Algebraic multigrid for directed graph Laplacian linear systems (NS-LAMG): Nonsymmetric Lean Algebraic Multigrid

We propose nonsymmetric lean algebraic multigrid (NS-LAMG), a new algebraic multigrid algorithm for directed graph Laplacian systems that combines ideas from undirected graph Laplacian multigrid solvers and multigrid algorithms for Markov chain stationary distribution systems. Low-degree elimination, proposed in LAMG for undirected graphs, is generalized to directed graphs and is a key component of NS-LAMG. In the setup phase, we propose a simple stationary-aggregation multigrid algorithms for Markov chain stationary distribution systems solver enhanced by low-degree elimination to find the right null-space vector that is used for the intergrid transfer operators. Numerical results show that low-degree elimination improves performance and that NS-LAMG outperforms generalized minimal residual method with restart and stable bi-conjugate gradient method for real-world, directed graph Laplacian linear systems.