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 inner 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 note develops analytic formulae on how to construct ``compatible'' transfer operators in nonsymmetric AMG such that $\|\Pi\| = 1$ in any standard matrix-induced norm. Discussion is provided on different options for norm in the nonsymmetric setting, the relation between ``ideal'' transfer operators in different norms, and insight into the convergence of nonsymmetric reduction-based AMG.

翻译：标准有效的代数多重网格(AMG)算法目标是开发能衰减互补误差模态的松弛和粗网格校正策略。在非对称设定下，粗网格校正$\Pi$在任何已知内积中几乎必然是非正交（且发散）的，即$\|\Pi\| > 1$。这引入了一个新的考量：需要在适当范数下使粗网格校正尽可能接近正交。此外，由于非正交性，$\Pi$实际上可能放大位于插值值域内的某些误差模态。此时松弛不仅要与插值互补，还必须快速消除被非正交校正放大的任何误差，否则算法可能发散。本文推导了在非对称AMG中构造"相容"转移算子的解析公式，使得$\|\Pi\| = 1$在任何标准矩阵诱导范数下成立。讨论了非对称设定下范数的不同选择、不同范数下"理想"转移算子之间的关系，以及非对称约化型AMG收敛性的深刻见解。