Learning incomplete factorization preconditioners for GMRES

Abstract

In this paper, we develop a data-driven approach to generate incomplete LU factorizations of large-scale sparse matrices. The learned approximate factorization is utilized as a preconditioner for the corresponding linear equation system in the GMRES method. Incomplete factorization methods are one of the most commonly applied algebraic preconditioners for sparse linear equation systems and are able to speed up the convergence of Krylov subspace methods. However, they are sensitive to hyper-parameters and might suffer from numerical breakdown or lead to slow convergence when not properly applied. We replace the typically hand-engineered algorithms with a graph neural network based approach that is trained against data to predict an approximate factorization. This allows us to learn preconditioners tailored for a specific problem distribution. We analyze and empirically evaluate different loss functions to train the learned preconditioners and show their effectiveness to decrease the number of GMRES iterations and improve the spectral properties on our synthetic dataset. The code is available at https://github.com/paulhausner/neural-incomplete-factorization.