N (n) = d r=1 nr, O(N (n)) operations are required by our technique, and therefore the corresponding method is optimal. Some numerical experiments concerning linear systems arising in applications, such as elliptic PDEs with mixed boundary conditions and image restoration problems, are considered and discussed.

We introduce a new iterative scheme for solving linear ill-posed problems, similar to nonstationary iterated Tikhonov regularization, but with an approximation of the underlying operator to be used for the Tikhonov equations. For image deblurring problems such an approximation can be a discrete deconvolution that operates entirely in the Fourier domain. We provide a theoretical analysis of the new scheme, using regularization parameters that are chosen by a certain adaptive strategy. The numerical performance of this method turns out to be superior to state of the art iterative methods, including the conjugate gradient iteration for the normal equation, with and without additional preconditioning.

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We analyze the convergence rate of a multigrid method for multilevel\ud linear systems whose coefficient matrices are generated by a real\ud and nonnegative multivariate polynomial $f$ and belong to multilevel\ud matrix algebras like circulant, tau, Hartley, or are of Toeplitz type.\ud \ud In the case of matrix algebra linear systems, we prove that the\ud convergence rate is independent of the system dimension even in\ud presence of asymptotical ill-conditioning (this happens iff $f$\ud takes the zero value). More precisely, if the $d$-level coefficient\ud matrix has partial dimension $n_r$ at level $r$, with $r=1,\dots,d$,\ud then the size of the system is $N(\mi{n})=\prod_{r=1}^d n_r$,\ud $\mi{n}=(n_1, \dots, n_d)$, and $O(N(\mi{n}))$ operations are\ud required by the considered $V$-cycle Multigrid in order to compute the solution\ud within a fixed accuracy. Since the total arithmetic cost is\ud asymptotically equivalent to the one of a matrix-vector product, the\ud proposed method is optimal. Some numerical experiments concerning\ud linear systems arising in 2D and 3D applications are considered\ud and discussed

We consider fast solvers for large linear systems arising from the Galerkin approximation based on B-splines of classical ddimensional\ud elliptic problems, d ≥ 1, in the context of isogeometric analysis. Our ultimate goal is to design iterative algorithms\ud with the following two properties. First, their computational cost is optimal, that is linear with respect to the number of degrees of\ud freedom, i.e. the resulting matrix size. Second, they are totally robust, i.e., their convergence speed is substantially independent of\ud all the relevant parameters: in our case, these are the matrix size (related to the fineness parameter), the spline degree (associated to\ud the approximation order), and the dimensionality d of the problem.We review several methods like PCG, multigrid, multi-iterative\ud algorithms, and we show how their numerical behavior (in terms of convergence speed) can be understood through the notion of\ud spectral distribution, i.e. through a compact symbol which describes the global eigenvalue behavior of the considered stiffness\ud matrices. As a final step, we show how we can design an optimal and totally robust multi-iterative method, by taking into account\ud the analytic features of the symbol. A wide variety of numerical experiments, few open problems and perspectives are presented\ud and critically discussed.\ud ⃝c 2014 Elsevier B.V. All rights reserved

We consider the stiffness matrices coming from the Galerkin B-spline isogeometric analysis approximation of classical elliptic problems. By exploiting specific spectral properties compactly described by a symbol, we design efficient multigrid methods for the fast solution of the related linear systems. We prove the optimality of the two-grid methods (in the sense that their convergence rate is independent of the matrix size) for spline degrees up to 3, both in the 1D and 2D case. Despite the theoretical optimality, the convergence rate of the two-grid methods with classical stationary smoothers worsens exponentially when the spline degrees increase. With the aid of the symbol, we provide a theoretical explanation of this exponential worsening and by a proper factorization of the symbol we provide a preconditioned conjugate gradient 'smoother', in the spirit of the multi-iterative strategy, that allows us to obtain a good convergence rate independent both of the matrix size and of the spline degrees. A selected set of numerical experiments confirms the effectiveness of our proposal and the numerical optimality with a uniformly high convergence rate, also for the V-cycle multigrid method and large spline degrees.

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 Crankâ\u80\u93Nicolson 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

Anti-reflective boundary conditions (BCs) have been introduced recently in connection with fast deblurring algorithms. In the noise free case, it has been shown that they substantially reduce artefacts called ringing effects with respect to other classical choices (zero Dirichlet, periodic, reflective BCs) and lead to O(n2log(n)) arithmetic operations, where n2 is the size of the image. In the one-dimensional case, for noisy data, we proposed a successful approach called re-blurring: more specifically, when the PSF is symmetric, the normal equations product ATA is replaced by A2, where A is the blurring operator (see Donatelli et al 2005 Inverse Problems 21 169–82). Our present goal is to extend the re-blurring idea to nonsymmetric point spread functions (PSFs) in two dimensions. In this more general framework, suitable for real applications, the new proposal is to replace AT by A′ in the normal equations, where A′ is the blurring matrix related to the current BCs with PSF rotated by 180°. We notice that, although with zero Dirichlet and periodic BCs the re-blurring approach is equivalent to the normal equations scheme, since there A′ = AT, the novelty concerns both reflective BCs and anti-reflective BCs, where in general A′ ≠ AT. We show that the re-blurring with anti-reflective BCs is computationally convenient and leads to a large reduction of the ringing effects arising in classical deblurring schemes. A wide set of numerical experiments concerning 2D images and nonsymmetric PSFs confirms the effectiveness of our proposal.