Krylov Subspace Methods

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Overview

Krylov subspace methods build low-dimensional spaces from repeated applications of a matrix, then use those spaces to approximate solutions, eigenvalues, or matrix functions. Conjugate gradients, GMRES, and Lanczos are core methods, often paired with preconditioning or stochastic estimates to tackle large sparse systems, PDEs, and statistical models. Recent approaches also learn useful subspaces or use Krylov approximations for graph filtering, spectral features, and influence estimation. A central challenge is improving speed and memory use while retaining reliable convergence, especially when guarantees must hold beyond the problem instances used to design or train a solver.

Latest papers

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10 papers0 papers in the last four weeks, against 2 the four weeks before. 0.0% of all new papers.

  1. RINS: Residual-Image Neural Subspace Solvers for Large Sparse Linear Systems

    Sep 9, 2026Zhongyan Ouyang, Weixin Liao, Mingquan Feng +2Neural PDE SolversLearning to Optimize

    Shanghai Innovation Institute, Shanghai, China · Faculty of Computation, Harbin Institute of Technology, Harbin, China · School of Artificial Intelligence and School of Computer Science, Shanghai Jiao Tong University, Shanghai, China +1

  2. Subspace Levenberg Marquardt Algorithms in Training Neural Networks

    Sep 1, 2026M. Duc HoangNeural Network OptimizationKrylov Subspace Methods

    Department of Mathematics, University of California, Davis, Davis, CA, USA

  3. Eigenbasis-Independent Learnable Spectral Positional Encodings for Directed Graphs via Hermitian Block Krylov Subspaces

    Jul 8, 2026Jiaqing Xie, Yuxin WangPositional EncodingSpectral Methods

    Fudan University

  4. Memory Is No Longer a Bottleneck: Memory-Efficient Graph Filtering for Scalable Collaborative Filtering

    Jun 19, 2026Jin-Duk Park, Won-Yong ShinMemory-Efficient InferenceKrylov Subspace Methods

    School of Mathematics and Computing (Computational Science and Engineering), Yonsei University, Seoul 03722, Republic of Korea

  5. Causal Unlearning in Collaborative Optimization: Exact and Approximate Influence Reversal under Adversarial Contributions

    May 19, 2026Ali Mahdavi, Azadeh Zamanifar, Amirfarhad Farhadi +1Federated UnlearningKrylov Subspace Methods

    Department of Computer Engineering SRC, Islamic Azad University Tehran, Iran · School of Computer Engineering Iran University of Science and Technology Tehran, Iran · Meta CA, USA

  6. Adaptive Diagonal Loading using Krylov Subspaces for Robust Beamforming

    May 11, 2026Manan Mittal, Ryan M. Corey, John R. Buck +1Microphone Array ProcessingKrylov Subspace Methods

    Stony Brook University1 · University of Illinois Chicago2 · University of Massachusetts Dartmouth3

  7. NSPOD: Accelerating Krylov solvers via DeepONet-learned POD subspaces

    May 8, 2026Francesc Levrero-Florencio, Youngkyu Lee, Jay Pathak +1PDE SolvingStructural Mechanics

    Synopsys UK Ltd, UK · Division of Applied Mathematics, Brown University, Rhode Island, USA · Synopsys Inc, California, USA

  8. Sparse Random-Feature Neural Networks with Krylov-Based SVD for Singularly Perturbed ODE

    May 8, 2026Kevin Kurian Thomas Vaidyan, Siddharth RoutNeural PDE SolversSingular Value Decomposition

    Institute of Applied Mathematics University of British Columbia Vancouver, BC V6T 1Z4

  9. Towards Universal Convergence of Backward Error in Linear System Solvers

    Apr 17, 2026Michał Dereziński, Yuji Nakatsukasa, Elizaveta RebrovaKrylov Subspace Methods

    University of Michigan, Ann Arbor, MI · University of Oxford, Oxford, UK · Princeton University, Princeton, NJ

  10. Scalable Krylov Subspace Methods for Generalized Mixed-Effects Models with Crossed Random Effects

    May 14, 2025Pascal Kündig, Fabio SigristLatent Variable ModelsKrylov Subspace Methods

    Lucerne University of Applied Sciences and Arts · University of Basel · Seminar for Statistics, ETH Zurich