A V-cycle Multigrid for multilevel matrix algebras: proof of optimality

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