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 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

Anti-reflective boundary conditions have been studied in connection with fast deblurring algorithms, in the case of d-dimensional objects (signals for d=1, images for d=2). Here we study how, under the assumption of strong symmetry of the point spread functions and under mild degree conditions, the associated matrices depend on a symbol and define an algebra homomorphism. Furthermore, the eigenvalues can be exhaustively described in terms of samplings of the symbol and other related functions, and appropriate O(ndlog(n)) arithmetic operations algorithms can be derived for the related computations. These results, in connection with the use of the anti-reflective transform, are of interest when employing filtering type procedures for the reconstruction of noisy and blurred objects

No abstract

No abstract