On Compatible Transfer Operators in Nonsymmetric Algebraic Multigrid

TL;DR

This paper develops methods to construct compatible transfer operators in nonsymmetric algebraic multigrid to ensure the coarse-grid correction is close to orthogonal, improving convergence and stability.

Contribution

It introduces analytic formulas for constructing transfer operators that keep the coarse-grid correction norm close to one in nonsymmetric AMG.

Findings

Compatible transfer operators can be constructed to control the norm of coarse-grid correction.

Different norms influence the ideal transfer operators and convergence behavior.

The approach provides insight into the convergence of nonsymmetric reduction-based AMG.

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 $\Pi$ will almost certainly be nonorthogonal (and divergent) in any known standard product, meaning $\|\Pi\| > 1$. 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, $\Pi$ 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 $\|\Pi\| = 1$ 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.