Algebraic multigrid (AMG) is known to be an effective solver for many sparse symmetric positive definite (SPD) linear systems. For SPD systems, the convergence theory of AMG is well-understood in terms of the $A$-norm but in a nonsymmetric setting such an energy norm is non-existent. For this reason, convergence of AMG for nonsymmetric systems of equations remains an open area of research. Existing nonsymmetric AMG algorithms in this setting mostly rely on heuristics motivated by SPD convergence theory. In the SPD setting, the classical form of optimal AMG interpolation provides a useful insight in determining the two grid convergence rate of the method. In this work, we discuss a generalization of the optimal AMG convergence theory targeting nonsymmetric problems by constructing a $2\times 2$ block symmetric indefinite system so that the Petrov-Galerkin AMG process for the nonsymmetric matrix $A$ can be recast as a Galerkin AMG process for a symmetric indefinite system. We show that using this generalization of the optimal interpolation theory, one can obtain the same identity for the two-grid convergence rate as that derived in the SPD setting for optimal interpolation. We also provide supporting numerical results for the convergence result and nonsymmetric advection-diffusion problems.

翻译：代数多重网格（AMG）被公认为求解许多稀疏对称正定（SPD）线性系统的有效方法。对于SPD系统，AMG的收敛理论在$A$-范数下已得到充分理解，但在非对称设定中此类能量范数并不存在。因此，非对称方程组AMG的收敛性仍是开放研究领域。现有非对称AMG算法多基于SPD收敛理论启发的启发式方法。在SPD设定中，经典形式的最优AMG插值为确定该方法的两网格收敛率提供了重要洞察。本文通过构造一个$2\times 2$块对称不定系统，将非对称矩阵$A$的Petrov-Galerkin AMG过程重构为对称不定系统的Galerkin AMG过程，从而讨论面向非对称问题的最优AMG收敛理论的推广。研究表明，利用此最优插值理论的推广，可获得与SPD设定中最优插值相同的两网格收敛率恒等式。文中还提供了支持数值结果及非对称对流扩散问题的验证。