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.

翻译：有效的代数多重网格（AMG）算法的标准目标是开发能互补衰减误差模式的松弛与粗网格校正方案。在非对称设置下，粗网格校正$\Pi$在任意已知标准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：在适当的范数下，应使粗网格校正尽可能接近正交。此外，由于非正交性，$\Pi$实际上可能放大插值范围内的某些误差模式。此时松弛不仅需要与插值互补，还必须快速消除由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，使得在某种标准矩阵诱导范数下$\|\Pi\| = 1$。文中讨论了非对称设置下范数的不同选择、不同范数下“理想”转移算子之间的关系，以及对基于约化的非对称AMG收敛性的深入见解。