A Theory of Relaxation-Based Algebraic Multigrid

Abstract

Algebraic multigrid (AMG) methods derive their optimal efficiency from the interplay between a relaxation process and a corresponding coarse grid correction. In many standard formulations, relaxation and coarse-graining are analyzed and treated as largely separate of one another. Here we propose an alternative theoretical approach centered entirely on the relaxation process, which exposes its fundamental role in the coarse-graining of the fine-scale problem. By treating the relaxation of the error as a dynamical system and applying a dimensional-reduction procedure analogous to the Mori-Zwanzig-Nakajima formalism, we derive exact expressions for the coarse-level equations and the interpolation operations, as well as a natural way of computing complementary transfer operators. We illustrate the unifying nature of this framework by recovering several well-known results for general non-symmetric systems, including ideal and optimal restriction and interpolation, as well as the limiting case of exact elimination. We further emphasize the pivotal importance of compatible-relaxation and identify dynamical corrections that naturally arise in our theory, which have the potential to enhance the convergence, robustness, and adaptivity of future algebraic multigrid methods.

Figures (5)

• Figure 1: Sample layers at time-slices $\ell-1$, $\ell$, and $\ell+1$ of the spatio-temporal structure $\mathcal{N}\times\mathcal{T}$, on which the degrees of freedom $e_i^{(\ell)}$ of the discrete-time dynamical process are defined and interconnected.
• Figure 2: Illustration of causal neighborhoods of a $(i,k)$, showing the nodes of $G_{T}$ which it depends on. Note, that variables on the same time slice are causally unrelated by the relaxation process since information cannot propagate within the same time-slice.
• Figure 3: Given $e_{i}^{(k)}$ on time slice $k$ the shift relation \ref{['eq:shift']} yields $e_{i}^{(\ell)}$ for $\ell < k$, depicted by the arrows pointing backwards in time. Application of a compatible relaxation then propagates this information back to time $k$. Effectively resulting in an interpolation relation between $e_{i}^{(k)}$ and $e_{j}^{(k)}$ for all $j$ in the cone of influence of $i$.
• Figure 4: Illustration of some of the terms appearing in \ref{['eq:interpolation']}. Each term is a path in $G_{T}$ originating at coarse nodes (squares) and ending at the fine node of interest while passing only through fine points (circles) on its way. The colors of the level sets represent the size of the contribution provided by the corresponding terms associated with coarse DOFs as suggested by the principle of compatible relaxation.
• Figure 5: Transition from a fine-level description of relaxation to a coarse-level description. The coarse nodes (squares) interact with each other through intermediate fine nodes (circles). The effective interaction between the coarse nodes in the coarse representation is then equivalent to the original interaction between the coarse nodes in the original fine representation through fine-only paths which originally connect them. Here only a subset of these paths are drawn.