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

Abstract

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

An algebraic generalization of local Fourier analysis for grid transfer operators in multigrid based on Toeplitz matrices

“…However, thanks to Proposition 6, the zero at the next level moves to the origin and, on the coarser grids, the problem becomes spectrally equivalent to the discretization of a constant coefficient elliptic PDE. Finally, we recall the V -cycle optimality conditions for Toeplitz matrices given in [8] for d = 1 and in [14] for d > 1.…”

“…Proposition 7 (Arico et al [8] and Arico and Donatelli [14]) Let A n (i) = C n (i) ( f i ) be the coefficient matrix at the level i, with f i having a unique zero at…”

“…. , n. The pre-smoother and the post-smoother, following [14], are one step of relaxed Richardson with relaxation parameter equal to 1.5/ f i ∞ and 1/ f i ∞ .…”

“…An MGM for circulant matrices was introduced in [13]. Furthermore in [8,14], generalizing the techniques used in [7], and using the Ruge and Stüben theory [15] and the Perron-Frobenius theorem, a complete proof of the optimality of the V -cycle for multilevel circulant and matrices was proposed. This analysis leads to a stronger condition with respect to the previous two-grid analysis.…”

Compact Fourier Analysis for Multigrid Methods based on Block Symbols

“…The main properties of a structured matrix can be reduced to a simple function and to its range [12,15,20]. We use CFA to -derive necessary conditions for smoothing property and convergence expressed as characteristics of the underlying block symbol, -determine all projections and smoothers that lead to MG as a direct solver, -identify acceptable projections for an efficient MG method [1], -design practicable MG components and improved MG algorithms, e.g such that MG acts nearly as direct solver. It is essential to emphasize that the block symbol reflects in the case of periodic boundary conditions exactly the behavior of the full matrices.…”