Data-driven geometric parameter optimization for PD-GMRES
Abstract
Restarted GMRES is a robust and widely used iterative solver for linear systems. The control of the restart parameter is a key task to accelerate convergence and to prevent the well-known stagnation phenomenon. We focus on the Proportional-Derivative GMRES (PD-GMRES), which has been derived using control-theoretic ideas in [Cuevas N\'u\~nez, Schaerer, and Bhaya (2018)] as a versatile method for modifying the restart parameter. Several variants of a quadtree-based geometric optimization approach are proposed to find a best choice of PD-GMRES parameters. We show that the optimized PD-GMRES performs well across a large number of matrix types and we observe superior performance as compared to major other GMRES-based iterative solvers. Moreover, we propose an extension of the PD-GMRES algorithm to further improve performance by controlling the range of values for the restart parameter.
Cite
@article{arxiv.2503.09728,
title = {Data-driven geometric parameter optimization for PD-GMRES},
author = {Lennart Duvenbeck and Cedric Riethmüller and Christian Rohde},
journal= {arXiv preprint arXiv:2503.09728},
year = {2026}
}