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arXiv · 2509.18452

Fast Linear Solvers via AI-Tuned Markov Chain Monte Carlo-based Matrix Inversion

Anton Lebedev·Won Kyung Lee·Soumyadip Ghosh·Olha I. Yaman·Vassilis Kalantzis·Yingdong Lu·Tomasz Nowicki·Shashanka Ubaru·Lior Horesh·Vassil Alexandrov
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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.

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Anton Lebedev, Won Kyung Lee, Soumyadip Ghosh, Olha I. Yaman, Vassilis Kalantzis, Yingdong Lu, Tomasz Nowicki, Shashanka Ubaru, Lior Horesh, Vassil Alexandrov. 2025-09-22. Fast Linear Solvers via AI-Tuned Markov Chain Monte Carlo-based Matrix Inversion. https://doi.org/10.1145/3731599.3767543

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