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

2025-09-22

AI Summary

To address slow convergence of Krylov solvers for large, sparse, ill-conditioned linear systems, this paper proposes an AI-driven framework for automatic parameter optimization of MCMC-based matrix inverse preconditioners. Our method innovatively integrates graph neural networks (GNNs) and Bayesian optimization: GNNs model preconditioner performance as a function of the coefficient matrix’s graph structure to construct an efficient surrogate model; Bayesian acquisition functions then guide parameter search, eliminating costly manual or grid-based tuning. Evaluated on unseen ill-conditioned systems, our approach achieves superior preconditioning with only 50% of the conventional computational budget, reducing Krylov iterations by approximately 10% on average—significantly improving generalizability and computational efficiency. To the best of our knowledge, this is the first work to jointly leverage GNNs and Bayesian optimization for learning MCMC preconditioner parameters, establishing a scalable, data-efficient paradigm for intelligent preconditioning of sparse linear systems.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: Social recommender systems and personalization for the social Web
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Optimizing MCMC parameters for preconditioning sparse linear systems
• Reducing Krylov solver iterations via AI-tuned matrix inversion
• Automating parameter search to accelerate ill-conditioned matrix solutions

Innovation

• AI-tuned MCMC parameters for matrix inversion
• Graph neural network predicts preconditioning speed
• Bayesian acquisition minimizes iterations via parameter selection