{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37140",
  "author_total": 3,
  "authors": [
    {
      "name": "D. M. Li"
    },
    {
      "name": "Shang-Tian Yang"
    },
    {
      "name": "Xin Qiu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e3w8ze536rbb9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37140",
  "fulltext_url": null,
  "id": "e3w8ze536rbb9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37140",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7214917972",
  "pdf_url": "https://arxiv.org/pdf/2609.37140",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e3w8ze536rbb9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.37140",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37140"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "BandPC: Learning Residual-Band Preconditioner Combinations for Flexible Conjugate Gradient Solvers",
  "title_text": "BandPC: Learning Residual-Band Preconditioner Combinations for Flexible Conjugate Gradient Solvers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/e3w8ze536rbb9",
  "venue": "arXiv",
  "versions": []
}