Fast Linear Solvers via AI-Tuned Markov Chain Monte Carlo-based Matrix Inversion
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| Main Authors: | , , , , , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915508294516736 |
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| author | Lebedev, Anton Lee, Won Kyung Ghosh, Soumyadip Yaman, Olha I. Kalantzis, Vassilis Lu, Yingdong Nowicki, Tomasz Ubaru, Shashanka Horesh, Lior Alexandrov, Vassil |
| author_facet | Lebedev, Anton Lee, Won Kyung Ghosh, Soumyadip Yaman, Olha I. Kalantzis, Vassilis Lu, Yingdong Nowicki, Tomasz Ubaru, Shashanka Horesh, Lior Alexandrov, Vassil |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_18452 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fast Linear Solvers via AI-Tuned Markov Chain Monte Carlo-based Matrix Inversion Lebedev, Anton Lee, Won Kyung Ghosh, Soumyadip Yaman, Olha I. Kalantzis, Vassilis Lu, Yingdong Nowicki, Tomasz Ubaru, Shashanka Horesh, Lior Alexandrov, Vassil Machine Learning Numerical Analysis D.2.0; G.4; B.8.2 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. |
| title | Fast Linear Solvers via AI-Tuned Markov Chain Monte Carlo-based Matrix Inversion |
| topic | Machine Learning Numerical Analysis D.2.0; G.4; B.8.2 |
| url | https://arxiv.org/abs/2509.18452 |