Using reinforcement learning to vary the m in GMRES(m)

Abstract

While the original GMRES(m) iterative solver assumes the restart parameter m stays fixed throughout the solve, in practic varying m can improve the convergence behavior of the solver. Previous work tried to take advantage of this fact by choosing the restart value at random for each outer iteration or by adaptively changing the restart value based on a measure of the progress made towards computing the solution in successive iterations. In this work a novel application of reinforcement learning to the problem of adaptively choosing values of m is described and then compared to the two existing strategies on matrices from a range of application areas.

Machine Learning for numerical weather and climate modelling: a review

“…Many earlier studies focused on using ML to select the best preconditioner and/or PDE solver from a set of possible choices (e.g. Holloway & Chen, 2007;Kuefler & Chen, 2008;George et al, 2008;Peairs & Chen, 2011;Huang et al, 2016;and Yamada et al, 2018). Ackmann et al (2020) approached the preconditioner part of the system more directly, using a variety of ML methods to directly predict the pre-condition of a linear solver, rather than using a standard preconditioner.…”

Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows

“…There have been some advances to apply machine-learning methods within the context of linear solvers. So far, work has focused on using machine learning to either select the best solver-preconditioner setup from a set of preconditioners and/or linear solvers for a given linear problem (Holloway & Chen, 2007;Kuefler & Chen, 2008;Xu & Zhang, 2005;George et al, 2008;Yamada et al, 2018;Huang et al, 2016;Peairs & Chen, 2011), to help improve efficiency for Block-Jacobi type preconditioners (Götz & Anzt, 2018), to reduce the time-to-solution by interspersing linear solver iterations with neural-network based correction steps (Rizzuti et al, 2019), or to replace the linear solver entirely (Tompson et al, 2017;Yang, Yang, & Xiao, 2016;Ladický et al, 2015). This paper will try a fundamentally new approach by using supervised machine learning to derive the preconditioner directly.…”

Solver Recommendation For Transport Problems in Slabs Using Machine Learning

“…The accuracy of the chosen transport model and the efficiency of the corresponding solver/preconditioner pair depend on these physical properties. Numerous studies have been carried out in the computer science and mathematics communities addressing the use of machine learning algorithms for choosing optimal solvers for linear systems [1,2,3,4,5]. None of these works, however, specifically target particle transport problems, and the authors are unaware of any such study addressing transport solvers.…”