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.

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The solution of large Toeplitz systems with nonnegative generating functions by multigrid methods was proposed in previous papers [13,14,22]. The technique was modified in [6,36] and a rigorous proof of convergence of the TGM (two-grid method) was given in the special case where the generating function has only a zero at x0 = 0 of order at most two. Here, by extending the latter approach, we perform a complete analysis of convergence of the TGM under the sole assumption that / is nonnegative and with a zero at x0 = 0 of finite order. An extension of the same analysis in the multilevel case and in the case of finite difference matrix sequences discretizing elliptic PDEs with nonconstant coefficients and of any order is then discussed

Recently, the class of Generalized Locally Toeplitz (GLT) sequences has been introduced as a generalization both of classical Toeplitz sequences and of variable coefficient differential operators and, for every sequence of the class, it has been demonstrated that it is possible to give a rigorous description of the asymptotic spectrum in terms of a function (the symbol) that can be easily identified. This generalizes the notion of a symbol for differential operators (discrete and continuous) or for Toeplitz sequences for which it is identified through the Fourier coefficients and is related to the classical Fourier Analysis. The GLT class has nice algebraic properties and indeed it has been proven that it is stable under linear combinations and products: in this paper we prove that the considered class is closed under inversion as well when the sequence which is inverted shows a sparsely vanishing symbol (sparsely vanishing symbol = a symbol which vanishes at most in a set of zero Lebesgue measure). Furthermore, we show that the GLT class virtually includes any Finite Difference or Finite Element discretization of PDEs and, based on this, we demonstrate that our results on GLT sequences can be used in a PDE setting in various directions: (1) as a generalized Fourier Analysis for the study of iterative and semi-iterative methods when dealing with variable coefficients, non-rectangular domains, non-uniform gridding or triangulations, (2) in order to provide a tool for the stability analysis of PDE numerical schemes (e.g., a necessary von Neumann criterium for variable coefficient systems of PDEs is obtained, uniformly with respect to the boundary conditions), (3) for a multigrid analysis of convergence and for providing spectral information on large preconditioned systems in the variable coefficient case, etc. The final part of the paper deals indeed with problems (1)-(3) and other possible directions in which the GLT analysis can be conveniently employed

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We introduce a multigrid technique for the solution of multilevel circulant linear systems whose coefficient matrix has eigenvalues of the form f (x [n] j ), where f is continuous and independent of n = (n 1 , . . . , n d ), and xThe interest of the proposed technique pertains to the multilevel banded case, where the total cost is optimal, i.e., O(N ) arithmetic operations (ops), N = d r=1 nr, instead of O(N log N ) ops arising from the use of FFTs. In fact, multilevel banded circulants are used as preconditioners for elliptic and parabolic PDEs (with Dirichlet or periodic boundary conditions) and for some two-dimensional image restoration problems where the point spread function (PSF) is numerically banded, so that the overall cost is reduced from O(k(ε, n) , n) is the number of PCG iterations to reach the solution within an accuracy of ε. Several numerical experiments concerning one-rank regularized circulant discretization of elliptic 2q-differential operators over one-dimensional and two-dimensional square domains with mixed boundary conditions are performed 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