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. A particular aspect missing from theory of nonsymmetric and indefinite AMG is the incorporation of general relaxation schemes. In the SPD setting, the classical form of optimal AMG interpolation provides a useful insight in determining the best possible two-grid convergence rate of a method based on an arbitrary symmetrized relaxation scheme. In this work, we discuss a generalization of the optimal AMG convergence theory targeting nonsymmetric problems, using a certain matrix-induced orthogonality of the left and right eigenvectors of a generalized eigenvalue problem relating the system matrix and relaxation operator. We show that using this generalization of the optimal convergence theory, one can obtain a measure of the spectral radius of the two grid error transfer operator that is mathematically equivalent to the derivation in the SPD setting for optimal interpolation, which instead uses norms. In addition, this generalization of the optimal AMG convergence theory can be further extended for symmetric indefinite problems, such as those arising from saddle point systems so that one can obtain a precise convergence rate of the resulting two-grid method based on optimal interpolation. We provide supporting numerical examples of the convergence theory for nonsymmetric advection-diffusion problems, two-dimensional Dirac equation motivated by $\gamma_5$-symmetry, and the mixed Darcy flow problem corresponding to a saddle point system.

翻译：代数多重网格（AMG）是求解稀疏对称正定线性系统的有效方法。对于对称正定系统，AMG的收敛理论在$A$范数意义下已有完善阐述；然而在非对称情形中，此类能量范数并不存在。因此，非对称方程组的AMG收敛性仍是开放的研究领域。现有非对称与不定AMG理论尤其缺乏对一般松弛格式的系统整合。在对称正定框架下，经典形式的最优AMG插值为分析任意对称化松弛格式可能达到的最佳两层网格收敛速率提供了理论依据。本研究针对非对称问题，通过建立系统矩阵与松弛算子广义特征值问题左右特征向量间的矩阵诱导正交性，提出了最优收敛理论的广义形式。我们证明：基于该广义最优收敛理论，可推导出两层网格误差传递算子谱半径的度量，其在数学上等价于对称正定情形中基于范数推导的最优插值结果。此外，该广义最优收敛理论可进一步拓展至对称不定问题（如源于鞍点系统的方程），从而精确计算基于最优插值的两层网格方法收敛速率。我们通过数值实验验证了该收敛理论对非对称对流-扩散问题、受$\gamma_5$对称性启发的二维狄拉克方程以及对应鞍点系统的混合达西流问题的适用性。