{"ID":2867985,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.18452","arxiv_id":"2509.18452","title":"Fast Linear Solvers via AI-Tuned Markov Chain Monte Carlo-based Matrix Inversion","abstract":"Large, sparse linear systems are pervasive in modern science and engineering, and Krylov subspace solvers are an established means of solving them. Yet convergence can be slow for ill-conditioned matrices, so practical deployments usually require preconditioners. Markov chain Monte Carlo (MCMC)-based matrix inversion can generate such preconditioners and accelerate Krylov iterations, but its effectiveness depends on parameters whose optima vary across matrices; manual or grid search is costly. We present an AI-driven framework recommending MCMC parameters for a given linear system. A graph neural surrogate predicts preconditioning speed from $A$ and MCMC parameters. A Bayesian acquisition function then chooses the parameter sets most likely to minimise iterations. On a previously unseen ill-conditioned system, the framework achieves better preconditioning with 50\\% of the search budget of conventional methods, yielding about a 10\\% reduction in iterations to convergence. These results suggest a route for incorporating MCMC-based preconditioners into large-scale systems.","short_abstract":"Large, sparse linear systems are pervasive in modern science and engineering, and Krylov subspace solvers are an established means of solving them. Yet convergence can be slow for ill-conditioned matrices, so practical deployments usually require preconditioners. Markov chain Monte Carlo (MCMC)-based matrix inversion c...","url_abs":"https://arxiv.org/abs/2509.18452","url_pdf":"https://arxiv.org/pdf/2509.18452v1","authors":"[\"Anton Lebedev\",\"Won Kyung Lee\",\"Soumyadip Ghosh\",\"Olha I. Yaman\",\"Vassilis Kalantzis\",\"Yingdong Lu\",\"Tomasz Nowicki\",\"Shashanka Ubaru\",\"Lior Horesh\",\"Vassil Alexandrov\"]","published":"2025-09-22T22:14:13Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.NA\",\"stat.ML\"]","methods":"[]","has_code":false}