Spectral Preconditioning from Predicted Invariant Subspaces

Krylov subspace methods are indispensable for solving large sparse linear systems, but their cost can be dominated by slowly converging spectral components. Preconditioning is widely used to accelerate Krylov subspace methods, and several preconditioners have been implemented in scientific computing packages. Neural Krylov Iteration (NeurKItt) showed that a neural operator can amortize the prediction of an invariant subspace and use it to augment a Krylov process for PDE-induced linear systems. The key idea behind NeurKItt is to address the slowly converging components separately. We argue that the invariant subspace can also define a problem-specific preconditioner. Using the invariant subspace predicted by NeurKItt, we define a symmetric positive definite (SPD), low-rank structured spectral preconditioner. The construction requires only matrix–vector products and dense algebra at the dimension of the predicted subspace. It clusters the spectrum of the preconditioned system and thereby reduces the number of Krylov iterations in methods such as GMRES. Theoretical analysis and experiments demonstrate the feasibility and effectiveness of the learned preconditioner.