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.