Inexact Gauss Seidel and Coarse Solvers for AMG and s-step CG

Communication-avoiding Krylov methods require solving small dense Gram systems at each outer iteration. We present a low-synchronization approach based on Forward Gauss--Seidel (FGS), which exploits the structure of Gram matrices arising from Chebyshev polynomial bases. We show that a single FGS sweep is mathematically equivalent to Modified Gram--Schmidt (MGS) orthogonalization in the A-norm and provide corresponding backward error bounds. For weak scaling on AMD MI-series GPUs, we demonstrate that 20--30 FGS iterations preserve scalability up to 64 GPUs with problem sizes exceeding 700 million unknowns. We further extend this approach to Algebraic MultiGrid (AMG) coarse-grid solves, removing the need to assemble or factor dense coarse operators