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arXiv · 2512.09642

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

Stephen Thomas·Pasqua D'Ambra
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Abstract

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

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Stephen Thomas, Pasqua D'Ambra. 2025-12-10. Inexact Gauss Seidel and Coarse Solvers for AMG and s-step CG. https://arxiv.org/abs/2512.09642

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