On Compatible Transfer Operators in Nonsymmetric Algebraic Multigrid

TL;DR

This work addresses convergence challenges in nonsymmetric algebraic multigrid by defining an $M$-orthogonal coarse-grid correction and deriving analytic conditions to couple transfer operators $R$ and $P$ so that $\|\Pi\|_M = 1$. It develops closed-form constructions for multiple choices of $M$ (e.g., $M=I$, $M=A^*A$, $M=(A^*A)^{1/2}$, $M=A_{sym}$, and $M=A^*A_{sym}^{-1}A$) and introduces the notion of ideal transfer operators to characterize compatible pairs that yield stable projections. The framework connects $M$-orthogonality to practical design via $P_{ideal}$ and $R_{ideal}$ and shows how to enforce compatibility through relationships like $R=R_{ideal}(AM^{-1}Q^*)$ or $P=P_{ideal}(Q^*A^{-*}M)$. These insights underpin a compatibility-driven approach to nonsymmetric AMG, with implications for methods like AIR and for robust convergence across advection-dominated and non-Hermitian problems. Overall, the paper provides a principled pathway to jointly design relaxation and transfer operators to achieve stable coarse-grid corrections in nonsymmetric multigrid.

Abstract

The standard goal for an effective algebraic multigrid (AMG) algorithm is to develop relaxation and coarse-grid correction schemes that attenuate complementary error modes. In the nonsymmetric setting, coarse-grid correction $Π$ will almost certainly be nonorthogonal (and divergent) in any known standard product, meaning $\|Π\| > 1$. This introduces a new consideration, that one wants coarse-grid correction to be as close to orthogonal as possible, in an appropriate norm. In addition, due to non-orthogonality, $Π$ may actually amplify certain error modes that are in the range of interpolation. Relaxation must then not only be complementary to interpolation, but also rapidly eliminate any error amplified by the non-orthogonal correction, or the algorithm may diverge. This paper develops analytic formulae on how to construct ``compatible'' transfer operators in nonsymmetric AMG such that $\|Π\| = 1$ in some standard matrix-induced norm. Discussion is provided on different options for the norm in the nonsymmetric setting, the relation between ``ideal'' transfer operators in different norms, and insight into the convergence of nonsymmetric reduction-based AMG.

Figures (1)

• Figure 1:

Theorems & Definitions (14)

• Lemma 1
• proof
• Corollary 2
• Lemma 3
• proof
• Lemma 4
• Lemma 5
• proof
• Remark 6: Symmetry between $P$ and $R$ pairs
• Remark 7: Uniqueness