Coalesced Matrix-Free Geometric Multigrid on Persistent Cell-Wise Storage

What the math gives to ML

The paper's transferable asset is its redundant cell-wise representation: local tensors remain unassembled, while a valence-weighted projection enforces consistency only at selected synchronization points. The identity \(\bar{S}=\mathcal{G}(\mathcal{G}^{T}\mathcal{G})^{-1}\mathcal{G}^{T}\) gives a simple differentiable interface between independently processed cell copies and a continuous global field, while active-subspace masking supports adaptive hierarchies. This suggests neural operators and graph networks for adaptive meshes that compute local blocks without assembling global sparse feature vectors. The likely benefit is higher GPU throughput and simpler batching on irregular meshes, not a universal improvement for dense neural networks.