用AI自动调优MCMC参数，加速稀疏线性方程求解。
Fast Linear Solvers via AI-Tuned Markov Chain Monte Carlo-based Matrix Inversion
- 构建图神经网络预测不同参数下的预处理速度
- 通过贝叶斯优化仅用一半预算达到更优收敛效果
- 适合需要快速求解大规模病态矩阵的工程场景
大规模稀疏线性系统在现代科学与工程中普遍存在，克里洛夫子空间求解器是常用方法。但病态矩阵会导致收敛缓慢，实际应用常需预处理。基于马尔可夫链蒙特卡洛（MCMC）的矩阵求逆可生成预处理矩阵，加速克里洛夫迭代，但其性能依赖于参数，而最优参数因矩阵而异，手动或网格搜索成本高。本文提出一种由AI驱动的框架，为给定线性系统推荐最优的MCMC参数。该框架利用图神经网络代理模型，根据矩阵A和MCMC参数预测预处理速度；再通过贝叶斯采集函数选择最可能减少迭代次数的参数组合。在先前未见的病态系统上，该框架仅用传统方法50%的搜索预算，实现更好的预处理效果，使收敛迭代次数减少约10%。结果表明，MCMC基预处理可有效集成到大规模求解系统中。
原文摘要 · Abstract (English)
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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