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 note 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.

翻译：有效代数多重网格（AMG）算法的标准目标是开发能够衰减互补误差模式的松弛和粗网格校正方案。在非对称设定下，粗网格校正Π在任何已知标准内积下几乎必然是非正交（且发散）的，这意味着∥Π∥ > 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，Π实际上可能放大插值值域内的某些误差模式。此时松弛不仅需要与插值互补，还必须快速消除因非正交校正而放大的任何误差，否则算法可能发散。本文针对非对称AMG中如何构造"相容"转移算子，使得在某种标准矩阵诱导范数下满足∥Π∥ = 1的问题，给出了解析公式。文中讨论了非对称设定下范数的不同选择、不同范数中"理想"转移算子之间的关系，以及对基于非对称约化AMG收敛性的深入见解。