Fast Linear Solvers via AI-Tuned Markov Chain Monte Carlo-based Matrix Inversion
This work addresses the need for efficient preconditioning in large-scale linear systems, offering an incremental improvement over manual parameter tuning methods.
The paper tackles the problem of slow convergence in Krylov subspace solvers for ill-conditioned linear systems by developing an AI-driven framework to optimize MCMC-based preconditioner parameters, achieving a 10% reduction in iterations with 50% less search budget.
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.