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