BandPC: Learning Residual-Band Preconditioner Combinations for Flexible Conjugate Gradient Solvers

Abstract

The conjugate gradient (CG) method is a classical iterative solver for sparse symmetric positive definite (SPD) linear systems, but its convergence strongly depends on the spectral properties of the system matrix. Preconditioning can improve these properties; however, designing preconditioners that generalize across diverse and irregular sparsity patterns remains challenging. We propose BandPC (Band Preconditioner Combinations, where "Band" denotes residual-norm bands), a data-driven framework that leverages graph neural networks (GNNs) to predict multi-stage preconditioning strategies for the flexible conjugate gradient (FCG) method. BandPC partitions the FCG iteration into three residual-norm-based bands and defines a structured search space of 125 candidate combinations over five classical preconditioners. By exploiting FCG's ability to switch preconditioners across iterations, BandPC learns to map matrix structure directly to a promising preconditioner sequence. To improve training and generalization, we introduce a soft-labeling mechanism that retains near-optimal combinations and normalizes their scores into a label distribution, together with a hierarchical feature representation that captures node-level attributes, edge-level algebraic coupling strengths, and global matrix statistics. Experiments on a hybrid benchmark of synthetic SPD problems and diverse matrices from the SuiteSparse Collection show that BandPC achieves the best iteration count and solution time on 41.3% and 33.1% of test matrices, respectively, even when compared with each matrix's individually best traditional preconditioner. These results demonstrate that learned residual-band preconditioner scheduling can effectively accelerate flexible CG solvers.