Inexact Gauss Seidel and Coarse Solvers for AMG and s-step CG
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908704299810816 |
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| author | Thomas, Stephen D'Ambra, Pasqua |
| author_facet | Thomas, Stephen D'Ambra, Pasqua |
| contents | 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09642 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inexact Gauss Seidel and Coarse Solvers for AMG and s-step CG Thomas, Stephen D'Ambra, Pasqua Numerical Analysis 65N55, 65F08, 65F10, 65Y05 G.1.3; F.2.1 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 |
| title | Inexact Gauss Seidel and Coarse Solvers for AMG and s-step CG |
| topic | Numerical Analysis 65N55, 65F08, 65F10, 65Y05 G.1.3; F.2.1 |
| url | https://arxiv.org/abs/2512.09642 |