Learned smoothing in operator-adapted multigrid: controlled accuracy gains and computational limits
Abstract
Learned multigrid components are often evaluated at a fixed iteration budget, but such comparisons do not distinguish local approximation gains from practical solver efficiency. We study this distinction for high-contrast elliptic problems on unstructured finite element meshes. A learned V-cycle uses operator-dependent transfers, Galerkin coarse operators, near-kernel deflation and an energy-norm safeguard; only its smoothers are trained. The architecture represents a specified class of fixed polynomial smoothers and is positively homogeneous in the load. On fresh conductive-channel problems, one learned pass reduces the energy error by factors of 5.7-6.1 relative to the same cycle with four-step Jacobi-preconditioned conjugate-gradient smoothing; the corresponding factors are 1.38-1.45 for insulating barriers. In a separate fixed-input test on channel problems, the fine-level learned smoother attains a smaller energy-norm error than the exact minimum over a five-dimensional Krylov space of its own input on 97-100 of 100 fresh inputs for each position and training initialization; this does not establish superiority over all polynomial-smoothed cycles. When the classical smoothing budget is increased, the paired geometric-mean accuracy crosses that of the learned cycle between 8 and 14 steps per smoothing call on channels and between 5 and 8 on barriers. The measured CPU implementation offers no computational advantage over the tested classical solvers, and load changes and floating islands further restrict its accuracy.
Related
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BibTeX
@misc{ranjan2026learned,
title = {{Learned smoothing in operator-adapted multigrid: controlled accuracy gains and computational limits}},
author = {Kumar Kaushik Ranjan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30692},
eprint = {2610.30692},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30692},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Learned smoothing in operator-adapted multigrid: controlled accuracy gains and computational limits
AU - Kumar Kaushik Ranjan
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30692
UR - https://arxiv.org/abs/2610.30692
AB - Learned multigrid components are often evaluated at a fixed iteration budget, but such comparisons do not distinguish local approximation gains from practical solver efficiency. We study this distinction for high-contrast elliptic problems on unstructured finite element meshes. A learned V-cycle uses operator-dependent transfers, Galerkin coarse operators, near-kernel deflation and an energy-norm safeguard; only its smoothers are trained. The architecture represents a specified class of fixed polynomial smoothers and is positively homogeneous in the load. On fresh conductive-channel problems, one learned pass reduces the energy error by factors of 5.7-6.1 relative to the same cycle with four-step Jacobi-preconditioned conjugate-gradient smoothing; the corresponding factors are 1.38-1.45 for insulating barriers. In a separate fixed-input test on channel problems, the fine-level learned smoother attains a smaller energy-norm error than the exact minimum over a five-dimensional Krylov space of its own input on 97-100 of 100 fresh inputs for each position and training initialization; this does not establish superiority over all polynomial-smoothed cycles. When the classical smoothing budget is increased, the paired geometric-mean accuracy crosses that of the learned cycle between 8 and 14 steps per smoothing call on channels and between 5 and 8 on barriers. The measured CPU implementation offers no computational advantage over the tested classical solvers, and load changes and floating islands further restrict its accuracy.
ER -